Multiobjective Design of Groundwater Monitoring Network Under Epistemic Uncertainty

Multiobjective Design of Groundwater MonitoringNetwork Under Epistemic Uncertainty

Anirban Dhar & Rajvardhan S. Patil

Received: 22 May 2010 /Accepted: 13 January 2012 /Published online: 28 January 2012# Springer Science+Business Media B.V. 2012

Abstract A methodology is proposed for optimal design of groundwater quality monitoringnetworks under epistemic uncertainty. The proposed methodology considers spatiotemporalpollutant concentrations as fuzzy numbers. It incorporates fuzzy ordinary kriging (FOK)within the decision model formulation for spatial estimation of contaminant concentrationvalues. A multiobjective monitoring network design model incorporating the objectives offuzzy mass estimation error and spatial coverage of the designed network is developed.Nondominated Sorting Genetic Algorithm-II (NSGA-II) is used for solving the monitoringnetwork design model. Performances of the proposed model are evaluated for hypotheticalillustrative system. Evaluation results indicate that the proposed methodology performsatisfactorily under uncertain system conditions. These performance evaluation resultsdemonstrate the potential applicability of the proposed methodology for optimal groundwatercontaminant monitoring network design under epistemic uncertainty.

Keywords Monitoring network design . Optimization . Groundwater pollution

1 Introduction

Management of groundwater pollution is a major concern of modern times. Industrialdevelopment and modern practices have worsened the groundwater pollution scenario.However, due to lack of funds it is hard to implement proper monitoring strategies at thefield level. Generally the amount of information available from the water supply wells andfield are very limited. Process control, performance measurement requires time consuming

Water Resour Manage (2012) 26:1809–1825DOI 10.1007/s11269-012-9988-1

A. Dhar (*) : R. S. PatilDepartment of Civil Engineering, Indian Institute of Technology Kharagpur,Kharagpur, WB 721302, Indiae-mail: [email protected]

A. Dhare-mail: [email protected]

R. S. Patile-mail: [email protected]

and costly data collection effort. Tracking the transient pollutant plume is a challenging taskdue to the uncertainties in predicting the complex subsurface flow and transport processes.The contaminant plume is not known due to its inherent epistemic uncertainty (Datta et al.2009a; Datta et al. 2011). The design of groundwater monitoring network requires properdefinition of objectives. The objective can be defined as monitoring the actual state ofgroundwater systems, i.e., estimatimation of the state variables in space and detection oftrends. The general problem of groundwater monitoring network design can be stated asoptimization of design objective(s) subject to budgetary limitation. If no uncertainty is foundin predicting the pollutant plumes and no economical restrictions are there, then there wouldbe no need for optimal monitoring network design. An efficient and optimally designedmonitoring network can save significant amount of long term monitoring cost.

Comprehensive review of groundwater monitoring network design can be found inLoaiciga et al. (1992), ASCE Task Committee (2003), and Dhar and Datta (2010). Due todynamic nature of the pollutant plumes, monitoring network design is a time varyingproblem. Only a few studies have addressed the issue of designing monitoring networksbased on multiple time steps, e.g., Mugunthan and Shoemaker (2004), Herrera and Pinder(2005), Dhar and Datta (2007), Chadalavada and Datta (2008).

Uncertainty based monitoring network designs are very limited in literature,e.g., Meyerand Brill (1988), Datta and Dhiman (1996), Mugunthan and Shoemaker (2004), Herrera andPinder (2005), Zhang et al. (2005), Wu et al. (2006), Dhar and Datta (2007), Chadalavadaand Datta (2008), Datta et al. (2009b). Uncertainty can be categorized into two basic types:aleatory uncertainty and epistemic uncertainty. It is impossible to reduce the aleatoryuncertainty with higher amount of information. However, reduction is possible in case ofepistemic uncertainty. Monitoring network design can be classified as an approach forreducing epistemic uncertainty about the subsurface system. Fuzzy representation of pollut-ant concentration is one of the approaches of handling the epistemic uncertainty. Fuzzytechniques are widely used in the field of subsurface hydrology to assess the vulnerability ofpollution (Ozbek and Pinder 2006). Use of fuzzy set theory with imprecise parameters (Douet al. 1995), neuro-fuzzy techniques (Dixon 2004; Affandi and Watanabe 2007; Duarte andRosario 2007; Kholghi and Hosseini 2009), fuzzy logic approach (Muhammetoglu andYardimci 2005; Afshar et al. 2007; Bisht et al. 2009), fuzzy modeling and fuzzy relationanalysis (Qin et al. 2006; Huang et al. 2007) are common for subsurface systems analysis.

