ABSTRACT: A combinatorial optimization procedure for best man-agement practice (BMP) placement at the watershed level facili-tates selection of cost effective BMP scenarios to control nonpointsource (NPS) pollution. A genetic algorithm (GA) was selected fromamong several optimization heuristics. The GA combines an opti-mization component written in the C++ language with spatiallyvariable NPS pollution prediction and economic analysis compo-nents written within the ArcView geographic information system.The procedure is modular in design, allowing for component modifi-cations while maintaining the basic conceptual framework. Anobjective function was developed to lexicographically optimize pol-lution reduction followed by cost increase. Scenario cost effective-ness is then calculated for scenario comparisons. The NPSpollutant fitness score allows for evaluation of multiple pollutants,based on prioritization of each pollutant. The economic componentconsiders farm level public and private costs, cost distribution, andland area requirements. Development of a sediment transport func-tion, used with the Universal Soil Loss Equation, allows the opti-mization procedure to run within a reasonable timeframe. Theprocedure identifies multiple near optimal solutions, providing anindication of which fields have a more critical impact on overall costeffectiveness and flexibility in the final solution selected for imple-mentation. The procedure was demonstrated for a 1,014-ha water-shed in the Ridge and Valley physiographic region of Virginia.(KEY TERMS: watershed management; genetic algorithm; spatialoptimization; modeling; geographic information systems; nonpointsource pollution; sediment delivery.)

Veith, Tamie L., Mary Leigh Wolfe, and Conrad D. Heatwole, 2003. OptimizationProcedure for Cost Effective BMP Placement at a Watershed Scale. Journal ofthe American Water Resources Association 39(6):1331-1343.

INTRODUCTION

In the last few decades there has been increasingconcern over water and soil borne pollutants thatinfluence human or aquatic health or that restrict

human activities. In particular, nonpoint source(NPS) pollution from agricultural lands contributessignificantly to water quality degradation. Govern-ment regulations, such as the Clean Water Act, areplacing growing emphasis on NPS pollution control.One method of control is through implementation ofbest management practices (BMPs) – structural, veg-etative, or cultural methods by which NPS pollution is eliminated or reduced sufficiently to meet waterquality criteria (Novotny and Olem, 1994).

Implementation of all applicable BMPs on eachfield within a watershed would help control NPS pol-lution to the greatest extent possible with the currentlevel of scientific knowledge. However, extensive,widespread BMP applications can be cost prohibitiveand redundant in pollution control. Best managementpractice implementation often increases both farmerand public costs. For example, many BMPs requireusing different machinery, building structures, andlearning new techniques, and might also reduce yield.Additionally, BMPs might be subject to governmentalcontracting and inspection to meet legal or cost sharerequirements.

Success in locating the most cost effective BMP sce-nario for a specific watershed depends on the abilityto consider the complete range of possible scenarioswithin a watershed, accounting for spatial variation,field specific BMP effectiveness, and BMP interactionwithin and among fields. However, the number ofways to allocate BMPs throughout a watershed isexponential with regard to the number of fields. Forexample, for 50 fields and four nonmutually exclusiveBMPs, there are (24)50 possible placement scenarios.Evaluation of all possible BMP scenarios becomes an

1Paper No. 03024 of the Journal of the American Water Resources Association. Discussions are open until June 1, 2004.2Respectively, Agricultural Engineer, USDA-ARS Pasture Systems and Watershed Management Research Unit, 3702 Curtin Road, Univer-

sity Park, Pennsylvania 16802-3702; and Associate Professors, Department of Biological Systems Engineering (0303), Virginia Tech, Blacks-burg, Virginia 24061 (E-Mail/Veith: Tamie.Veith@ars.usda.gov).

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JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATIONDECEMBER AMERICAN WATER RESOURCES ASSOCIATION 2003

OPTIMIZATION PROCEDURE FOR COST EFFECTIVEBMP PLACEMENT AT A WATERSHED SCALE1

Tamie L. Veith, Mary Leigh Wolfe, and Conrad D. Heatwole2

intractable problem, one that is computationally diffi-cult or impossible to solve for an exact solution in afinite amount of time.

Evaluating and comparing a small number ofpotential BMP scenarios through expert judgment,even with the aid of a geographic information system(GIS) and analysis software, are time consuming andsubject to judgment inconsistencies. Advances in com-putational speed and software now enable evaluationof a large sample of possible scenarios for a watershedin a reasonable timeframe. Using an optimizationheuristic to determine scenario effectiveness elimi-nates the laborious task of individual evaluation andlessens computational errors and evaluation inconsis-tencies. In particular, Srivastava et al. (2002) haveshown the potential of the genetic algorithm (GA) inlocating scenarios that reduce pollution or farmercosts as compared to multiple random scenarios.Additionally, because the optimization procedure eval-uates the complete BMP scenario, each scenario cantheoretically be limited to precisely the combinationof BMPs necessary to meet the water quality criteria.

