Ptld(1、 `ヽVatcr En、 .ilon(2009,7:163-175

DO110 1()07/、 10333()(),()1603

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Fuzzy optimization model for integrated management

and nonpoint sources in watershed

Shigeya Maeda' Toshihiko Kawachi' Koichi Unami .

Junichiro Takeuchi . Tomoki Izumi . Syunsuke Chono

Rcccivcd: 27 lヽt、 y 200ヽ /Rcviscdi 19 ,、 ltlich 2()()9/Acccptcdi 19 1ヽ lttrch 20()9/Publishccl olllil、 c: 14rヽ pli1 20()9

● Splingcl Vcrlag 2()()り

of total nitrogen loads from distributed point

Abstract A fizzy optimization model is developed toallocate allowable total nitrogen (T-N) loads to distributednonpoint sources (NPSs) and point sources (PSs) in a

watershed for river water quality management using the

linear programing technique. The watershed is divided intouniform grid cells on which T-N loads issuing from NPSs

such as paddy fields, upland crop fields and cities are

controlled. A geographic information system integrated

with the digital elevation model facilitates computation ofroute lengths of surface and subsurface flows from cel1s toa river running through the watershed. The T-N loads

discharged from their sources are assumed to decay, sub-ject to distance-related Iirst-order kinetics. As managementgoals, maximizations of total allowable NPS loads, totalallowable PS loads and total yield of rice are consideredfrom environmental and economic viewpoints. A primeconstraint is an effluent limitation standard for the aggre-gate amount of loads that ar:rive at the downstream end ofthe river. The fuzzy sets theory helps appropriatelydescribe vague attitudes of decision-makers (i.e., stake-

holders and management authorities) in terms of con-straints and conflicting goals. An application of the fuzzyoptimization model, developed as an improvement over ourlast nonfuzzy model, to a real watershed in Shiga prefec-

ture, Japan, demonstrates that the fuzzy model embodies

our last model, and is capable of creating management

alternatives for T-N load allocation in a more practical and

flexible manner.

S. l\'laecla (,4).T. Kauachi .K. [-lnarrri .J. Takeuchi .

'1. Izrrtti . S. ('hono

Cracluate School of Aglicr-rlturc. Kyoto [Jniversity.KitushiliLkau a oiuake-cho. Sukro-ku.Kr oto 60(r-8-501. .lapan

c-rt)itil : \rttilcrlit{o kiri\.1\\ r,l('-u.ire.-il)

Keywords Watershed management . River waterquality 'Pollutant load allocation 'Fuzzy sets .

Decision suppor-t model

Introduction

Pollutant loads stemming fiom agricultural and/or muni-cipal wastewaters are the main causes of water quality

deterioration in many rivers, lakes and estuaries. However,difflculty of diminishing such loads from nonpoint sources

(NPSs) compared with those from industrial plants and

sewage works (i.e., point sources, PSs) retards the

improvement of water qualities in these bodies of water. InShiga prefecture, Japan, various actions have been taken tocontrol wastewater discharges from NPSs. For example,

since 2004, a system of direct payments has been enforced

to encourage farmers to reduce the amount of chemical

fertilizers applied to crop fields. Farmers may receive

monetary aid linked to the area under crops in payment for507o or more reduction of ferlilizer application. In Kusatsu

City and Moriyama City, some facilities have been con-

structed since 1998 to reduce effluent loads from urban

districts by trapping the first flash of direct runoff water and

treating it physically and biologically. Any of these actions

is not based on an advance quantitative estimate of itssubstantial effects in the improvement of water qualities inrivers or lakes. In this respect, such actions, usually taken

in other places of Japan as well as in the world, are onlyspeculative and suboptimal. To take nonspeculative and

optimal actions, watershed-based goal-directed strategies

should be established which are strongly motivated and

highly organized for meeting the water quality standards

designated for the bodies of water, taking a global view ofthe situation. This supervisory top-to-down strategy is to

a springer

Pailclr, \\'uter Lnr i|r,rr rl00i)t 7:l(rj-l- j

optimally anocate the total allowable pollutant load among

dispcrsed wastcwater dischargcrs, and thcn to require the

overloading dischttgcrs tO take appropriate actions for

reducing the culTent loads up to their own anocated linlits

Pollutant load (waStCload) al10Cation problems have

been handlcd in a framework of thc mathcmatical pro―

grallling The problcms arc typicaHy modcled M/ith a sin―

glc― or multi otteCtiVC fllnction and a sct of constraints in

the context of minimizing the total cost for wasteM/ater

treatincnt cquipmcnts and their operations,or maxilnizing

the total allowablc pollutant 10ad of COD(chcmiCa1 0xy―

gen dcmand)Or T_P(total phosphorus)tO a bOdy of v/atcr

From a vicwpoint of the v/ater quality managcmcnt intcn―

ded for watersheds, Inodeling can bc diversincd into twO

types of modeling′ 72′θ″―and′ 71′″α―Watershed managemcnt

problcms. The ル:′ιr―v/atcrshed management model con―

sidcrs individual sub― watershed of the watershed as a

lumped(nonSCgmental)System t0 0btain sub― watershcd―

bascd managcment options in a g10bal scnse.On the othcr

hand,thc′ 71′′α―ヽvatCrshed managcment inodcl considers the

watcrshcd as a distributed(sCgmental)syStem to obtain

scgment― based managcment options.Thc area of watcrshed

to which the intra― watcrshed model is applied is gcnerally

smaller than that where the intcr― watershed model is

operated

Thc inter― watcrshed managcment problcms have been

formulatcd v/ith thc aid of the mathcmatical prograllling to

dcvelop modcls fOr sccking optimal wasteload allocation

strategies br river water quality managemcnts(e.g。,

Ka、vachi and Maeda 2004a, b; Jia and Culver 2006;