Classical geostatistical spatial interpolation techniques are helpful for finding out deter-ministic attribute values (e.g., concentration) at unknown locations based on the availableinformation. Moreover, spatial interpolation techniques with capability of handing theimprecise information are important from monitoring network design point of view. Impreciseinformation based interpolation techniques are reported in Bardossy et al. (1987), Diamond(1988), Diamond (1989), Bardossy et al. (1990a), Bardossy et al. (1990b), Huang et al. (1998),Bandemer and Gebhardt (2000), Sunila et al. (2004).

Multiobjective formulations involving different design objectives for monitoring networkdesign problems are presented in Reed and Minsker (2004), Kollat and Reed (2007), Kollatet al. (2008), Dhar and Datta (2009). These studies focus on the improvement of theobjective function(s). However, less attention has been given to individual solutions. Theobjective of the groundwater monitoring network design is to provide the sufficient infor-mation with minimal cost to fulfill the management goals. In reality, feedback informationare needed for effective implementation of any operation policy. Compliance monitoring isrequired to judge effectiveness of the prescribed management strategy. To account for theepistemic uncertainties involved in the design process a fuzzy kriging based groundwatermonitoring network design methodology is proposed. This design framework combines

1810 A. Dhar, R.S. Patil

groundwater flow and transport simulation in conjunction with imprecise spatial interpolationand optimization model. The fuzzy ordirnary kriging algorithm (Diamond 1989) is imple-mented as logical external module to the optimization algorithm.

2 Monitoring Network Design Model

Groundwater system is complex system due to lack of knowledge about systems parameter(s) which can be termed as epistemic uncertainty. Epistemic uncertainty can be dealt withinthe framework of interval analysis, fuzzy set theory, possibility theory, evidence theory andimprecise probability theory. In the present work fuzzy set theory is utilized to frame theepistemic uncertainty of groundwater system.

2.1 Fuzzy Representation of Pollutant Concentration

Fuzzy number is a fuzzy set with convex membership function, which assumes a maximumvalue of 1 and minimum value zero. Spatiotemporal (x, t) concentration value can beassumed to be a fuzzy numbers and associated membership function can be represented astriangular fuzzy membership function (Fig. 1). Triangular fuzzy membership functionrequires specification of lower, modal and upper values. The lower (c), modal (cm) andupper (c) values are defined in terms of 5, 50, and 95 percentile values (A percentile is thevalue of a variable below which a certain percent of observations fall).

A fuzzy number ec (concentration value) is a special fuzzy subset on the set < of realnumbers which satisfy the following conditions:

& There exists a c x; tð Þ 2 <, so that the degree of its membership μ ec x; tð Þ½ � ¼ 1.& Membership function μ ec x; tð Þ½ � is left and right continuous.

Thus the fuzzy number ec ¼ c; cm; cð Þ can be written as

μ ec x; tð Þ½ � ¼ L c x; tð Þ½ � ; c � c x; tð Þ � cmR c x; tð Þ½ � ; cm � c x; tð Þ � c

�ð1Þ

where, L(.) is an increasing function of c 2 c; cm½ � and right continuous 0 � L :ð Þ � 1, R(.) isa decreasing function of c 2 cm; c½ � and left continuous, 0 � R :ð Þ � 1

Fig. 1 Representation of spatio-temporal concentration values asfuzzy number

Multiobjective Design of Groundwater Monitoring Network 1811

2.2 Fuzzy Ordinary Kriging (FOK)

Spatial interpolation schemes are needed to estimate the concentration at all unmonitoredlocations. Kriging is commonly used interpolation method in Geostatistics. A generaloverview about kriging methods is available in (Goovaerts 1997). Fuzzy kriging is proposedby Diamond (1989). It is a method to determine the Modal, Lower and Upper imprecisevalues at an unknown location with the help of Modal, Lower and Upper imprecise values atneighboring locations. This approach can be utilized to find out unknown concentrationvalues at spatial locations.

In the first stage of fuzzy kriging, fuzzy experimental variogram is created from the inputconcentration data with imprecise information. Theoretical variogram is determined basedon the fuzziness of the experimental variogram. In the final step of kriging calculation, thefuzzy input values are used for interpolation. This procedure gives kriging results which arefuzzy numbers. Detailed derivation and implementation of the fuzzy kriging are given inDiamond (1989).

