Cost-effective allocation of watershed management practices

using a genetic algorithm

Mazdak Arabi,1 Rao S. Govindaraju,2 and Mohamed M. Hantush3

Received 30 January 2006; revised 12 May 2006; accepted 12 June 2006; published 31 October 2006.

[1] Implementation of conservation programs are perceived as being crucial for restoringand protecting waters and watersheds from nonpoint source pollution. Success of theseprograms depends to a great extent on planning tools that can assist the watershedmanagement process. Herein a novel optimization methodology is presented for derivingwatershed-scale sediment and nutrient control plans that incorporate multiple, and oftenconflicting, objectives. The method combines the use of a watershed model (SWAT),representation of best management practices, an economic component, and a geneticalgorithm-based spatial search procedure. For two small watersheds in Indiana locatedin the midwestern portion of the United States, selection and placement of bestmanagement practices by optimization was found to be nearly 3 times more cost-effectivethan targeting strategies for the same level of protection specified in terms of maximummonthly sediment, phosphorus, and nitrogen loads. Conversely, for the same cost, theoptimization plan reduced the maximum monthly loads by a factor of 2 when compared tothe targeting plan. The optimization methodology developed in this paper can facilitateattaining water quality goals at significantly lower costs than commonly used cost shareand targeting strategies.

Citation: Arabi, M., R. S. Govindaraju, and M. M. Hantush (2006), Cost-effective allocation of watershed management practices

using a genetic algorithm, Water Resour. Res., 42, W10429, doi:10.1029/2006WR004931.

1. Introduction

[2] Best management practices (BMPs) are widelyaccepted as effective control measures for agriculturalnonpoint sources of sediments and nutrients. The 2002Farm Bill provided up to $13 billion for conservationprograms aimed at protecting water quality from agriculturalnonpoint source (NPS) pollution (Natural ResourcesConservation Service, U.S. Department of Agriculture,Farm Security and Rural Investment Act of 2002Conservation provisionsSection by section description,available at http://www.nrcs.usda.gov/about/legislative/pdf/SectionbySection57.pdf). In addition, under theClean Water Act Section 319 Nonpoint Source NationalMonitoring Program and wetland protection programs, theU.S. Environmental Protection Agency (USEPA) supportsprograms to reduce the negative impacts of runoff fromagricultural, urban, and industrialized areas. Similarly, theNatural Resources Conservation Service (NRCS) provideshundreds of millions of dollars in federal funds to supportagricultural best management practices (BMPs) in aneffort to reduce the movement of pollutants into ourwaterways. Success of such programs, however, is contingentupon availability of efficient watershed-scale planning tools.

[3] Implementation of BMPs is challenged by complexi-ties in incorporation of conflicting environmental, economic,and institutional criteria. Environmental assessments inwatersheds hinge on resolving social benefits such as achiev-ing the goal of swimable and fishable water bodies under theUSEPAs total maximum daily load (TMDL) agenda. WhileBMPs facilitate achievement of such targets, their establish-ment bears additional cost for watershed management and/oragricultural producers. Since management practices are usu-ally implemented under a limited budget, costs associatedwith unnecessary/redundant management actions may jeop-ardize attainability of designated water quality goals. Identi-fying optimal combinations of watershed managementpractices requires systematic approaches that allow decisionmakers to quickly assess trade-offs among environmental andeconomic criteria.[4] Cost sharing with landowners is promoted by gov-

ernment agencies for BMP implementation in agriculturalfields (USEPA, Introduction to the Clean Water Act, http://www.epa.gov/watertrain/cwa/). BMPs are implementedthrough site investigation, monitoring, and field-scale mod-eling. While cost share programs may improve water qualitystandards at a field scale, their impact at a watershed scaletypically remains unknown for two main reasons. First,impact of BMPs may duplicate or overlap each other,thereby reducing the potential benefit for the watershed.Interactions between BMPs may also significantly affecttheir individual performances at a watershed scale. Second,water quality impacts of BMPs are site-specific, greatlyinfluenced by landscape characteristics such as land use,soils, and management actions. Thus the same BMP islikely to have varying efficacies at different locations within

1Department of Agricultural and Biological Engineering, PurdueUniversity, West Lafayette, Indiana, USA.