The overall goal of the paper is to describe develop-ment of an optimization procedure that identifiesBMP combinations at a watershed scale that meetspecified pollutant reduction levels while minimizingcosts. The specific objective of this research was tocreate an optimization procedure that would: (1) placesufficient BMPs on the watershed to meet water qual-ity criteria; (2) limit each scenario, as much as possi-ble, to the combination of BMPs necessary to meet thewater quality criteria; and (3) identify the lowest costscenario possible.

PROCEDURE DEVELOPMENT

Heuristic Selection

Existing optimization heuristics for solvingintractable problems include gradient and nongradi-ent based neighborhood searches as well as methodsdeveloped from studies of natural systems. To deter-mine a basic heuristic well suited to this problem, fiveheuristics for solving intractable problems were con-sidered: GA, response surface methodology, shuffledcomplex evolution, simulated annealing (SA), andtabu search. Several factors were compared amongthe heuristics, including performance for similartypes of problems in previous studies, proof of conver-gence, and ease of formulation. Next, each heuristic’scontinuity and differentiability requirements, conver-gence rate, and relative efficiency were considered, as were sensitivity of the heuristic to the problem

formulation and the number of points needed as astarting requirement. In addition to the above heuris-tics, the use of a classical method, such as integer pro-gramming or nonlinear optimization (as used byBraden et al., 1989), was considered briefly.

Overall, the problem was determined to be mostsimply suited to characterization as a combinatorialoptimization problem (Lawler, 1976; Grötschel, 1982).Compared to the other techniques considered, the GAand SA seemed more straightforward to formulate ina manner that could accommodate evaluation of dif-ferent watersheds. Because the GA and SA do notrequire continuity or differentiability, they are wellsuited to the combinatorial aspect of this problem.Both the GA and SA have been proven to convergearbitrarily close to the optimum under certainassumptions (Lundy and Mees, 1986; Siegelmann andFrieder, 1991). Additionally, unlike the tabu search,the GA and SA do not require problem specific selec-tion rules.

At each generation, the GA evaluates multiple sce-narios, often from different areas of the search space.This parallelism decreases susceptibility to becomingfixed at local minima (Buckles and Petry, 1992). Addi-tionally, by looking at the most fit scenarios in a givengeneration, a policy maker can determine which fieldstend to be managed in the same way across scenariosand which fields vary in management across scenar-ios. Since the most fit scenarios are selected fromacross the search space, it is likely that the fields thatare managed the same across scenarios have agreater impact on total watershed loadings than thefields for which management varies across scenarios.

Additionally, subsequent evaluations, drawing fromscenarios across the breadth of the search space, helpidentify fields within the watershed that appear tohave a greater impact on the total watershed qualityand those that can vary in management practice withless overall impact.

Convergence rate and relative efficiency of the GA,in comparison with the other heuristics, were notclear. Performance of the GA in these two areasappeared no better or worse than that of the majorityof the other heuristics and to, perhaps, be dependenton the specific problem and/or problem formulation.The GA, like the other heuristics, was seen to be sen-sitive to problem formulation. Previous work with theGA in placement of management practices (Srivasta-va et al., 2002) was available to provide some insightinto a possible problem formulation. Based on a sub-jective comparison of the heuristics with regard to thecited factors (Veith, 2002), the GA was selected for usein the optimization procedure.

The GA is conceptually based on natural selectiontechniques seen in biological evolution (Goldberg,

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VEITH, WOLFE, AND HEATWOLE

1989; Chambers, 1995; Mitchell, 1999). The probabili-ty of an individual surviving to the next generationincreases with increasing fitness, where fitness is ameasure of how well an individual meets the goal ofthe optimization. Theoretically, the GA is intended tofind one or more near optimal solutions within a rea-sonable amount of time. With regard to this research,a near optimal solution suggests a watershed scenariothat meets the cost effectiveness criteria. This sce-nario may then be fine tuned by policy makers tomeet individualized farmer needs or used as a start-ing point for more detailed predictive modeling.

Problem Formulation

To formulate the BMP location problem for GAoptimization, each watershed scenario can be thoughtof as a chromosome. A possible solution to the prob-lem is represented as a chromosome and each landuse area, including crop/cover and BMPs, is repre-sented as a gene on that chromosome. The value ofeach gene, representing the land cover and manage-ment practice or combination of practices, along thechromosome is chosen from a set of possible values, oralleles, for that gene.

In the watershed scenario representation, the chro-mosome is written as an array of numbers. Each posi-tion of the array holds the identification value for acorresponding field or management unit. The value inthat position represents the specific managementpractice on that field, while the list of all acceptablemanagement practices for that field forms the alleleset for that array position.

Each scenario, evaluated for pollution reductionand cost increase, is assigned an ordered value of costeffectiveness. The BMP location problem can then beformulated as an unconstrained optimization problemwith an objective function that maximizes cost effec-tiveness.

The GA is initialized with a random population ofscenarios where each scenario is subject to the con-straints of the allele array. Thus, any land use areanot in production is assigned a single allele and main-tains a fixed set of management practices. Land useareas in production are assigned a set of alleles, corre-sponding to the set of acceptable BMPs.