2.3 Design Objectives, Constraints, and Model Formulation

Design objectives always depend on the decision maker. Design objectives and constraintscan be mathematically formulated as follows:

2.3.1 Fuzzy Mass Estimation Error Objective

Contaminant plume characterization is a major objective from remediation point of view.Without a properly designed network, it is not possible to asses the spatiotemporal extent ofcontaminant plume. Spatial moments signify the overall distribution of concentration in theaquifer. Zeroth moment provides the information about the contaminant mass in the aquifer.It can be expressed as:

Mt ¼ðΩn c x; y; tð Þ d4 ð2Þ

Where, Mt is the mass of the contaminant in the aquifer at time instant t; n is the porosityof aquifer material and Ω represents space. Due to uncertainty in concentration mass is also afuzzy number and it can be represented as:

eMt ¼ðΩn ec x; y; tð Þ d4 ð3Þ

The monitoring objective can be stated as the minimization of normalized deviation offuzzy mass based on all monitoring wells from estimated fuzzy mass based on presentsampling plan.

Minimize Mtest ¼

d eMt

D Eall; eMt

D Eest

� �d eMt

D Eall;e0� � � 100 ð4Þ

where, eMt

D Eall

is fuzzy mass estimated using all potential sampling wells, eMt

D Eest

is fuzzy

mass estimated using potential sampling wells according to present sampling plan, d :; :ð Þcalculates the distance between two fuzzy numbers (eA, eB) which can be defined as


d eA; eB� �¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia� bð Þ2 þ am � bmð Þ2 þ a� b

� �2� �3=

rð5Þ

and e0 is the fuzzy representation of the pollution free state of the aquifer. e0 can be defined as:μ e0 Mtð Þh i

¼ 1 ;Mt ¼ 00 ; otherwise

�ð6Þ

2.3.2 Spatial Coverage Objective

Spatial coverage of the network is also a major concern. A distance criterion is the mostcommon for quantifying spatial coverage. If inter-well distances are summed up andmaximized, then it can ensure the desired goal. This is in other words, maximization oftotal inter-well distance for all possible combinations. This objective can be written as:

Maximization : dest ¼XNw�1

i¼1

XNw

j¼iþ1

di;jest ð7Þ

Where, di;j is the distance between potential monitoring locations i and j, Nw is thenumber of wells that has to be monitored.

2.3.3 Monitoring Cost Constraint

Monitoring networks are long-term investments and cost associated with a particularnetwork design is important. If the wells are having equal monitoring cost then costobjective can be represented as an inequality constraint.

XNw

i¼1

ci � P ð8Þ

Where, ci is a binary decision variable indicating whether a monitoring well is to besampled at location i, a value of 1 indicates sampling; P is the maximum number of wellsthat can be monitored.

2.3.4 Formulation of Optimal Design Model

Multiple-objective optimization model for the monitoring network design can be formulatedas:

Min : M1est;M

2est; . . . ;M

Nest

� ð9Þ

where, N is the number of monitoring times steps.

Max : dest ð10ÞSubject to: Constraint (8) and fuzzy interpolation constraints described in Diamond

(1989).


2.4 Monitoring Network Design Methodology

The proposed methodology combines flow and transport simulation in subsurface systemwith the imprecise spatial interpolation method and optimization algorithm. This methodologyis based on feedback information for sequential monitoring strategy. Schematic representationis of the overall methodology shown in Fig. 2.

3 Application of the Developed Methdology

Monitoring network design methodology needs to be evaluated for groundwater system.Generally, evaluations of methodologies are performed on illustrative hypothetical aquifers.This is due to the fact that use of real aquifer data with anthropogenic interventions may leadto erroneous evaluation of methodology. The monitoring network design methodologydeveloped considers the objective of design and installation of a new monitoring network.

In order to demonstrate the application potential, the proposed methodology is applied forthe design of monitoring networks in an illustrative study area (300 m×100 m×5 m). Theaquifer is having impervious boundaries along north and south direction. The aquifer isconfined and governed by two-dimensional flow and transport processes in the principaldirection (oriented in coordinate axes direction). The study area is shown in Fig. 3. Single

Fig. 2 Schematic representation of the proposed monitoring network design methodology


pollutant is assumed to be active throughout the simulation horizon. Flux from the pollutantsource during a planning horizon is assumed to be constant. The pollutant of concentration500 mg/l is assumed to leak from the source location. Injection rate of polluted water into theaquifer is assumed to be 40 l/hr. In this study it is assumed that the flow and transport modelsare based on the finite difference flow code, MODFLOW (McDonald and Harbaugh 1988;Harbaugh et al. 2000), and its solute transport companion MT3DMS (Zheng and Wang1999). The illustrative area is discretized as 5 m×5 m×5 m. The entire study area is coveredby 1200 cells. Initially the aquifer is assumed to be contaminant free. Initial head values aregenerated using steady state simulation of flow model for imposed boundary conditions.