2School of Civil Engineering, Purdue University, West Lafayette,Indiana, USA.

3National Risk Management Research Laboratory, U.S. EnvironmentalProtection Agency, Cincinnati, Ohio, USA.

Copyright 2006 by the American Geophysical Union.0043-1397/06/2006WR004931

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WATER RESOURCES RESEARCH, VOL. 42, W10429, doi:10.1029/2006WR004931, 2006

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a watershed. The direction of many regulatory programssuch as the USEPAs Clean Water Act (CWA) has nowshifted from a source-by-source, pollutant-by pollutantapproach to a more holistic watershed-scale strategy(USEPA, Introduction to the Clean Water Act). An exampleof watershed scale approaches is targeting where pollutioncontrol measures are allocated to critical source areas.Critical sources are portions of the watershed that arebelieved to contribute intensively to nonpoint source pollu-tion. Previous studies in the same watershed, watershed data(i.e., land use, soils, and management characteristics),physical characteristics such as topography and proximityto a stream, and in-stream monitoring data are typicallyanalyzed to identify critical sources. This approach has beenendorsed in the recent draft of USEPAs Handbook fordeveloping watershed plans to restore and protect ourwaters [USEPA, 2005]. However, identification of anoptimal pollution control strategy through targetingbecomes infeasible in large complex watersheds, becausethe number of possible scenarios increases exponentiallywith the number of fields. The search for an optimalwatershed pollution control plan needs to be conducted ina more efficient manner.[5] Development of increasingly powerful computers has

facilitated application of mathematical programming heu-ristics for solving complex and computationally cumber-some problems. A few previous studies have usedevolutionary algorithms (EA) to optimize placement ofwatershed management actions. Srivastava et al. [2002]linked a genetic algorithm (GA) with a continuous NPSmodel (ANNAGNPS) to optimize selection of crop rotationpractices in a 7.25 km2 USDA experimental watershed inPennsylvania. The optimized crop management schemeobtained after 150 GA generations decreased pollution loadby nearly 56% and increased net annual return by nearly110% from a random selection of crop rotation scenarios.The objective function of the optimization procedure wasformulated such that either pollution reduction or net return,but not both at the same time, was maximized. Veith et al.[2004] developed a GA-based optimization model for cost-effective allocation of land use and tillage practices toupland fields that minimized cost (economic criteria) whilesediment load at the outlet of watershed was constrained toa predefined value. The model was tested in a 10.14 km2

case study watershed in Virginia. Results of the studyshowed that the optimization plan achieved the samesediment reduction as a targeting plan at a lower cost.The major setback with optimizing only one of the decisionobjectives (economic, environmental, and/or institutional) isthat the procedure cannot demonstrate the tradeoff betweenmultiple objectives.[6] Incorporation of multiple objectives in the search for

alternative agricultural landscapes was studied by Bekeleand Nicklow [2005] and Muleta and Nicklow [2005]. Amutiobjective evolutionary algorithm (strength Pareto evo-lutionary algorithm (SPEA)) [Zitzler and Thiele, 1999] waslinked with a watershed model for selection of crop rotationand tillage operations in a 133 km2 watershed in southernIllinois. Multiple objectives included minimizing pollutantloads, and maximizing net profit. Though both studiesshowed the effectiveness of an integrated approach inproviding tradeoff between multiple objectives, explicit

incorporation of water quality and budget constraints wasnot examined. These two studies do not offer any clearguidance for selection of operating parameters and termina-tion criteria as they rely on the previously developed SPEA.Moreover, neither optimization results were compared withthe more commonly used targeting strategies, nor wasconvergence of pollutant loads/net profit tested. Henceappraisal of efficiency of the optimization method in deriv-ing solutions better than targeting strategies was curtailed.[7] Research to date indicates the promise of heuristic