The baseline scenario is the scenario to which eachnew scenario is compared. Baseline managementpractices can come from the current land uses andmanagement practices in the watershed, from theprofit maximizing scenario (most profitable manage-ment practice for each field), or from any other sce-nario of choice.

In Srivastava et al.’s (2002) problem formulation,scenarios were formulated as two-dimensional binary

strings where a binary string for each field identifiedthe field’s rotation. Fifteen cropland rotations wereconsidered. In the formulation described here, use ofallele sets in each scenario representation allowsincreased specificity of management practice optionsfor each land use type. That is, management practicesappropriate only to cropland, or even to a specific typeof crop, can be excluded from consideration on hay-land and other areas.

PROGRAM STRUCTURE

To facilitate evaluation of the objective function,the optimization procedure is comprised of threeparts: an optimization component based on the GAheuristic, an NPS prediction component, and an eco-nomic analysis component (Figure 1). At each genera-tion of the optimization procedure, the optimizationcomponent forms a number of scenarios to considerfor addition into the GA population. Each scenario issent to the NPS and economic components wherewatershed level pollutant load and cost are calculat-ed. These values are returned to the optimizationcomponent, converted into fitness scores, and evaluat-ed. After the evaluation process determines whichscenarios, both new and existing, to transfer to thenext population, the entire process repeats.

The optimization process ends upon reaching sometermination criterion, which can be defined in a num-ber of ways. For example, termination can be set tooccur after a predetermined number of iterations ofthe optimization process. The termination criterioncan also be defined as a minimal improvement in themaximum fitness score; that is, termination occurseither when the change in fitness score is below a pre-determined tolerance or when the score increase hasremained below a tolerance for a predetermined num-ber of generations.

The optimization procedure was programmed usingthe C++ language and ArcView GIS (ESRI, 1999). Theoptimization component was written as a dynamiclink library (DLL) in C++, using the GALib geneticalgorithm package (Wall, 1999).

The pollutant loading and cost components werewritten as ArcView scripts. A GIS was used to providea spatial structure within which the NPS componentcould function. The GIS facilitates spatial calculationsin addition to enabling within-cell calculations to bemade simultaneously for all cells in a watershed.Additionally, ArcView scripts format input data.

The main ArcView script calls the DLL that runsthe GA. For each scenario evaluation, a function inthe DLL calls a script in ArcView and passes themanagement practices for each field to that script.

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OPTIMIZATION PROCEDURE FOR COST EFFECTIVE BMP PLACEMENT AT A WATERSHED SCALE

Through ArcView scripts, the NPS and economic com-ponents determine the pollutant load at the outletand the scenario cost. This information is then passedback to the DLL where the scenario fitness isassigned. When the GA has met the termination crite-rion, control is returned to the main ArcView script.

Optimization Component

The optimization component translates the resultsof the NPS prediction component into a pollutant fit-ness score for each scenario based on pollution reduc-tion. The optimization component also translates theeconomic analysis component results into an economicfitness score. The combined fitness score for each sce-nario is compared with the fitness scores of other sce-narios.

Pollution Reduction. In the interest of assigningBMPs to decrease NPS pollution from a baseline, pol-lution increase at the watershed outlet, as a result ofaltering a BMP assignment, was not an acceptable

option. Such a scenario would be no better than thebaseline scenario in terms of pollution. Thus, a pollu-tion fitness score was developed that focused on posi-tive pollution reduction. The pollution fitness scorescales the pollutant loading of each scenario relativeto the loading goal. As a result, all scenarios thatincrease pollution as compared to the baseline aregiven a pollution fitness score of zero. Scenarios thatreduce pollution are given a positive fitness score. Thefitness score for a single pollutant is calculated by

where pi = fitness score of pollutant i, zb = pollutantloading from baseline scenario (Mg), zw = pollutantloading from working scenario (Mg), and zt = maxi-mum pollutant loading goal (Mg).

In the case of multiple pollutants, a unique pollu-tant loading criterion can be set for each pollutantand the individual pollutant fitness scores weightedrelative to each other in terms of importance

where P = total pollutant fitness score, βi = weightingfactor of pollutant i, pi = fitness score of pollutant i,

goals as inputs, the optimization procedure maintainsflexibility to numbers and types of pollutants, pollu-tant weightings, and reduction goals.

Cost Increase. While it was anticipated that costswould increase from the baseline as BMPs wereadded, a scenario meeting the pollution reduction cri-terion and decreasing cost would certainly be accept-able. Thus, in modeling the scenario cost, theeconomic fitness score had to allow for both increaseand decrease in cost as compared to the baseline. Eco-nomic fitness score calculations are relative to oppor-tunity costs (i.e., the costs of not adopting themanagement practice with the highest net return).Thus, a scenario’s economic fitness score remains pos-itive even if the cost decreases below the baseline sce-nario. However, cost increase, expressed as change incost relative to the baseline, may be positive or nega-tive.

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Figure 1. Flow Chart Showing Linkage of theThree Optimization Procedure Components.

p

z z

z zz z

z z z

z z

i

b w

b w

b tt w b

w t

=

≤−−

< <

≤

0

1

for

for

for

P

pi ii

ii

=∑∑

β

β

and βii

∑ = 1.