Groundwater flows from left to right with constant head boundaries on left and right endsof the aquifer, and no flow boundaries at top and bottom. Some of the parameter values aretaken from Dhar and Datta (2009). Contaminant transport is modeled for advection, disper-sion and diffusion processes under general hydrologic conditions. Representative contami-nant plumes produced by numerical simulation model represents future scenario that has tobe monitored. In order to investigate contaminant plume behaviour, simulated randomhydraulic conductivity fields are required to be generated over the discrete domain with aspecified variance, mean and correlation structure. Random hydraulic conductivity fields canbe generated using a specific semivariogram type and integral scale (correlation length).During repetitive simulation 100 hydraulic conductivity fields are generated from sequentialgaussian simulation (Deutsch and Journel 1998) algorithm with an exponential semivariogramstructure. The mean value of the log conductivity field is 2.0127 and the semivariogram which

Fig. 3 Illustrative study area with pollution source location

Table 1 Different parameters forillustrative study area Parameter Unit Value

Length m 300.00

Width m 100.00

Depth m 5.00

Hydraulic Conductivity (lnK) m/day ln4–ln14

Porosity - 0.30

Longitudinal Dispersivity m 1

Transverse Dispersivity m 0.5


is representing its spatial correlation structure is an exponential model (Herrera and Pinder2005):

gFðhÞ ¼ σ2F 1� exp � h

lF

�� ð11Þ

where,

F(x) ln K(x)σ2F variance of F(x)

lF correlation scale.Ababou (1988) concluded that for the variable F(x), the effects of local integration over

discretization of length Δx can be avoided if λF and the variance of F(x), σ2F satisfy thefollowing relationship:

η ¼ lF$x

� 1þ σ2F ð12Þ

Table 2 Different types of restrictions for selection of monitoring wells

Risk category Concentrationrange (mg/l)

Number of wellsfor 1st year

Number of wells to be selectedfor monitoring

Type-A Type-B

11 > 0.5, ≤10 173 10 Unknown

10 > 10, ≤20 48 10 Unknown

9 > 20, ≤30 23 4 Unknown

8 > 30, ≤40 9 2 Unknown

7 > 40, ≤50 4 1 1

6 > 50, ≤60 2 1 1

5 > 60, ≤70 3 1 1

4 > 70, ≤80 0 0 0

3 > 80, ≤90 0 0 0

2 > 90, ≤100 0 0 0

1 > 100, ≤110 1 1 1

Total 263 30 30

Fig. 4 Pollution zones for monitoring time step: 01


In the present work integral scale is assumed to be equal to 15 m (λF). The value of σF istaken as 1. From Eq. 12 it can be shown that Δx should be always less than 7.5 m. Thus ourassumption regarding discretization, i.e., Δx05 m is valid. Also, longitudinal dispersivityvalue is taken as 1 m. The aquifer parameters are given in Table 1.

During evaluation process, it is assumed that within planning horizon of 3 years reme-diation work can be completed. Monitoring time step of 1 year is proposed to evaluate theperformance of the remediation work. Sometimes information are needed from all differentlevel of pollution zones to see the effectiveness of remediation strategy. In the present study,restrictions are imposed on selection of wells from different pollution zones (Table 2) andreferred to as Type A. Type B restrictions are less rigid than Type A (Table 2). Differentzones are shown in Figs. 4, 5 and 6 for three monitoring time steps. To evaluate theperformance of the developed methodology four different scenarios (A2, B2, A4, B4) areconsidered. If the modal concentration in a particular cell is greater than 0.5 mg/l it is definedas a polluted cell. Numbers of polluted cells are found to be equal to 263 (N 1

P) for 1st year,

598 (N 2P) for 2

nd year and 703 (N 3P) for 3

rd year. In all cases 30 (Nw) monitoring wells areselected out of 263 potential locations. The resulting scenarios are solved using NSGA-II(Deb 2001). Semivariograms for different time steps are presented in Table 3. Cell indices (i.e.,row, column numbers) are used as coordinates for obtaining the theoretical semivariograms.Thus, lag distance h is actually h/Δx (as Δx0Δy).