optimization for cost-effective allocation of watershed man-agement practices. Unlike gradient-based approaches, heu-ristic techniques do not require linearity, continuity, ordifferentiability either for objective/constraint functions orfor input parameters. Thus they are well suited for cost-effective allocation of watershed management plans. How-ever, several questions still defy answers. A decision makingtool that can clearly accommodate economic, environmentaland institutional criteria is still lacking. The means forimposing target values for pollutant loads, and total water-shed cost of implementation of management plans needs tobe explored. As discussed by Srivastava et al. [2002], a moreversatile formulation is needed to maximize pollution reduc-tion and minimize cost at the same time. Moreover, previousstudies have only focused on alternative agricultural land-scapes. Research on economic evaluation of structural BMPsthat are installed both in upland areas and within the channelnetwork is lacking. In such cases, the interaction between theBMPs could be critical to reach water quality goals. Finally,selection of GAs operating parameters, and terminationcriteria need further study.

2. Objectives

[8] The main goal of this study is to develop an optimi-zation framework that enhances decision makers capacityto evaluate a range of agricultural and environmentalmanagement alternatives. The tool will be designed toidentify near optimal watershed plans that reduce pollutantloads at a watershed outlet to below regulatory or targetvalues with minimum cost. We hypothesize that reductionsof pollutants at watershed outlets can be attained at signif-icantly lower cost by optimized implementation of conser-vation practices than by current cost share and targetingapproaches. This overall goal is achieved by the followingspecific steps: (1) development of a novel genetic algorithm-based spatial search model, which will focus on formulatingversatile objective and constraint functions for the optimiza-tion model that can handle multicriteria and landscape char-acteristics; (2) integration of an NPS model (soil and waterassessment tool (SWAT)), a newBMP representationmethod,and a cost-benefit economic relationship with the GA-basedspatial optimization model to identify optimal spatial alloca-tion of best management practices; (3) provision of guidancefor selection of GA operating parameters and terminationcriteria; and (4) conducting case studies to determine howoptimization plan compares to plans from cost share andtargeting strategies.

3. Case Study Watersheds

[9] The utility of the optimization framework was exam-ined for two subwatersheds in the Black Creek basin. The

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Black Creek watershed (Figure 1) located in northeastIndiana is a typical watershed in the upper Maumee Riverbasin in the midwestern portion of the United States. In mid1970s and early 1980s, several BMPs were implemented inthe watershed and detailed water quality monitoring wascarried out at various locations within the watershed toexamine short-term water quality impacts of soil and waterconservation techniques [Lake and Morrison, 1977, 1978].Data collected from automated samplers at the outlets ofDreisbach (6.23 km2) and Smith Fry (7.3 km2) within theBlack Creek watershed were the most complete and wereused in this study. Black Creek watershed has been listed asan impaired water body in Indiana with nutrients, algalgrowth, and impaired biotic communities as major con-cerns (USEPA, 2002 Section 303(d) list fact sheet forIndiana, available at http://oaspub.epa.gov/waters/state_rept.control?p_state=IN). The dominant hydrological soilgroup in the study watersheds is type C. Land use inDreisbach is mostly pasture in the upper portion, while rowcrops are wide spread in the remainder of the watershed.Smith Fry is a predominantly agricultural watershed. Fieldborders, parallel terraces, grade stabilization structures, andgrassed waterways were the structural BMPs installed in thewatershed by targeting (see Figure 1). Available data, land usedistribution, and other information for the watersheds can beobtained from Arabi et al. [2004, 2006].