(1)

(2)

By incorporating pollutant reduction

Agricultural BMPs can affect the crops and foragesproduced by farms in a watershed. Thus, it wasimportant that solutions not be chosen based on pollu-tion reduction and cost alone, but that they also con-form to reasonable farming practices. To facilitatethis, two additional criteria were addressed in formu-lating the objective function.

First, preference was given towards farms meetingfeed production and nutrient management require-ments. Implementation of a farm area requirementensures that the optimization program assigns fieldsinto cropland and hay/pasture as necessary for thefarm to produce sufficient amounts of feed and havesufficient grassland available for manure/litterspreading. Allocation amounts vary based on farmtype (such as beef, dairy, or poultry).

Second, farm level cost fairness was incorporated toprevent, as much as possible, a solution from placingextreme cost burdens on a few farmers. The optimiza-tion procedure minimizes cost increase per pollutionreduction at the watershed scale. Spurlock andClifton (1982) demonstrated that an NPS pollutioncontrol strategy based on cost increase per unit of pol-lution reduction is economically more equitable tofarmers than a strategy based on pollution reductionper unit area. To incorporate this information, theeconomic fitness score was developed such that forscenarios of near equivalent cost increase, the sce-nario involving the most farmers and distributingcosts most equally across the affected farmers is pre-ferred.

The economic fitness score was developed specifi-cally for this optimization procedure to address theidentified issues and criteria. It was structured toconsider public and private costs as well as farm levelarea requirements and cost fairness, and is expressedas

where E = economic fitness score; Co = total opportu-nity cost for all farms in scenario($); xi = cost of working scenario for farm i ($); xi = (cw + dw)i; cw =private cost of working scenario for farm i ($); dw =public cost of working scenario for farm i ($); ai =

meets area requirements of farm i.

where n = the number of requirements r for farm i,aow = area in working scenario contributing towardrequirement r for farm i (ha), ae = area requiredunder requirement r for farm i (ha), and ab = extentto which baseline scenario meets area requirements offarm i.

where aob = area in baseline scenario contributingtoward requirement r for farm i (ha), and n and ae asabove. The opportunity cost of the baseline scenario isused to scale the function to allow for different rangesof costs in each optimization run.

The Euclidean distance metric was used in the eco-nomic fitness function to help distribute the impact ofcost increase among farms. Using this metric insteadof simply adding private costs across farms results ina more preferable score when several farms eachincur a little cost than when a single farm incurs theequivalent cost. However, since each farm incurs thepublic cost once for one or more BMPs adopted, lesstotal public cost is incurred if the management prac-tice changes are distributed over as few farms as pos-sible. The result is, roughly, that the economic fitnessfunction tends to prefer change in a limited number offarms while preferring cost increases to be distributedas equally as possible among those farms.

A farm may or may not meet area requirements inthe baseline or in a working scenario, depending onthe farm type and size and on the management of pro-duction land. For example, a dairy farm that requiresall production land to be in corn to supply sufficientfeed will no longer meet the area requirement if oneof the fields is changed to hay in the working sce-nario. When the area requirement percentage met bya farm in the working scenario is less than that metby the baseline, the farm’s contribution to the eco-nomic fitness function increases, resulting in adecrease in the economic fitness function.

Cost Effectiveness. The cost effectiveness ratio –pollution reduction over cost increase – can be writtenas p/c where p and c are real numbers. Mathematical-ly, however, (-p)/(-c) = p/c, where p and c are positivereal numbers. This relationship implies, for example,that reducing 10 Mg/ha of sediment for a cost increase

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OPTIMIZATION PROCEDURE FOR COST EFFECTIVE BMP PLACEMENT AT A WATERSHED SCALE

EC

xa

o

i

ii

= +

+∑1

12

minawab i

, ;1 aw = extent to which working scenario

(3)

aw naaow

e rr=

∑11min ,

1 for n = 0.

for n > 0, r = 1 to n(4)

ab naaob

e rr=

∑11min ,

1 for n = 0

for n > 0, r = 1 to n (5)

of $100 is equal in cost effectiveness to increasing sed-iment by 10 Mg/ha but decreasing costs by $100. Inthe first situation, both terms are positive with regardto the cost effectiveness definition, whereas in the sec-ond situation, both terms are negative. Thus, bothratios equal 0.10. Additionally, it is implied mathe-matically that both situations are more cost effectivethan reducing 10 Mg/ha of sediment while decreasingcosts by $100 (0.10 Mg/ha/$ versus -0.10 Mg/ha/$).However, the latter situation is certainly both envi-ronmentally and economically preferable. Thus, usingthe cost effectiveness ratio as a single objective func-tion for the GA does not clearly define the responsesurface to the research problem.