Kamlaker and MLliumdar 2006).Thc maximum allowablc

load a1located to an outfaH,obtained frolll the model of this

type, can also be a maximum aHowablc lilllit of pollutant

rclcase fronl thc co■ esponding sub― watcrshcd The inter―

watershed management model thus scrves to provide a

constraint(rclated tO water quality standard to be mct)for

the intra― watershcd managcment modcl

The intra― watershcd management model lends itsclf to

fornling a sound schcme for deliberatcly controlling NPS―

born v′ aste、vatcrs Compared with the reccnt advanccd

intcr― watcrshed managemcnt rnodcls cited abovc,thc intra―

watershed management rnodels v/hich havc bccn presentcd

sccm primitivc Jenq ct al.(1983)emp10ycd the lincar

progranling for thc problenl of rnccting the T― P standard of

a rccciving water of a lakc at rninimutt cost.Both PSs and

NPSs wcre taken into account, but no― use of GIS (gcO―

graphic information system)made it dimcult to apply the

model to a rcal lake watershed.Randhir et al.(2000)

devcloped a GIS― aided integrated frame、 vork WISDOM foragricultural NPS pollution (sedilnent) contr01, which

comprises surfacc water quality and crop growth sirnula―

tion modcls, and their interfaces, employing a spatial

multiattribute dynanlic progranling algorithm The mOdcl

2 SPringer

\\'as applicd tcl u w,atershed. cuvcrecl b1 83 grid cclls L,:

2(X) rn x 200 rn. in Indiana. L-lS,\. iLncl oprimal sprLri.,.

crollpin-s plan. its cultivation pnrcricc\ ancl f'ertilization:,achicVe econ0nric trncl Yn,ater qLllrlit\ ol-r.jectiVes \\,efe !i '.

eratecl. Kumar ct al. (2(X)2a. b) clerclopecl a tu,o-ob.jeert,;littear pro-uran'ring nrrlclel that helps allocate the allou:,:. ;total nitrocen (1.-N) loads to \PSs riithin a \vatcr'\lt.!lriVer. The watershecl $as ll'usurentecl into 51111111 .q:r.,:.;,

cells. ancl the loacl allouecl ro NPSs u,as allocutctl rl,\.,: .

the cell so-callecl lancl ntanu-ecntent unit (L\lL \1.,;.r..et al. (2006) rnoclitic'd thc optintizatiort rnodel Pr's'.s, 1;...1 r1

KLrrnrr et al. (2(X)lb) so that the et'flr-rcnts tlirnt P\. ,.,:t he

contmllecl as nell. and ntlLrirrrizlLtiort ol riic r t:..i -:l'i he

consiclerecl as an aclclitional ob.jectir c critcr.i,,n. C1,it.i!lerin-sintfa-\\atefshe.cl u,'ater qLralitl, contfoi ()\.i iL l\r;i-{ period oftirne (slr1'. Ir'ear long). thc rnoclel s;1.;1|Jrl1g.l to a par-ticular n,atershecl in Konan Citr..luP111i. r') \)lltuin lL sct ofthe cli['ltrent noninf'eriol' solutionr r ('lrrrnkrinS and HaintesIt)ll3)that -uive the clilterent ailorirhle tllilr,aveliiged -l'-N

loacls fhrrn [-N{Us ancl PS'. .\ttrrirtntenr leIels of thc- conr-petin-g rltarla-gcrlrent ob jee tir c\ \\ cre eontrollcd b1, thei:-constraint o1'the n'rorlcl tirrntultitecl bl the ii-u()nstrlintrrcthocl. u'hich is one ol'tltt p()pular niethocls of ohtaining i.t

scalar problcnr ll!nt lr \cctor {or rni-rltiobjcctir.. I ,r1.p111,,-

zation problcrn (Chaitkonu and Haintcs l9ll3).Conventionall-r. ci en slight I'iolations of the utrn:rt'.rinis

in the rnttlienratical pro-{r'arnin-u prclblcm ilrc n()t pe|ntittecllor 1'easible solutions. i.c., onlr,'crisp' or'halcl' eorr:rlrLi rrts

are cousiderecl. Horvever. this too rigid leasiblc lcsion is

ofien r-rnsuitable because the r-rpper or lorier bound: ol'sonre constraints arc Vague in rcalin. uncl thLr' Lhc

ctttpltrrtttettt rrl hltt'tl !r)tl\tt'llilll\ i\ 1('t .rl)lrr{'lltiiltc i1

ntodel building. Furthelntotc. lt()nLlrriuttiirLri\ L' {oirl\ are

otien lar,orable firr practical cases tll'uutcr qualitr ntun-agerncnt. For exiir-nple. the .loal 'treiltntent co:t :holLld bc

recluced' is t1,pical. but it is r-rnclear rhar ro u har clegree the

ccrst should be clccrensecl . Fvzzt scts theor\ l-rls tltr,rs bccn

emplclvccl to express vusuene\\ irt etrnstririrrt: artcl ob.jec-

til'es in rnathernatical progran'ring problerns on ri\cl' \\ltel'qual itl' rnana-scrlrcnt (e. g.. N'lu j unrclar ancl SasikLr ntar' l00l :

Karrnakar arrcl N{r-r.jr-rr.ndar 2006) ancl on water resources

allocation (Kincller lt)c)2). Chu ancl Chan-e (2(X)9.r haleacloptecl firzzy, int'erencc li)r dvnantic sroun(l\vittcr rente-cliation clesign. It is notecl thiit the application ol'thc fuzz1,,

sets theorv har"e, hou,cver. never lrccn conrluctecl in thestuclies on intra-u,atershecl pollr.rtant colttrols llentionedaboVc.