4. Theory

4.1. NPS Model

[10] Soil and water assessment tool (SWAT) [Arnold etal., 1998; Arnold and Fohrer, 2005] is a process-baseddistributed parameter simulation model, operating on a dailytime step. The model was originally developed to quantifythe impact of land management practices in large, complexwatersheds with varying soils, land use, and management

conditions over a long period of time. SWAT uses readilyavailable inputs and has the capability of routing runoff andchemicals through streams and reservoirs, and allows foraddition of flows and inclusion of measured data from pointsources. Moreover, SWAT has the capability to evaluate therelative effects of different management scenarios on waterquality, sediment, and agricultural chemical yield in large,ungauged basins. Major components of the model includeweather, surface runoff, return flow, percolation, evapo-transpiration (ET), transmission losses, pond and reservoirstorage, crop growth and irrigation, groundwater flow, reachrouting, nutrient and pesticide loads, and water transfer.[11] For simulation purposes, SWAT partitions the water-

shed into subunits including subbasins, reach/main channelsegments, impoundments on main channel network, andpoint sources. Subbasins are divided into hydrologic re-sponse units (HRUs) that are portions of subbasins withunique land use, management, and soil attributes. SWATuses a modification of the SCS curve number method [SoilConservation Service, 1972] or Green and Ampt infiltrationmethod [Green and Ampt, 1911] to compute surface runoffvolume for each HRU. Peak runoff rate is estimated using amodification of the rational method. Daily or subdailyrainfall data is used for calculations. Flow is routed throughthe channel using a variable storage coefficient methoddeveloped by Williams [1969] or the Muskingum routingmethod.[12] Erosion and sediment yield are estimated for each

HRU with the modified universal soil loss equation(MUSLE) [Williams, 1975]. Sediment deposition and deg-radation are the two dominant channel processes that affectsediment yield at the outlet of the watershed. Whetherchannel deposition or channel degradation occurs dependson the sediment loads from upland areas and transportcapacity of the channel network. If sediment load in achannel segment is larger than its sediment transport capac-

Figure 1. Location of BMPs in Dreisbach and Smith Fry watersheds.

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ity, channel deposition will be the dominant process.Otherwise, channel degradation (i.e., channel erosion)occurs over the channel segment.[13] Movement and transformation of several forms of

nitrogen and phosphorus over the watershed are accountedfor within the SWAT model. Nutrients are introduced intothe main channel through surface runoff and lateral subsur-face flow, and transported downstream with channel flow.Major phosphorous sources in mineral soil include organicphosphorus available in humus and mineral phosphorus thatis not soluble. Phosphorus may be added to the soil in theform of fertilizer, manure, and residue application. Surfacerunoff is deemed as the major mechanism of phosphorusremoval from a field. Unlike phosphorus that has lowsolubility, nitrogen is highly mobile. Major nitrogen sourcesin mineral soil include organic nitrogen available in humus,mineral nitrogen in soil colloids, and mineral nitrogen insolution. Nitrogen may be added to the soil in the form offertilizer, manure, or residue application. Plant uptake,denitrification, volatilization, leaching, and soil erosion arethe major mechanisms of nitrogen removal from a field.[14] It is worthwhile mentioning that the planning tool

proposed here does not necessarily require using the SWATmodel as the NPS component.We have chosen SWATbecausethe case studies that will be performed will deal with sedi-ments, and nutrients. Soundness of mathematical representa-tion of sediment, nutrient, and pesticide processes in SWAThas been validated in previous research [Santhi et al., 2001;Kirsch et al., 2002; Arabi et al., 2004; Vazquez-Amabile,2005]. Moreover, SWAT has been linked with optimizationroutines for automated calibration of themodel [vanGriensvenand Bauwens, 2003] and for evaluation of efficiency ofnonpoint source pollution regulatory programs [Whittaker etal., 2003]. The optimization framework herein can be easilyintegrated with other hydrologic/water quality models todevelop management plans for other types of pollutants.