As pollutant reduction criteria are introduced, thescenario preference becomes dependent on which ofthe scenarios, if any, meet the reduction criteria. Tosolve this problem, the single cost effectiveness ratioobjective function was split into a two-part multi-objective problem: (1) meet or exceed the pollutionreduction criterion (P from Equation 2) and (2) mini-mize cost increase (E from Equation 3). The two objec-tives were reconciled into a single set of objectivefunctions using a lexicographic method (Roumasset,1976; Rentmeesters et al., 1996; Coello, 2000). In thismethod, the objective functions are prioritized insome manner, such as by desirability or importance,and solved sequentially.

The objective function combines the pollutant andeconomic fitness scores to evaluate each scenario:

where F = objective function (combined fitness score).Each scenario is first examined to see if its pollutantload meets all pollutant targeting criteria. All scenar-ios that meet the pollutant targeting criteria have atotal pollutant fitness score of one and are rankedbased on their economic fitness scores. Thus, theircombined fitness scores equal their economic fitnessscores, which range from one to (1 + Co). All scenariosnot meeting the pollutant targeting criteria areranked by their total pollutant fitness scores so thattheir combined fitness scores equal their total pollu-tant fitness scores, which range from zero to one.Hence, for each population and for the GA as a wholethe scenario that meets all pollutant targeting criteriaand farm area requirements for the least cost has thehighest combined fitness score.

NPS Component

Two criteria were established for the NPS compo-nent of the optimization procedure:

1. Sufficient within field variation should be incor-porated to compare the impacts of BMPs with regardto their location in the watershed and to utilize thespatial data available.

2. Computer run time should not exceed one dayfor small watersheds with few (less than 10) manage-ment alternatives per field when using a 1.6 Ghz com-puter.

To meet the first criterion, each watershed was dis-cretized into cells smaller than most fields (0.09-hacells). Current NPS models (e.g., ANSWERS-2000;Bouraoui and Dillaha, 1996), with adequate levels ofdiscretization require prohibitive amounts of comput-er run time for the number of evaluations needed byan optimization heuristic. Thus, to meet the secondcriterion under this level of discretization, an NPScomponent was developed to determine cell level grosserosion and route eroded sediment to the watershedoutlet through downstream overland and channelcells. Use of a GIS enabled the desired level of dis-cretization and facilitated routing and simultaneouscell level calculations across the watershed.

Gross Erosion. Gross erosion is modeled in theGIS using the Universal Soil Loss Equation (USLE)(Schwab et al., 1993).

A = RKSLCP

where A = average annual soil loss (Mg/ha), R = com-

L = slope length factor, C = cover-management factor,and P = supporting practices factor.

The S and L factors are calculated as (Schwab etal., 1993)

S = 10.8 sin Θ + 0.03 for Θ < 5.14 degreesS = 16.8 sin Θ - 0.50 for Θ ≥ 5.14 degrees

where Θ = slope steepness in degrees, and

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FP PE P

=<=

forfor

11

(6)

(7)

bined rainfall and runoff erosivity MJ mmha h y

⋅⋅ ⋅

, K =soil

Mg ha hha MJ mm

⋅ ⋅⋅ ⋅

erodibility , S = slope steepness factor,

(8)

Ll m

=

22

(9)

where l = slope length (meters), and m = L-factor

Required data for the gross erosion model includethe USLE R and K factors, a digital elevation model(DEM), management unit boundaries, and land useand management practices for each unit. Requireddata for the USLE S and L factors include slopesteepness, obtainable from a DEM, and characteristicfield slope length, obtainable from a local resourceconservationist or from field measurements. Addition-ally, the USLE C and P factors must be defined foreach crop management practice to be considered.

Sediment Routing. A sediment routing compo-nent was developed to account for downstream effectson sediment delivery. To account for interactionsamong neighboring BMPs, spatial variation in sedi-ment delivery was considered at the smallest avail-able level, the GIS cell. A delivery ratio for each GIScell was calculated and applied both to gross erosiongenerated within a cell and to sediment flowing into acell. Delivery from each cell was then routed along theflow path to obtain the net sediment yield of each cellto the watershed outlet.

Separate sediment delivery equations were devel-oped for overland flow and for two types of channels:shallow concentrated flow and stream flow throughephemeral and perennial streams. Sediment deliverythrough overland flow cells is modeled as a function ofland use cover, slope steepness, and flow length:

where d = sediment delivery ratio through an over-land cell, α = land use coefficient (dimensionless), s =slope steepness across cell (m/m), and f = length offlow path across cell (m).

Equation (10) was developed as follows. Becausethe USLE predicts average annual gross erosion, con-sideration of nonstorm specific delivery factors wasappropriate. Land cover and slope are key, nonstormspecific factors affecting delivery rates (Novotny andOlem, 1994). Thus, sediment delivery was related tooverland flow velocity by modification of the flowvelocity equation (Haan et al., 1994)

v = as1/2

where v = velocity (m/s), s = slope (m/m), and α = landuse coefficient. This equation is nonstorm specific and

applicable to overland and shallow channel flow. Also,it considers the effects of land use and slope.

Watershed level sediment delivery is a complexfunction of individual watershed characteristics. Inparticular, multiple studies, summarized by Walling(1983), have shown sediment yield at the watershedoutlet to decrease as watershed area increases. Addi-tionally, Walling (1983) summarized sediment deliv-ery prediction equations developed for several regionsof the United States. These equations proposed thatsediment delivery ratios at the watershed level alsodecrease as watershed area increases. The predictionequations are functions of watershed area, relief,length, and slope.