Since our last nonfuzzl, rnoclcl (Maecla et al. 1006) riasfirrmulatccl using thc /i-constraint rnethocl. the ntodel is

heleaiier nanrccl as the'ii-constraint ntoclcl'(ECNl). In theprescnt stuclr,. thc L,CN4. clcVclopecl as nn iutfl-\\'atcrshednrallagenrent rnoclel. is lcfirrmulated b1, crrploy ing thefuzzy optirnization frante\\,ork to cor.rsicler tl'rc ritglrcness

Paci(l、 Vヽatcl Ln、 =i「 oll(200tl,7:163-175

inherent in setting up the management goals and lower orupper limits of some variables. Through an application ofthe fizzy optimization model (FOM) so developed to a realwatershed, it is demonstrated that the model can success-fully reflect decision-makers (i.e., management authoritiesand stakeholders) attitude to the limitations and the goals,and be reduced to the ECM by discarding its fuzziness.

Modeling total nitrogen (T-N) transport

Consider a river watershed where controlling T-N loadsfrom PSs and NPSs is intended for water quality conser-vation in the river. The watershed is segmented withnumbers of uniform square (say, 50 m x 50 m) grid cellseach of which is regarded as a unit (LMU) for controllingT-N loads from NPSs. The watershed is supposed to con-tain one main stream at most which is represented by asequence of cells, and its tributaries are neglected even ifthey exist in reality. The GIS and the digital elevationmodel (DEM) are employed for delineating the watershedboundary, determining the flow direction on a LMU and

identifying the routed flow on the surface of watershed.T-N originating from a LMU is assumed to be conveyedwith direct runoff and baseflow which are differently butvertically in parallel routed.

The discharged waters from a LMU are self-puriliedwith removal of T-N on the ways to an outlet of thewatershed by denitrification, or retention in biomass orsediment. Modeling the self-purification mechanism iscrucial to evaluate the allowable discharged load from a

LMU. Since the modeled mechanism is intended to be

directly embodied in the constraints of the mathematicalprograming model in this study, less parameterized andgenerally-used description of the T-N reduction is favor-able. The following are thus assumed based on the study bySkop and Sprensen (1998), which requires only easy-to-obtain data.

1. T-N aniving at a point is expressed by an exponentialdecay function with respect to the travel (or transport)distance of T-N, which contains an adjustable coeffi-cient referred to as a self-purilication coefficient.

2. Another primary effect on T-N removal, the travel timeeffect, can be reflected by appropriately estimatingvalue of the self-purification coefficient. The baseflowthat gains ample opportunity of T-N removal with a longtravel time takes larger value of the coefficient than thedirect runoff with a short travel time.

3. The occurrence ratio of baseflow- to direct runoff-induced T-N loads from a LMU is simply related to the

soil texture of the LMU.

Optimization problem

e-Constraint model (ECM)

In the earlier work (Maeda et al. 2006), a nonfuzzydeterministic optimization model (or ECM) with threeobjective criteria was presented for allocating allowableT-N load among LMUs (or NPSs) and PSs in a riverwatershed. Before describing an advanced fuzzy multiob-jective optimization model, this nonfuzzy model (the

ECM) is reviewed.The selected objective criteria of the intra-watershed

model for river water quality management are described

with the following watershed-scale environmental and

economic goals:

Goal Gl: Maxirrization of total allowable load fromLMUs (i.e., controllable load from NPSs).Goal G2: Maximization of total allowable load fromPSs.

Goal G3: Maximization of total yield of rice by

controlling T-N discharged from paddy fields.

As an initial stage of incorporating an economic objectiverepresented by crop yield into the wasteload allocationmodel, rice, which is one of the most popular agriculturalproducts in Japan, is taken into consideration as in <GoalG3>. For simplicity, other influencing tactors on the

growth of rice such as irrigation and drainage conditionsare assumed perfectly managed in the paddy fields, and

therefore rice yield is exclusively controlled by the amountof nitrogen fertilizer applied to the fields. Since T-Ndischarged from a piece of paddy field has a close relationwith that applied to the piece, <Goal G3> could be a

criterion suitable for meetins the demand from the

agricultural sector.

The ECM for the present optimization problem can be

formulated as

Miniruize :t

suL.lject to

(1)

1. Effluent limitation standard for controllable T-N loadtransported to an outlet of a watershed

2. Relation between direct runoff- and baseflow-inducedT-N loads from a LMU (differently related fordifferent soil textures)

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Padd、 Vヽtllcr Environ(2009, 7:163-175

3. Deviation ol'actual T-N load frorn an optinLllstanclarcl (erplained lrelou') of'T-N kracl frotn a

pacldl'LMtJ

L't/, +Llt', -\,',',t't - ut;,- Ltt,. i: l.).....:t, (1)

.+. Upper lirnits tbr pararneters tl; and l1l,

tt,,-u,,. ,,,,t\. i-lr.).....|t, (5t

7.