4.2. Genetic Algorithm

[15] Genetic algorithms (GAs) belong to the evolutionaryclass of artificial intelligence (AI) techniques. GAs arebased on natural selection of chromosomes from a popula-tion for mating, reproduction of offspring by crossover, andmutation to ascertain diversity: ideas borrowed from biology.Each chromosome string in the population corresponds toa solution for the problem at hand, with each variablebeing represented by a gene (a specific position in thestring). The values of the genes, known as allele, can bebinary, real-valued, or character-valued.[16] The GA begins with an initial population containing

randomly generated chromosomes or strings, each repre-senting a possible solution to the problem. The nextpopulation is generated by mating the fittest solutions(i.e., chromosomes) in the previous population. On the basisof the principle of survival of the fittest, the higher thesolutions fitness, the more likely it will be chosen forreproduction. The search for the optimal solution continuesuntil a predefined termination criterion is reached.[17] The GA is essentially a search algorithm for solving

difficult nonlinear optimization problems and is notintended to examine all possible solutions. Regardless ofhow long the algorithm searches, its convergence to anoptimum cannot be guarantied. However, it has been shown

that the GA converges to near optimal solutions for a varietyof problems [Winston and Venkataramanan, 2003]. Effi-ciency of the algorithm depends on the optimizationsoperating parameters and the convergence criterion. Thehigher the number of individual evaluations for convergingto the optimum, the less efficient is the procedure. Thevalues of the operating parameters are problem-dependentand can be determined by performing a sensitivity analysis.[18] The GA is a well-suited optimization technique for

spatial allocation of BMPs, because unlike gradient-basedmethods it does not require linearity, continuity, or differ-entiability of either the objective function or the constraintfunction.

5. Methodology

[19] The optimization model developed in this study iscomprised of the SWAT model for simulating pollutantloads, a BMP representation tool, an economic component,and a GA-based spatial optimization technique. AMATLAB computer program was developed to providethe linkage among various components of the model asshown in Figure 2. The model was tested for optimization ofthe location of field borders, parallel terraces, grassedwaterways, and grade stabilization structures in the Dreis-bach and Smith Fry watersheds.[20] SWAT simulations were performed for a 10-year

period from 1 January 2000 through 31 December 2009.In the analysis, 19912000 precipitation data, 2000 USDANational Agriculture Statistics Service (NASS) land use,and 2002 Soil Survey Geographical Database (SSURGO)were utilized to establish a base case SWAT run. Parametervalues in the base case run were selected from a manualcalibration [Arabi et al., 2006]. Portions of the watershedclassified as urban and forested areas were not consideredfor implementation of BMPs.

5.1. BMP Representation

[21] For this study, a method presented by Arabi et al.[2006] and Bracmort et al. [2006] was utilized to evaluatethe water quality impacts of grassed waterways, gradestabilization structures, field borders and parallel terraces.The method was developed based on published literaturepertaining to BMP simulation in hydrological models andconsidering the hydrologic and water quality processessimulated in SWAT. On the basis of the function of theBMPs and hydrologic and water quality processes that aremodified by their implementation, corresponding SWATparameters were selected and altered. Table 1 summarizesthe SWAT parameters and their corresponding values forrepresentation of the BMPs. Arabi et al. [2004] provides adetailed description of the method used for representation offield borders, parallel teraces, grassed waterways, and gradestabilization structures.

5.2. Economic Component

[22] An economic component was developed for theoptimization model that is comprised of a cost function inaddition to an economic return (benefit) function. Both costand benefit are a function of watershed characteristics andtime. The benefit function reflects the impact of BMPs onsediment and nutrient reductions.

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5.2.1. Cost Function[23] The total cost of implementation of BMPs was

evaluated by establishment, maintenance, and opportunitycosts. Establishment costs included the cost of BMP instal-lation, and technical and field assistance. Maintenance costis usually evaluated as a percentage of establishment cost.The opportunity cost is a dollar value that would beproduced over the BMP design life as a result of investingthe establishment and maintenance costs by purchasingsaving bonds. For each individual BMP, the total cost (ct)was evaluated by the following equation:

ct x; t; td c0 1 s tdc0 rmXtdt1

1 s t1 " #

1

c0 c0 x; t 2

where c0 is the establishment cost that is a function of stateof the watershed (i.e., x, landscape and physical character-istics) at time t; s is the interest rate; td is BMP design life;and rm is the ratio of maintenance cost to establishment

cost. Interest rate (s) of 6.5% was used in computations. Thedesign life of a BMP is defined by NRCS as the intendedperiod of time that the practice will function successfullywith only routine maintenance; determined during designphase (National conservation practice standardsNHCP,available at http://www.nrcs.usda.gov/Technical/Standards/nhcp.html). Design life, establishment cost, and mainte-nance rate for BMPs in this study are summarized in Table 2.Table 2 also provides information with regard to thenumber and area (i.e., quantity of each type of BMP)allocated within the study watersheds through targeting(see Figure 1). For each management plan, establishmentcost (c0) of each type of BMP was obtained by multiplyingits unit establishment cost (c) by its quantity (abmp). Thetotal watershed cost of BMPs was computed by summingup individual costs from equation (1).5.2.2. Benefit Function[24] The economic return of implementation of BMPs

was determined by assigning monetary values to onsite andoffsite benefits of sediment and nutrient reductions. Reduc-tion of sediments and nutrients as a result of implementation

Table 1. Representation of BMPs With SWAT

BMP Function

Representing SWAT Parameter

Parameter (SWAT Input File) Range Value When BMP Implementeda

Field border increase sediment trapping FILTERW (.hru) 05 (m) 5 (m)Parallel terrace reduce overland flow CN2 (.mgt) 0100 A

reduce sheet erosion USLEP (.mgt) 01 0.1 (terraced)reduce slope length SLSUBBSN (.hru) 10150 B

Grassed waterway increase channel cover CH_COV (.rch) 01 0.0 (fully protected)reduce channel erodibility CH_EROD (.rch) 01 0.0 (non-erosive)increase channel roughness CH_N2 (.rch) 00.3 0.24

Grade stabilization Structure reduce gully erosion CH_EROD (.rch) 01 0.0 (non-erosive)reduce slope steepness CH_S2 (.rch) - C

aLetters A, B, and C are as follows: A, estimated based on land use and hydrologic soil group of the HRU where it is installed for terraced condition.B, estimated for each parallel terrace based on its features and SWAT assigned overland slope of the HRU where it is installed, SLSUBBSN =(A S + B) 100/S, where S is average slope of the HRU, A = 0.21, and B = 0.9 [American Society of Agricultural and Biological Engineers, 2003]. C,estimated for each grade stabilization structure based on its features and SWAT assigned slope, CH_S2old, and length of the channel segment where it isinstalled, CH_S2new = CH_S2old D/CH_L2, where D is height of the structure (1.2 m) and CH_L2 is length of the channel segment.

Figure 2. Schematic of the proposed optimization procedure.

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of a management practice is a function of landscapecharacteristics. Likewise, benefits gained by BMP imple-mentation vary with landscape properties of the site wherethe BMP is installed. While offsite benefit of BMPs weredetermined based on cost of offsite damage that would becaused by sediment and nutrient loads, onsite benefits wereestimated based on the quantity of reduced nutrient loadsfrom fields under the influence of BMPs.[25] Ribaudo et al. [1989] equated the benefit of reducing

soil loss to offsite damage in $ per ton of eroded soil basedon freshwater recreation, marine recreation, water storage,navigation flooding, roadside ditches, irrigation ditches,freshwater commercial fishing, marine commercial fishing,municipal water treatment, and municipal and industrial use.A $1.15/t damage cost estimate was suggested for the CornBelt region in the United States. Additionally, Cangelosi etal. [2001] evaluated the benefit of reducing soil erosion fordredging in the Maumee River basin to be $0.87 for a ton oferoded soil. In a study by Fang and Easter [2003], a socialcost of $2.65/kg of removed phosphorous was used toquantify the cost-effectiveness of a water quality trade insoutheastern Minnesota. Onsite water quality benefit ofBMP implementation was evaluated by the monetary valueof nutrient reduction from upland areas. Buckner [2001]estimated the benefit of reducing phosphorus from agricul-tural lands by the bulk rate cost of triple super phosphate(00-46-00) that, in 2003, was $0.26/kg (Heartland Co-op,personal communication, 2004). Table 3 summarizes themonetary benefits (bm) of unit reduction of sediment,phosphorus, and nitrogen deliveries expressed in 2000dollars used for the Dreisbach and Smith Fry watersheds.The values used in this study are site specific and may varyfor other regions.[26] After on-site and off-site benefits of BMPs were