The research summarized by Walling (1983) indi-cates that both slope and flow length are significantfactors in predicting sediment delivery. Additionally,the inverse relationship between sediment deliveryand watershed area suggests an inverse relationshipbetween sediment delivery and overland flow length.Thus, to create a cell level delivery function, the rightside of Equation (11) was divided by the square root ofthe flow length on a per cell basis. Next, a new landuse coefficient, α, appropriate for determining sedi-ment delivery rates, was developed to replace the land use coefficient, a, from Equation (11), which isappropriate for determining velocity. The resultingequation was used to calculate cell level deliveryratios:

where d, α, s, and f are as described in Equation (10).The slope steepness and length of flow path acrosseach overland cell are determined by a GIS.

As an empirical coefficient, α can be determinedusing two approaches. One approach includes use ofmeasured sediment yield or delivery data along withslope and length. Another approach is to predict sedi-ment delivery using an NPS model. After collectingdata with either method, Equation (12) can then besolved for α.

Data available in the literature were insufficient todetermine α values. Thus, the field scale NPS model,RUSLE2 (University of Tennessee, 2001) was used toestimate a delivery ratio for a slope/soil combination.First, a two-section slope profile (Figure 2) was mod-eled in RUSLE2. The soil of the lower section wasdefined as noneroding so that no gross erosion wassimulated for the lower section. Erosion leaving theupper section underwent deposition in the lower sec-tion. The amount of deposition was a function of themanagement practice, slope steepness, and slope

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OPTIMIZATION PROCEDURE FOR COST EFFECTIVE BMP PLACEMENT AT A WATERSHED SCALE

exponent =+ +

sin

sin . (sin ) ...

ΘΘ Θ0 269 0 050 8

dsf

=

min ,α 1 (10)

(11)

dsf

= α (12)

length of the lower section. This allowed the deliveryratio for the lower section to be estimated as the ratioof the net soil loss from the slope profile to the net soilloss from the upper section. Then Equation (12) wassolved for α, resulting in the values presented inTable 1.

Higher delivery is expected by channel than byoverland flow due to increased flow depth, velocity,and carrying capacity. Channel cells can be identifiedfrom a DEM in terms of the number of upstream cellsaccumulating to create a channel cell. The use ofDEMs and a flow accumulation threshold to representthe stream network is widely used in GIS applications(Garbrecht and Martz, 2000). For example, in theRidge and Valley physiographic region of Virginia,shallow concentrated flow was identified as flow accu-mulated from at least 60 cells but less than 200 cellsusing a 30-m DEM and stream flow as flow accumu-lated from at least 200 cells (Veith, 2002). The entirecell containing a stream is assigned the relevantstream delivery value. Overland sediment moving tothe channel is not treated separately for cells contain-ing streams. Delivery ratios of 0.98 and 0.9998 wereassigned for shallow concentrated flow and streamflow cells, respectively. These ratios were selected torepresent the low level of deposition expected in small

headwater, rural watersheds (T. A. Dillaha, personalcommunication, Biological Systems EngineeringDepartment, Virginia Tech, Blacksburg, Virginia,March 8, 2002). For different sized watersheds it maybe desirable to adjust the cell sizes or to modify thechannel definitions or delivery levels.

The sediment yield contribution of each cell isdetermined by routing sediment from the cell throughdownstream cells to the outlet. For each cell, the grosserosion is multiplied by the delivery ratios of cells inthe flow path from the cell to the outlet

where Yi = sediment loss of cell i reaching the outlet(Mg), Ai = gross erosion from cell i (Mg/ha), ai = areaof cell i (ha), dj = sediment delivery ratio of cell j, andj indexes all flow path cells between cell i and the out-let.

ArcView GIS (ESRI, 1999) does not currently pro-vide a function for multiplying cell values along a flowpath. However, the ArcView FlowLength function canbe used to closely approximate Equation (13) byrewriting the product of the delivery ratios as anadditive exponential function

where dj = sediment delivery ratio of cell j, fj = flowlength assigned to cell j, and tj = travel distance offlow between cell j and the next cell in the flow path.

The routing process is illustrated for a single cell inFigure 3. The arrows show the flow path from Cell 1,through Cells 2 and 5 to the outlet (Cell 9). UsingEquation (14), the sediment delivery to the watershedoutlet for Cell 1 is calculated as

Summing the sediment loss reaching the outlet (i.e.,the Yi’s) over all cells and dividing by the watershedarea gives the sediment yield of the watershed inMg/ha. This method is similar to that used by Koth-yari and Jain (1997) for routing sediment in forestedwatersheds.

Sensitivity to Spatial Placement of Manage-ment Practices. The effect of land use placementwithin a watershed on sediment yield was determinedto conform to expected trends in the routing portion ofthe NPS component. It was expected that sedimentyield at the watershed outlet would increase when

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Figure 2. Slope Profile Used in RUSLE2to Estimate Delivery Ratios.