Lo',r'er ancl r-rppcr litnits ol-T-N load fr<tnt a nonpatlcly

LMLI (uplancl crop fielcl ancl cit1,, LMLJs)

L,, !L',1 + I-',', <i. i - l.).....1,,:'r.aa r, + Lt,' < t- i : r.2. . . .. r, (6)

Low'er ancl Lrppcr lirnits of T-N concentration of\\jaste\\'ater fl'orl a PS

Cps,SCr,.,<(nf,. k:1.2.....K (1)

Nonnegatir,'itv conditions

t'! t' Cps.. t1, . tt,,, ) 0. Vi..l. A (8)

r:-Constrtints

こ2≦ ~ε2[3≦ ε3

、vherc

coetticient lbr liver flori, 1r1:76x1,): L : Lrpper linrit of totalcontrollable'f-N loacl convcl-'cd fl'ont all the LN{Us ancl PSs

throughout the u'atershcd to its olltlet (g/.la\ ): r'1 anrl

r'\ : occurrence rates ol' bascllow-inducecl 'l'-N loacl todirect runofT-induced loacl. lbl loarrtv ancl sanch soils.respectivelv: L)]l't - optirnurn stanclard of -t'-N

ioacl frorn a

paclcll,LML-r l,i'ith respect to rice l,iclcl (g/cla1,): r1;, ancl r/t; :l.rositive and nesative clei'iations (giclar') front L't',|'l

Iesl)ecti\el\: rr,, untl rr^ : upper limits ol' u,,. ';nttl rt,, (!lda1-). respectivelv: L,, and 1-,, : krn'er lirnits of T-N loadfrorn r.rplancl clop fielcl an.l .rtl f Uirr tg/du1 t. r'espectivell,l1," and L" : ul,t,.r. lirnits ol'T-N loacl tirrrn uplancl cropfield iind city l-MLJs (-t/dar,). respectivell,: (.'ps,urlr-l a.! -lor.vcr uncl r-rpper limits ol T-N e ()ncclltl i.rtiol'r uf wastewaterfr'orn PS (g/mr1. respectively: iir tnd i;r: llal'amctersrelevant tcr ,, ancl :.r. respectiYely: and .t : (t't . tt,' .

C1,s, . rr,1 . lr, ) : rnana-gelnellt altcrnative or vee tttt' rr hose

conrporlents are all the variablcs in the optintization rtrodel.

The objective critcrion ,I is entplovecl as a sirr-ule

objectivc function. corrcsl.roncling to <Cioal GI>. r,i,hile;1itt.tcl :: appear in tl.tc r;-constraints (Eq.9) renresentino<Cioal (12> and <Goal G-i>. respectivell,,.

Tlre tirst term at the lett-hancl sicle of Eq. I is produccdbasecl on the relation

(9) ιヽ Ps= Σ

ノ

←‐

(10)

where LNp5 : transported T-N load issuing from all the

controllable NPSs to the outlet. The parameters 2d, )b andi' could be identifled by the method developed by Haet al. (1998). That is, using Monte Carlo optimizationtechnique, optimal values for these parameters could be

selected at the minimum value of the error ratio, which iscalculated by the difference between the delivered T-Nload predicted with unit loading factors and the observedload at the outlet of the watershed. Ha et al. (1998)

showed that the basin-wide self-puriflcation coef{icientscan be obtained on monthly basis. Since the optimizationmodel considered in our study is, however, a deterministicmodel, a representative set of evaluated parameter values,e.9., a set of temporally-averaged values, must be

adopted.

Note that Z in eq. 2 excludes uncontrollable T-N loadoriginating from the watershed of interest (e.g., effluentfrom forests) and delivered T-N through river flow fiomupstream watersheds. Variation of discharge in the riverchannel Q is assumed negligible. Flow length xi,,x)', andx[ in Eq. 2 can be computed based on flow direction thatcan be determined using the DEM in the GIS and the

concept of steepest slope to one of the eight-neighboringcells (e.g., Kumar et al. 2002a).

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′|

Σl(4≠十イ)

′=1

.,(r) : - I′

κ ち

:r(-r) : -IOnr,Cr.. .r(r) : \(,,,, +,,,;)=々1

そ1,そ2 and ζ3=0切 CCtiVe criteria;subscriptノ =′ ,″ and c,

which denote paddy neld, upland crop neld and city,

respectively: subscript ′=idcntiflcation number of LMUof typeプ (′ =1,2,… ,み);み =tOtal number of LMUs oftypeプ ;た =identincation numbcr of PS:κ =total numbcrof PSs;I′ =T― N load discharged with diК ct runoff iom

LMU■ (g/day);考 =T― N load discharged whh base■ owiom LMUノ ,(g/day);CPst=T― N conccntration of effluent

fl・tDm PSた (g/m3):1/:=■OW length(overland travel

distance)■Om a LMUプ′tO the Outhll whcК thc load

discharged fl・ om LMUtt emptiesinto a river(m);ィ =10Wlength(in― riVer travcl distancc)frOm the outfall to the outlet

of the watershed(thC dOwnsicam end of the river)(m);