3.1 Scenario A2

Scenario-A2 refers to Type-A restriction with 2 objectives. It considers the objectives ofminimizing fuzzy mass estimation error at the end of first monitoring time step andmaximizing the inter-well distance. Mathematically fuzzy mass is calculated as zerothmoment using fuzzy concentration for all polluted cells (N1

P) of the aquifer at the end offirst monitoring time step. Fuzzy concentrations at all unmonitored polluted location areestimated using fuzzy kriging.

Figure 7 shows the nondominated front for Scenario-A2. In order to validate the resultstwo points A2-1 (1.651340, 3661.856179) and A2-3 (2.833587, 4208.897739) are chosenon the final front. In solution A2-1, 1.651340 percent of fuzzy mass estimation error ispresent, while the total inter-well distance is 3661.856179. If the total inter-well distance is

Table 3 Derived theoretical semivariograms for different monitoring time steps

Monitoringtime step

Semivariogram

Lower Modal Upper

01 3 Nug(0)+4 Pow(0.55) 15 Nug(0)+48 Pow(0.55) 160 Nug(0)+190 Pow(0.55)

02 1.9 Nug(0)+1.10 Pow(0.55) 10 Nug(0)+18 Pow(0.55) 75 Nug(0)+68 Pow(0.55)

03 1.5 Nug(0)+1 Pow(0.55) 12 Nug(0)+15 Pow(0.55) 54 Nug(0)+55 Pow(0.55)

• g1ðhÞ ¼ a1 Nugð0Þ ¼ 0 ; h ¼ 0a1 ; h > 0

�• g2ðhÞ ¼ a2 Pow b2ð Þ ¼ a2 h

b2 ; h � 0; 0 < b2 � 2

• gðhÞ ¼ g1ðhÞ þ g2ðhÞ

Fig. 7 Nondominated front for scenario-A2


increased to 4208.897739 (solution A2-3), consequently estimation error will increase to avalue of 2.833587. It is evident that for any improvement in one objective, the other has tobe sacrificed, as expected from a multiobjective problem with conflicting objectives. Atypical well configuration corresponding to A2-3 is shown in Fig. 8.

3.2 Scenario B2

Conditions and objectives for Scenario-B2 are same as Scenario-A2 expect the type ofrestriction imposed on selection of wells. It considers Type-B restriction. Figure 9 shows thenondominated front for Scenario-B2. It is evident that with less restrictive type-B constraint,scenario-B2 gives less mass estimation compared to scenario-A2. In order to validate theresults two points B2-4 (0.378836, 5334.371138) and B2-19 (2.547480, 6035.013738) arechosen on the final front. In solution B2-4, 0.378836 percent of fuzzy mass estimation error

Fig. 8 Configuration of monitoring wells for solution A2-3

Fig. 9 Nondominated front for scenario-B2


is present, while the total inter-well distance is 5334.371138. If the total inter-well distance isincreased to 6035.013738 (solution B2-19), consequently estimation error will increase to avalue of 2.547480. A typical well configuration corresponding to B2-19 is shown in Fig. 10.

3.3 Scenario A4

Real optimality of any policy depends on the time horizon considered. Monitoring for singletime step (i.e., for shorter time horizon) can result in solutions which are myopic in nature,and not optimal for whole planning horizon. However, it is difficult to prescribe a singlemonitoring plan for planning horizon, due to transient nature of the pollutant plume.Scenario A4 considers the objectives of minimizing fuzzy mass estimation error at the endof all three monitoring time step and maximizing the total inter-well distance. Scenario-A4refers to Type-A restriction with 4 objectives. Mathematically, fuzzy mass is calculated as

Fig. 10 Configuration of monitoring wells for solution B2-19

Fig. 11 Parallel axis plot for scenario-A4


zeroth moment using fuzzy concentration for all polluted cells (N1P ,N

2P ,N

3P) of the aquifer at

the end of all monitoring time steps. Figure 11 shows the parallel axis plot for resultingobjective functions from multiobjective optimization solutions. It consists of 4 different axesdenoting different objective function values. Solution presented in Fig. 11 shows mutualconflict of the objective functions by intersecting nature of the lines in between two axes. Inorder to validate the results two points A4-1 (1.270628, 72.688474, 107.702587,2594.182377) and A4-7 (7.639051, 64.053881, 87.796822, 3979.667022) are chosen fromthe final results. In solution A4-1, 1.270628%, 72.688474%, 107.702587% of fuzzy massestimation error are present for 1st, 2nd and 3rd year, while the total inter-well distance is2594.182377. If the total inter-well distance is increased to 3979.667022 (solution A4-7),consequently estimation error will increase to 7.639051%, 64.053881%, 87.796822% forthree different monitoring time steps. A typical well configuration corresponding to A4-7 isshown in Fig. 12.