quantified by monetary values from related literature, thetotal benefit (bt) was calculated for individual BMPs overtheir design life as

bt x; t; td; Xtdt1

Xi

Dyi;tbmi

1 s t1 h i

3

Dy y x; t; td y x; t; td; 4

where Dyi,t is annual reduction of constituent i (i.e.,sediment in t/yr, phosphorus in kg/yr, and nitrogen in kg/yr)for year t at the outlet of the study watershed, td (yr) isBMP design life, and bm is the monetary benefit of BMP asa result of reducing constituent i from Table 3. In (4), x isthe state of the watershed at any given time t, denotes thewatershed management plan, and function y reflects mathe-matical relationships in the SWAT model that are used forrepresentation of hydrologic and water quality processes.[27] It is worthwhile noting that bmi serves as a weighting

factor that incorporates relative importance of reduction ofpollutant constituent i in development of watershed man-agement plans. A higher value indicates that larger benefitscan be gained from unit reduction of the correspondingconstituent. In the absence of regional and site-specific datafor the sort of analysis that was used in this paper, this termcould be assigned by judgment of the analyst. If all con-stituents bear the same importance, bm should be 1 for allconstituents of concern.

5.3. Optimization Component

[28] A genetic algorithm (GA) was employed to optimizespatial allocation of BMPs. In this GA component, eachoptimization string (vector j

i) corresponds to a specificwatershed management plan. Figure 3 is a demonstration ofan optimization string for placement of BMPs. The lengthof each string (m) corresponds to the total number of genes,i.e., individual management actions (j,l

i ) that are consideredin optimization. For example, in a watershed with 50 fields(i.e., nhru = 50) considered for implementation of fieldborders and/or parallel terraces, and 20 reach segments (i.e.,nch = 20) considered for implementation of grassed water-ways and/or grade stabilization structures, the total numberof genes on each management string is equal to m = [2 50 + 2 20 =] 140. The alleles are binary values, with 1

Table 2. Unit Cost of Establishment (c), Maintenance Rate (rm), and Design Life (td) of BMPs in the Study Along With Number (nbmp)

and Quantity (abmp) of Each Type in the Study Watersheds as Shown in Figure 1 (i.e., Targeting Strategy)a

BMP c,a $ rm,a % td, years

Dreisbach Smith Fry

nbmp abmp nbmp abmp

Field border 4.6/m 1 10 7 2600 m 1 1800 mParallel terrace 26/m 3 10 4 2130 m 2 480 mGrassed waterway 6200/ha 3 10 5 3.50 ha 1 0.95 haGrade stabilization structure 10,000/structure 2 15 10 - 2 -

aAdapted from Indiana Environmental Quality Incentives Program (2004 EQIP cost list information, available at http://www.in.nrcs.usda.gov/programs/2004eqip/state_costlist.html).

Table 3. Monetary Benefits of Unit Reduction of Sediment and

Nutrient Loads in 2000 Dollars

Variable Benefit bm, $

Sediment 2.5/tPhosphorus 2.86/kgNitrogen 3.5/kg

Figure 3. Schematic of an optimization string (chromo-some) representing grade stabilization structures (gss), grassedwaterways (gww), parallel terraces (pt), and field borders (fb).Here nch and nhru refer to the number of channel segmentsand fields (HRUs) in the watershed, respectively.