χソ=■OW length iom tlle outね 1l whcrc the load dischargcd

from PSた emptics into a rivcr to the outlet ofthc watershed

(m);0=unifom rivcr discharge(m3/day);2Psた =Cffluentdischarge tom PSた (m3/day);λ

′and λ

み=watcrshcd― wide

self―purincation cocfflcients for direct runoff and baselo、 v

(1/m),respectively;λW=watershed―

widc sclipurincatiOn

2 SPringer

I)を t(1(1):ヽV〔ltcl Enviloll{2009,7:163-175

Fukayama (1990) have shown necessary nitrogen fertil-izer at several stages of rice growth enough to produce the

maximum yield. As shown in Fig. l, if soil nitrogen absor-

bed by the crop body and percentage of nitrogen f'ertilizerused for crop growth are known, discharged nitrogen orig-inating from discharged unused fertilizer corresponding tothe maximum yield can be estimated. lt is assumed that T-Ndischarge from a LMU p; is composed of the unused T-Nfertilizer and atmospheric input, which is considered as the

optimum standard of discharged T-N, Lilt ,in Eq. 4, and that

nitrogen in plant residue remaining in the LMU after har-

vesting is stored in the soil. The relation between rice yieldand T-N discharge from paddies, as shown in Fig. 2, is thus

presumed, and as in Eqs. 4 and 5 and the second constraint in

F,q. 9, L!, + L! isrendered as close as possible to 4pt *ith anTI

allowable range of lr;y' - rr,.Lil'+ rr,)

Lower limits fbr T-N loads issuing from LMUs con-

cerning upland crop flelds and cities in Eq. 6 should

t、理型 主竪ノ

Optimum stored nitrogenin crop body of rice

Ab、 olbcd l Ljscdsoil nitlogen nitlogen l'ertilizcr'

=④

[-ig. I Deterrninution of optirlunr stxnclltrd ol' T-N clischargecl ftotlpatldl l-N4t1. 1-)ir". corresporrdi:r-{ to rntrirnunr f iclcl

り「

practically be given as marginal emissions from agricul-

tural and urban sectors, respectively. Additionally since

effluent limitation standard regarding T-N concentration ofwastewater is provided for PSs by the Water PollutionControl Law and the stringent add-on effluent standard

imposed by the prefectural government in Japan, the con-

straints that represent the standard are imposed (Eq. 7).

The ECM (Eqs. 1 9) is solved with selected values of e2

and e3 which represent decision-makers' (i.e., management

authorities') preference for the three goals. As a result, anoninferior solution, i.e., allowable daily discharged loads

fiom each LMU, L:f and Il. and T-N concentration ofeffluent from each PS, Cp5,,, are determined, where.f2 - e2

and/. : e3 hold (Chankong and Haimes 1983).

Fuzzy sets and membership functions

The ECM (Eqs. l-9) is extended to a tuzzy linear pro-graming model by employing Zimmetmann's minimum-operator as finzy decision fbr the effluent control problem(Sakawa 19841. Values ol the upper limits un, md u,,

in Eq. 5 and the lower limits L,,,,L,, and Cps*in Eqs. 6 and

7 are presumed to be subjectively fixed by decision-

makers, because these are not related to official standards.

ln order to consider the inherent vagueness of the con-

straints including these limitation values, the constraiuts

are relaxed (i.e., allowed to be violated to a ceftain degree)

and converled into fuzzy constraints.

Six kinds of fuzzy sets are first linguistically defined so

that the imprecisely determined these upper and lower

bounds, as well as goals in the ECM, can be described inthe more practical manner. The fuzzy sets are expressed as

ri, : {xlu is smaller than or equal to r:.}.' L'P' ' t")

Fir: {rfu0,

is smaller than or equal to rlr}.' (

npr thon ^.

an,ol tn / \F'r: \r)

L', -f Li, is larger than or equal to L_LJ.

' ( nor thon

^" o^,,.1 rn / \f'^: lxlti, * Li is larger than or equal to !, J.

n{ : {xlCp5,

is larger than or equal to Qt,}G1 : {xlzt is smaller than or equal to zfl}, l: 1.2,3

where zl: maximum permitted value of zr (l : 1,2,3).The ftzzy sets are associated with fuzzy constraints and

fitzzy goals. The remaining constraints other than the fuzzy

constraints mentioned above are unihed to produce a crisp

set of x. C.

Generally defining nonlinear membership functionsassociated with the fuzzy sets requires collection of much

information on decision-makers' attitudes about the con-

straints in detail compared with the linear ones. In addition,

supposing linear membership functions in this context

こ

０一０】、

００嗣餞

Fig.2 Rice yield function with respect to discharged T-N

のSPringer

らF+ら,

Discharged unusednitrogen fertilizer

P`tcld、 ヽヽ/tltcr Hnviloll(2009)7:163-175

∀一与＋一与＜

一釦

一」

〓犠

一

一」一」一一、″・・．輛

１

　

　

１

　　０

ｒ

ｌ

ｌ

く

―

―

ヽ

〓

results in a linear programing problem that is easier to find aglobal optimal solution than a nonlinear programing problemthat includes nonlinear membership functions. It is practicalthat linear membership functions are assumed unless this iscefiainly wrong. Therefore, the linear membership functions

trtp.r(n : 1,2,3,4,forall 1t. trr6r (forallk)and1r", (l : 1,2,3)are defi ned for F'" (n : 1. 2. 3, 4.for all r.1. F! lfor all k) and G r

(l: 1,2.3), respectively, written as follows (typical threemembership functions are illustrated in Fig. 3).