Fig. 12 Configuration of monitoring wells for solution A4-7

Fig. 13 Parallel axis plot for scenario-B4


3.4 Scenario B4

Scenario-B4 is similar to scenario-A4 except the type of restriction. Scenario-B4 refers to Type-B restriction with 4 objectives. Figure 13 shows the parallel axis plot for resulting objectivefunctions from multiobjective optimization solutions. In order to validate the results two pointsB4-1 (0.369466, 72.474376, 100.544520, 4377.500949) and B4-14 (10.619083, 67.248951,103.783015, 5193.476565) are chosen from the final results. In solution B4-1, 0.369466%,72.474376%, 100.544520% of fuzzymass estimation error are present for 1st, 2nd and 3rd year,while the total inter-well distance is 4377.500949. If the total inter-well distance is increased to5193.476565 (solution B4-14), consequently estimation error will increase to 10.619083%,67.248951%, 103.783015% for three different monitoring time steps. A typical well configu-ration corresponding to B4-14 is shown in Fig. 14.

4 Discussions and Conclusions

The primary objective of this study is to develop a methodology for monitoring of contam-inated aquifers under epistemic uncertainty. Spatial interpolation is an important componentfor monitoring network design problem. The developed methodology utilizes fuzzy kriginglinked nonlinear formulation for the monitoring network design problem. In this study, thefuzzy kriging linked NSGA-II algorithm is run for a population size of 24 and for 500generations. Instead of using binary decision variables, real coded version of NSGA-II isused, as this version shows much faster convergence compared to the binary coded one.Moreover, by using real coded version constraint (8) can be avoided. It reduces the numberof variables from N 1

P to Nw. Different parameter values associated with the optimizationalgorithmNSGA-II are shown in Table 4. These parameters are varied for testing the sensitivity

Fig. 14 Configuration of monitoring wells for solution B4-14

Table 4 NSGA-II parametersParameter Value

Population size 24

Crossover probability 0.9

Distribution index for crossover 10.0

Mutation probability 0.1

Distribution index for mutation 20.0


of the solutions. But no significant improvement is observed. Linked simulation optimizationcarries the curse of computational burden with it. Each run, i.e., 24×500 function evaluationstook around 2/4 days (varies for different scenarios) in Intel® Core™ 2 Duo CPU E8400 @3.00 GHz with 2 GB RAM. Computational complexity dictated the choice of the number ofgenerations. Thus the resulting front may not be the Pareto optimal front or true nondominatedfront, but a near nondominated one.

The objective of monitoring may generally differ based on site specific requirements. Thesolutions for the design methodology are dependent on the objectives specified. Due toindependent nature of objectives, it is useful to analyze the system within multiobjectiveoptimization framework. These limited results and evaluations should help the decisionmaker to choose the proper solution for design of the monitoring network. Sometime resultspecifies less than 30 monitoring wells. This may be due to the fact that the configuration isoptimal in terms of objective function values. Dhar and Datta (2009) have showed that massestimation error is not only dependent on the configurations but also on the number of wells.Performance evaluation results of this methodology show substantial improvement in theperformance of the monitoring network design in terms of handling uncertainty in complexgroundwater system.

Multiobjective problem with spatial interpolation constraints results in a nonlinear for-mulation of the design problem. Nonlinear formulations faces problem with discrete varia-bles, non-convexity, and discontinuity (Dhar and Datta, 2009). They inherit the possibilitiesof the solutions getting stuck to some local optima. However, these results show potentialapplicability of the developed methodology. Monitoring network design is essentially amixed-integer/integer problem. Smaller changes in the concentration values can influencethe resulting solutions. Use of percentile values eliminates the chances of deviation. Thus,fuzzy approach is more robust in avoiding sensitivity problem compared to conventionaldeterministic approach. A methodology for groundwater monitoring network design isdeveloped to the epistemic uncertainty in complex groundwater system. Limited performanceevaluations show utility and applicability of the approach. The developed methodology isessentially generic in nature and should be applicable to various types of groundwater contam-ination problem with minor modifications.

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Multiobjective Design of Groundwater Monitoring Network Under Epistemic Uncertainty