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or zero indicating that the corresponding BMP be ornot be implemented. The ith optimization generation (Pi)with nstr solution vectors is expressed as

Pi

i1

i2

..

.

instr

26666666664

37777777775

i1;1 i1;mi2;1 i2;m

..

. . .. ..

.

instr;1 instr;m

26666666664

37777777775

5

where

8j 2 1; 2; . . . ; nstrf g; 8l 2 1; 2; . . . ;mf g : ij;l 0=1 6

where ji denotes the jth management vector in Pi, and

j,li reflects the lth gene (an individual management action)

in ji. For a total of ngen optimization generations, the entire

solution space (A) is obtained from union of solutions inoptimization generations:

A [ngen

i1 Pi 7

5.3.1. Mathematical Formulation[29] A general multiobjective optimization problem can

be formally expressed as [Zitzler and Thiele, 1999]

Minimize=Maximize z f f1 ; f2 ; . . . ; fn 8

subject to

1; 2; . . . ; m 2 W 9

z z1; z2; . . . ; zn 2 Z 10

where z denotes a vector function f that maps a set of mdecision variables to a set of n objectives; W and Z are,respectively, the decision parameter space and objectivespace (refer to Ringuest [1992] for more information).[30] Several common methods have been previously used

for handling multiobjective optimization problems [Deb,2001]. The most commonly used approach is the weightedsum approach. In thismethod, a set of objectives are aggregatedinto a single objective by multiplying each objective with auser-defined weight (w). These weights reflect the importanceof each objective in the context of the optimization problem:

z f Xni1

wi fi 11

[31] The BMP spatial allocation problem was formulatedwith an aggregate single-objective based on multiple crite-ria. The multiple criteria included minimizing implementa-tion cost of BMPs and maximizing pollutant loadreductions. Monetary benefits of unit reduction of sediment,phosphorus, and nitrogen loads (Table 3) were used todetermine the weighted sum of onsite and offsite benefits

gained by reducing pollutant loads (bt). Then, the objectivefunction was evaluated with a benefit (bt) to cost (ct) ratio atthe watershed level. Pollutant loads were constrained toascertain that sediment and nutrient yields would not exceedallowable values and/or that total available budget is notexceeded by entailed costs. Allowable pollutant loads aretypically specified by regulatory agencies. Since availablebudget for implementation of BMPs is usually limited, anadditional constraint was imposed to search for solutionsthat can be implemented with a cost less than the specifiedbudget. The mathematical representation of the objectivefunction used in this paper was

Maximize z btmax ct; 1

Ptdt1

Pi

Dyi;tbmi

1 s t1 h i

max c0 1 s tdc0 rmPtdt1

1 s t1

; 1

12

subject to water quality constraints

G1 x; t; td; ymaxi x; t; td; y; yti 0 13

and budget constraints

G2 x; t; td; ct x; t; td; w cos t 0 14where ymaxi is maximum delivery of pollutant constituent iafter implementation of BMP combination () estimatedwith SWAT simulations over period td; and yti is theallowable load of constituent i. ct is the total cost ofimplementation of estimated by applying equation (1),and wcost is the total available budget for implementation ofmanagement plans. The denominator in (12) was designedsuch that it will never be zero.[32] The optimization constraints in (13) and (14) are

typically defined by regulatory and implementation agen-cies. For example, allowable sediment and nutrient loads(yti) may be obtained from a total maximum daily load(TMDL) for a given watershed. While yti and ymaxi can beexpressed on a daily, monthly, or annual basis, as loads orconcentrations, their units should be consistent. The costconstraint represents available budget for implementation ofwatershed management scenarios and may be specified byimplementation agencies.[33] The fitness score ( fs) for each string was evaluated

by the objective function (z) associated with the string fromequation (12). Infeasible solutions, i.e., solutions that do notsatisfy the constraint functions G1 and G2 in (13) and (14),were penalized by applying a penalizing factor k as

fs k z;

k 105 ;G1 _ G2 > 0

1 ;G1 ^ G2 0

8