I l, .,( r) . -','I

1r,,t.r ) { - . '.:,<:.1r.t)':'it /:l.l..lI

I r): .-,( r ) ._i"

'"vhere ,l\.f 5"tlL,,.lL, arrcl r1(lps, : aclnrissihle

violutions of the associatecl original constraints: ancl

:i': minimunr acceptable value of -,t ([ :1.2.3). Notcthat cletinecl by the intersection of thc scts clctjnccl abo'u,e.

the set of firzzv clecision, Z. is ivritten as

,: h.,) n (1.,) n (nc,) n. ( )

The rnernbelship function of 7, is -gi\,en lrv

tt,t \) : i]li,l[,,, .ttt .tt,;] , l:r

The optinral clecision (or satisfiLctoly solr-rtion) of theproblerr considerecl hcre. .v'i'. is linally procr.rrcd by

1Lr(x ') :1 : ,,,,.11,r(.t)] ( 13)

r'vhere l. : nrinimLlm satisfaction level. ancl l,'i' : rnari-mizecl rnininrurn satisfuction lei,el. -l'he problenr search-

in.r lbr.r':'results in r r.na.x nrin plohlerr u'hich intcncls

to rnaxirttize the satisllLction lcvcl tbr the ob-jcctir.es bi'nraxirnizin-s tl.re rninir.nr.rrn level of the membershipfunctions firr thc constraints that ure in conflict ortracle-ol'1' r'clation. ln the nert section. this l'uzzi,' opti-mizltiou problern is exprcsscil in a nrorc tractablcnlart tlef .

Fr,rzzv optirnizatiorl mo.lcl (FON,I)

LIsing the mernbelship firnctions of the fuzz1, eou\trlilltsancl goals. the T-N kxrcl allocation ploblenr is cclr-rivalcrrth

reu,ritten as lbllorvs:

N4axinrize ).

sub.iect to

/ι /t(Cハ 1)≧ ′‐ ∀た

メιc.(ヽ )≧':

∀′

・0=|||1癬

=計

〆T‐辱Ⅵ(14)

０

一′

′

　

　

　

８

　

　

９

　

　

０

∀

気一

＜

一

＜

一

ｒ

ｌ

ｌ

ｌ

く

―

―

―

ヽ

二―ヽ″

Ｃ

ヽ

―

＞

！

ノ

ι

「′一げ・

一十　　　一ι

ノ．　　≦

、呵り　　　　　ヽ

■一　　　Ｇ

一

　

　

　

　

　

ＰＳ‐

イ〔　　

　

ω

／１１、

　

　

　

　

■

．

ど　　　　　，７ヽ

ｒｉｌく――ヽ

　

　

　

ど

４Σ［κΣ日

∀ノ

∀′

Ⅵ　　Ⅵ　　”・　　”・

＞＞

就ゼ

．与　

与　イ

　メ」

Σ

ツ

/,t t. J t1l-',' : LNIL i ,rtl Irrrtttt_r srrils *.. j

'' - I i.L',/: LNIUi;ott:ltttlt strils' Yt't\

９

´

イオ有},fF」 iilli に→

(1)s.≦ CPsI た二 12,… ∵κ (25)

イイCh・弔i4‐ ≧O Wたた に0

ThC fu77y Ol〕 ti1llizcttion lllodel.「 ()IνI.is ina‖ yl‐el)rescnted

i ll ■11lol c traCtclblc貴 )rin tts

●SPringer

I)を 1(ltl、 Vヽatcr Ln、 lrolぅ (20()|),7:163-175

I"ig. 3 T1'pical threernemttrshil.r furrctionsintroducetl to the fLrzzv

optinrization rlodel

Maxinli′c ' (27)

SuttCCt t()

平喜[|∵Jイレ∈・→→ 的

十ΣEご 夕CP、 CP、 グた_1

イ={1サ:嚇:I‖ :IIキ ::‖Ⅵノ に

"イ+′ 夕′―ιメ:=′与::― ι

「 ノ=12… …4, (3())

`与:.≦ ι与,十 (1-′ )`ノ

`41″「≦可+(1-′ )`海ア′=12… …4, (31)

′ll― (1-′ )`性″.≦ Ll+止「 ′=12… …′″:

ι`.― (1-′ )`′

Lて ,≦ イ|+Ll.、 ′=1、 2… …/( (32)

CPst― (1-′ )〃 (｀P、 |≦ CPsI、 た=12 … κ (33)

fll+L「 ≦耳, ′=12 1/″ :

41+′イ.≦可 ′=12_:ノて (34)CPs、 ≦εf)sI, た二 12,… .κ (35)

〔′(_T)≦ イ十(1-′ )(ず ―イ): ′=12‐ 3 (36)

イ ′‐l CPstヽ ″′

'`与

‐≧0‐ ∀′,た (37)o<′ <1 (38)Thc varitlblc ' play、 a rolc of a tl‐ claxtttion paralllctcr・

sincc it contl‐ ols thc illagllitudc()fl・ claxation of thc fuzzy

con、 trtlints in Eq、 31-33 ttnd 36 A、 ′ `lppl・()aches to

7Cl‐(). thC spacc 、ccliChCd lk)1・ thc feを lsiblc solutions

bcconlcs 、vidCr bccausc thc rt177y cOnstl・ aints bccolllC

l()()、 cr

Although arbitrary v〔llucs can bc givcn to[,'andこメ(ノ == 1, 2, 3). systCnlatic 、「ays or thc scttillg [trc there By

Zinlnlcrnnann'、 ■lctllod,thc valucs()fイ (′ =1,2.3)can bc、pccincd by 、。lvillg a sirlglc― ollCCtiVc pl‐ ()blcnl. ic,:11linilllizeこ

′、LlttCCt t()Eqs 28-38 cxccPt Eq 36',and thc、.aluc、 of[だ (′ =1,2.3)can bC COnlputcd by thcお llowing

cquatlons

ZF=鴨、レ′(χア)], ′=1,2,3 (39)

wheК χア=optimal solution of the single― o可 cctiVe

optimization problcm where thc otteCt市 e inction is ζ′

(SakaWa,1984)Inthe ECM,thc noninferior solutions can be gcnerated by

setting ε2 and ε3inEq.9whichreflectprcferenceofdecision―

makerstotheobieCt市 es Dctermination ofε 2 and ε3 ValuCsis,

ho、vever,not alv/ays an casy task due to lack of scicntinc

standards for them.In contrast,the FO卜 /1 never rcquires thc

decision― makers to spcciサ paniCular valucs to the ottcc―

tivcs.Their vague desires for the goals and linlitations havc

only to be exprcsscd as linguistic statements shown above.