TABLE 1. Sediment Delivery α-Factors byLand Use for a 30-m Flow Length

Description α

Farmstead 19.2

Conventional Tillage Corn Silage 9.7

Conventional Tillages Corn Silage With Winter Grain Cover 6.2

Minimum Till Corn Silage 4.9

Alfalfa/Grass Hay 3.3

Minimum Till Corn Silage With Winter Grain Cover 3.1

Pasture 1.6

Forest 1.1

∏ = ∑ ≈− ×

−

d e ejd

FlowLength td

fjj

j

jln( )

ln( )

(14)

Y A a d d d d1 1 1 1 2 5 9= (15)

Y A a di i i j= ∏ (13)

erodible land uses were located nearer to streams or nearer to the watershed outlet. To assess spatialsensitivity, a land use layer consisting of seven agri-cultural fields and two larger land use regions wascreated (Figure 4). Each agricultural field was 3.6 hain size. A constant USLE K factor of 0.042 Mg·ha·h/(ha·MJ·mm) was assigned to eliminate variability insoil erodibility. Slopes in each field ranged from two tofive percent. The USLE C factor was set at 0.003 forforest, 0.01 for grass hay, and 0.49 for conventionallytilled corn silage areas. For all regions, a USLE P fac-tor of one was used. The α value was set at 1.1 for for-est, 3.3 for grass hay, and 9.7 for corn silage.

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Figure 3. Routing of a Single Cell to the Watershed Outlet.

Figure 4. Fields and Flow Networks Used to Assess Sensitivity of NPS Module to Spatial Placement of BMPs.

The NPS component was used to calculate sedi-ment loading to the outlet for one reference and seventest runs. The upper region remained in forest for alltest runs. For the reference run, the lower region ofthe watershed, including all seven agricultural fields,was placed in grass hay. For each of the test runs, adifferent agricultural field was placed in convention-ally tilled corn silage. The remainder of the lowerregion was placed in grass hay.

Differences in gross erosion among test runs (Table2) were due to slope steepness and flow length charac-teristics of the cells in each field. These two factorscontributed to variations in the S and L factors of theUSLE, while all other factors of the USLE were con-trolled. As expected, differences in sediment yield didnot vary consistently with differences in gross erosionin the watershed (Table 2). For example, for Field 1,gross erosion increased about 11 percent relative tothe reference run, while sediment yield increasedabout 6 percent. In contrast, for Field 2, gross erosionincreased about 11 percent, while sediment yieldincreased 18 percent, three times more than for Field1.

Increases in sediment yield relative to the refer-ence run were as expected based on the placement ofthe fields and distribution of flow along the field edge.Also, relative differences in sediment yield due to dif-ferences in field locations within the watershed fol-lowed expected trends. For example, Field 1 waslocated just downstream of Field 7. Both fields werebordered by the same stream and had one to two cellwidths of hay buffer along most of the stream edge.As expected, watershed sediment yield when thedownstream field was in corn was greater than whenthe upstream field was in corn (0.094 Mg/ha versus0.089 Mg/ha).

Economic Component

The economic impact of a given watershed scenarioconsists of the sum of private costs, which reflect thefarmers’ compliance costs due to changing manage-ment practices, and public transaction costs, whichare incurred by the government in ensuring thatwater quality goals are being met (Carpentier et al.,1998). Private costs, incurred by each farmer as aresult of applying a management practice, are firstdetermined at the field level as opportunity costminus net return. Opportunity cost refers to the costof not choosing the management practice with thehighest net return. The private cost for each farm isthe sum of field costs for all fields in the farm:

where ci = private cost for farm i ($), oij = opportunitycost for farm i and field j ($), yijkl = yield for farm i,field j, crop k, and soil l (qty/ha), sijk = selling price ofcrop k on farm i and field j ($/qty), eij = enterprise pro-duction cost for farm i and field j ($/ha), and aij = fieldarea for farm i and field j (ha).

Public costs, the sum of contracting and enforce-ment costs for a given scenario, are calculated foreach farm for which a BMP has been added to one ormore fields. Contracting costs are incurred by govern-ment agencies while forming agreements with thosefarmers who are required to change managementpractices. Enforcement costs include expensesincurred by government agencies while ensuring con-tract agreements are met.

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TABLE 2. Output From NPS Model for Placement Test Fields.

Increase in Increase inWatershed Sediment Sediment

Gross Gross Gross Erosion Yield at YieldField in Erosion in Erosion Compared With Watershed Compared With

Test Corn Watershed Within Field Reference Run Outlet Reference RunRun Silage (Mg/ha) (Mg/ha) (percent) (Mg/ha) (percent)

Ref. None 1.400 N/A N/A 0.089 N/A

1 1 1.549 28.962 11 0.094 06

2 2 1.558 30.714 11 0.105 18

3 3 1.565 31.971 12 0.098 10

4 4 1.554 29.864 11 0.090 02

5 5 1.572 33.287 12 0.103 16

6 6 1.562 31.468 12 0.089 00

7 7 1.529 24.954 09 0.089 00

c o y s e ai ij ijkll

ijk ijk

ijj

= −

−

∑∑∑ (16)

Two additional types of public costs were consid-ered when developing the optimization procedure:cost share and information. In cost share programs,the farmer implements an appropriate BMP and isreimbursed, in part, by a government incentive. Con-sidered at a farm level, the impact of cost share pro-grams cancels out. That is, the total costs per farmequal the private costs (from which the cost shareamount is subtracted) plus the public costs (to whichthe cost share amount is added). Hence, the optimiza-tion program does not explicitly consider cost shareprograms.