Relation betwecn ECM and FOM

If thc F()M satisies all thc following conditions,then thc

FOn/1 becOmes cquivalcntto the ECMI(1)ZCrO iS givcn to

証 ぬc ad血∬Ыc宙d血 on、 グけ,煽,色,色 and

″CPsた ,(2)そ

`=ィ

=-82 andイ =イ =ε3,and(3)てf and

そr are determincd by thc Zimmermann's method.Notc thatthc condition(1)fOrbids relaxing thc constraints on lower

and uppcr lilnits of the variablcs, and(2)as wcll as(1),

converts the fuzzy constraints into crisp ones. The lowcr

and upper lilnits on_-l rcalizcd by the condition(3)guar―

antee a range for cxploring minimum そ1 、vhich is wide

cnough.Consequently,Eq 36 can be rewritten as

[1(V)≦ た1+(1-′ )こ7

こ2(・)≦ ―ε2

〔3(・ )≦ ε3

←0)

(41)

(42)

Maximization of i in the FOM in this context leads to

giving more weight to z? than zT in Eq.40, resulting inminimization of 21. Therefore the FOM encompasses the

ECM.

Application

Study area

The FOM (Eqs.27-38) developed for supporling deter-

mination of allowable T-N loads discharged from PSs and

●SPringer

L,,i t L,i'

170 Ptt(1(1、 | ヽヽ「`ttcl li11、

11(】112009)7:163-175

Di rection of river flow

Outlet ofwatershed

1,000 2

Point source

River

Paddy fieldUpland cropfield

City

Area not

considered

rrrlrrrl

Fig..l l-oclLtion ol LNIL s atrcl PSs in the ua{ershecl ol'intere:t

NPSs within a watershcd is tested herc A wttershed

(136° 4′ 40〃-136° 7′ 17″E, 34° 58′ 17〃 -35° 0′ 46′′N)in Konan

city,Shiga prefecturc,Japan,which is a pa■ ofthe M/atershcd

of thc Yasu River,is chosen as a study area.Average pre―

cipitation observed in 1 979-2000 atthe Higashiohnli station

located near the study arca (136° 11.4′E, 35° 3.7′N)isl,449 mnl～ ear.The arca ofthe watershcd is 9 2 km2 con―

taining 49 49ら offorest,1639ら ofpaddy neld,289ら ofupland

crop flcld,and 19.993 of city The elevation in thc study area

rangcs fron■ 128 to 449 m The watcrshed is fragmented into

grid cells at a rcsolution of 50 m × 50111,which is idcntical

to thc rcsolution of the DEM based on elevation data com―

piled by CIeological Survey lnstitutc in Japan.Allthe cells on

paddy nelds,upland crop flclds and cities arc rcgardcd as

LMUs.Thc T― N loads fl・ om other area in the watershcd are

trcated as uncontrollable.Therc arc l,436 LNIUs altogethcr,

breaking do、vn into 602 paddy ncld,lo2uplandcropand732

city units(Fig.4).Notc that one main rivcr(the Yasu River)

runs through the watershcd from southcastto no■ hwest,and

cities and agriculturallands arc distributed along the river.In

addition,flve PSs contribute to T― N loading into the outlct of

thc watcrshed There are no sewagc works and facilitics

raising livcstock.Maps of sOil texture and a distribution of

low length ttifor LMU,calCulatcd using AcrGIS 8 and thc

DEM are sho、vn in Figs.5 and 6,respectively.

2 SPringer

ＰＳ５☆隋一Ｍ朧〓

ｌＡＩＩＩＩ材〕――‐　　ｍ

０

Ｆ　　　　

，ｏｏ

靱魃一一一一〓一一一０

１ rrrlrrrl2,000rn

Fig. 5 Soil texture in LMU in the watershed of interest

River

Sandy

Loamy soil

Area not

considered

I Rivcr

11(nl)

150-500

1500- 1.000

1 1.000-1.500

11.5()0 2.0〔 )()

12.000-2.500

12.500-3.000

13.0()0 3.5()()

3.500-4.000

rヽ rca ll()1

con、ldclcd

|

|

2.0()()111

['ig. 6 Distrihution of flou Icnsthrir er- .r,

隋岬颯階資壼〓選通西□

Ｎ

Иｌｌｌｌ

lrorr a I-NIL l, to the orrtl'lll on the

1,000

1,000

lr

1)ad(1、 'ヽVtttcl Ellvil()n(200ヽ ))7:163-175

Table 1 Parameters related to PS

た OPs.(nl｀ /dtty)(lP、 .(g/nl｀ )(lP、 ,(g力 11い , Catcgor)()「 industry

'l-nble.l r\cceptahlc rrinirnunr untl rruxirnuur rlLlues ol ob-iectirelunctions iu firzzl optinrizution rroclel

2

3

4

90()

2260

1()()

2■ ()

160

600

80()

6()0

■()0

4()0

150

20()

150

1()()

1()()

Cileaning. hrrheratrcl birthltor:sc

Appliarrcc lactor'1'

I\letalloid rnining

Drirrk. lirrauc and

cigarette frLctorl

Plastic llcton,

Solution :i' :'](g/cla1 ) (g/cluy,) (3/(ltlゝ ) (g/(1を 1)) (g/(1を tゝ ) {g/dt｀ ゝ)

A

B

C

D

E

-36 122 1721■ 200

-36122 -11()()()5{)()()

―-37、101 -20_280 0

-36635 -202ヽ () 0

37■ 00 -20300 0

-20_8t13 17_241 200

-20.893 -11()()0 50()0

-21、 115 -tl_165 6560

-21173 -9_958 5_771

-21400 -9160 200

Table 2 Paranreters in conslraints ol lirzzl optinrizltion nroclcl

Z{g/rtal I

Q (rnr/cla1 )

).'t ( I lnt)

)-t'1llnlLL l, ,/ ( nl /(lil.\ )

1.)|'' (g/cla).)