Information costs represent the costs involved ingenerating the optimal solution from the baseline sce-nario through development and use of the optimiza-tion procedure. Since, by this definition, informationcosts do not vary by run of the optimization scenario,they were not considered within the optimization pro-cedure.

PROCEDURE DEMONSTRATION

Performance of the optimization procedure wasdemonstrated on a 1,104-ha watershed in the Ridgeand Valley physiographic region of Virginia (Veith,2002). Agricultural production in the watershed is

distributed across 775 ha and 18 farms: two large cat-tle farms, one dairy and one beef (150 cows each); sixmedium dairy (100 cows), one with poultry; one medi-um beef with poultry (70 cows); four small dairy (60cows); five small beef (40 cows), one with poultry.Farms were divided into 125 fields of which 51 per-cent was in cropland, 37 percent was in hay, and 12percent was in pasture. The remaining 239 ha of thewatershed are forested (19 percent) or for residentialuse (4 percent).

Three optimization runs were performed to assessthe procedure’s response under varying BMP place-ment strategies. The maximum acceptable pollutant(sediment) load was the same for all runs. However,the runs included variation in allowable choices andcombinations of cropland management (conventionalor minimum tillage, with or against the contour, andwith or without cover crop) and forage (pasture orgrass hay).

Results across the runs were similar, relative todifferent conditions in each run. Thus, results for asingle run demonstrate a typical progression of fit-ness score, sediment yield, and watershed cost valuesduring the run (Figure 5). In this run, sediment yielddecreased steadily from the baseline loading untilgeneration 140, when the maximum acceptable pollu-tant load of 0.64 Mg/ha was achieved. This corre-sponds to a pollution fitness score of 1.0. During this

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Figure 5. Comparison of Cost and Pollution Variables With Fitness Scores for a Single OptimizationRun on a 1,014 ha Watershed in the Ridge and Valley Physiographic Region of Virginia.

period of the run, fluctuation was seen in total water-shed cost and economic fitness, which were being cal-culated but not used to control the optimization. Afterthe maximum acceptable pollutant load was achieved,the economic fitness score increased to 4.78, the levelat which the GA ceased to find lower cost solutions.Total watershed costs decreased from above $120,000to a final value of $89,750.

The procedure performed as expected across allruns. The pollution fitness score (Equation 1) wasdeveloped to lead the optimization component to findsolutions for which the NPS component predicts sedi-ment yield equal to or less than the maximum accept-able load. Accordingly, pollution fitness scoresincreased as sediment yield decreased.

Once the maximum acceptable pollutant load wasmet, the economic fitness score (Equation 3) wasintended to drive the optimization process using theEconomic component. The Economic component incor-porates the two considerations for maintaining rea-sonable farming practices: farm area requirement andcost fairness. However, the economic fitness scoreshould also correspond with reduction of total water-shed cost, a more direct monetary measure of the eco-nomic system being modeled than that provided bythe Economic component. As expected, after the maxi-mum acceptable pollutant load was met, economic fit-ness scores increased as total watershed costdecreased.

Additionally, all runs displayed a convergencetrend within 700 generations. This demonstrated theGA’s ability to converge to a solution within a similaramount of time for runs of varying strategies and ran-dom initial populations but otherwise similar inputs.

SUMMARY AND CONCLUSIONS

A functional procedure was developed to optimizeBMP placement based on cost and NPS pollutionreduction for a watershed. This provides a computer-ized method for locating scenarios for which alterna-tive BMP placements increase watershed level costeffectiveness.

Among a range of optimization heuristics, the GAand SA heuristics have features most suited to thisproblem type. Because the GA provides multiple solu-tions that meet the objectives, there is flexibility inselection of the most suitable solution based on thepriorities of farmers and other stakeholders. Addition-ally, comparison of the final solutions can lead to anindication of those fields that are more critical to theoverall watershed cost effectiveness.

Development of this procedure revealed that repre-senting cost effectiveness as a ratio in a single objec-tive function does not define a clear response surfacefor this problem. An effective solution involved use ofa lexicographic technique to prioritize a multiobjec-tive function. Additionally, it was found that using theUSLE with a sediment transport function instead of amore detailed NPS model allows the procedure to runwithin a reasonable timeframe. The sediment routingroutine developed for the NPS component was foundto respond as expected to spatial changes in landmanagement.

The optimization procedure was demonstrated suc-cessfully on a 1,014-ha watershed. Increase in fitnesscorresponded with a sediment yield decrease to themaximum acceptable pollutant load, followed bydecreased costs.

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