5. ,;tgl.t",,r1-,, . 1- ig/da1,)

L, . t, tg/iltf i

31.7834

661821

()

()00()ヽ 50

()480

1 439 tl()()

281

852.ヽ 52

126616

180 104

Determination of parameter values

Parameter values used in the computations here are listed inTables I ,2 3 and 4. The upper limit of the T-N concentrationin discharged water from PSk, Cpsr. is determined consid-

ering the nationwide applied effluent standard (60 g/m3) and

the local stringent add-on effluent standard (only for

Qpsr) 10 m3/day) which are provided by the Water Pollu-tion Control Law and the prefectural bylaw (Shiga prefec-

ture, unpublished), respectively. Since a certain amount ofdischarged T-N load should, however, be allowed in a

practical sense, the value of Cp5*is determined as 257o ofCps, val ue. Watershed-wi de sel f-pu ri fi cati on coe ffi c ien ts Ld.

ib and 2u" arc assumed based on Ha et al. (1993) and Skop and

Sprensen (1998). The discharge of river, Q, is computed byflow analysis with monthly data in 1990-1998, observedat Yokotabashi and Ishibe, which are located on the

upstream and downstream sides ofthe river portion consid-ered, respectively (Ministry of Architecture (2000), Shiga

Table 3 Admissible vioiations in fuzzy constraints

prefecture (unpublished), Yasu River Land ImprovementDistrict (unpublished)).

Skop and Loaiciga (1998) formulated a linear regression

model to represent the ratio of annual baseflow to annual

total discharge at the olltlet of the subcatchment by theratio of sandy area to total area in the subcatchment withinthe Vejle f,ord catchment (136 km2;, Denmark. Based on

the work, Skop and Sprensen ( 1 998) deduced that if the soiltexture of a land is loamy, 4lVo of load discharged from the

land is transported by direct runoff and 597o of the load bybaseflow, while if sandy, corresponding values are 10 and

90, respectively. A regression model could be developed

fbr the study area in a similar way of Skop and Loaiciga(1998). In the present study, however, the results of Skop

and Loaiciga (1998) are directly adopted to determine 17

and r, in Eq.29.In case of Koshihikari variety of rice, necessary nitrogen

in basal dressing and that in topdressing were found 2.4-3and 3.0-3.5 kg/10a to realize the target yield 560 kg/l0a ina test field in Chiba prefecture (Fukayama 1990). Fukay-ama (1990) also reported that percentages of used nitrogenincluded in basal depressing and that in topdressing were

40-50 and 60-10Vo, respectively. Using the data, unused

nitrogen fertilizer released from a field is assumed 2.62 kg/10a in a year, i.e., 18.0 g/day from a LMU. In addition,supply of T-N through precipitation and falling dust is

estimated as 1,520 kg/km2 per year by observed data col-lected in every two weeks during 1975-1981 and 1990-1992 in Kusatsu city in Shiga prefecture (Kunimatsu and

Sudo 1994). Its daily-averaged value, 10.4 glday, is taken

for atmospheric input of nitrogen into a LMU. It is thus

assumed that Li!' = 28.4 elday (i: t, &) and

L,,,:10.4 g/day (r - 1,...,1.). The values of $ and u;

Srrltttiort ,t;(g/clat )

`五

′F(g/dtty)

dt,,,

( s/dir)')

dr,,

{ g/day')

ど('Ps

(g/111｀ )

r/Cl,s,

(g/ntrl`/(lP、

1

(g/!11｀ )

`/(「Ps.

(g/11デ )

r1Cps.

(g/rnr )

A,B

C,E

D

0

213

()852

()

213

()852

()

315

315

()

()450

()450

()

375

375

0

05()()

05()0

0

375

375

()

250

250

0

250

250

a sPringer

Pa(1(1、 Vヽtttcl En、 iron(200tl)7:163-175

depend on decision― makers' aversion to a bad harvcst.In

this study,the deviation of discharged T― N load at a paddy

型d f iOm弔,ぉ permited up to 30%of弔 1,iCぅ

け=砺 =85g/dayln tlpland crop flcld,a leaching ratc ofnitrogcn,deined as

(amOunt Of leaching)× 100/(amOunt of applied fertilizer),

is 20-30%in」 apan(Tabuchi 1994).Gcncrally nitrogen is

fertilized more than 300 kg/ha for vegetables(TabuChi

1994). SuppOSC that lcaching rate of nitrogen is 30% and

used fertilizer is 300 kg/ha.Then T― N of61.6g/day is cal―

culated to be dischttged frolll a LMl」 of upland crop fleld lt

is used as an assumcd value for ι″′.Lower liinits of cfflucnt

ι“′and Lcare taken as 20%valuc of the unit loading factor

[Site― Spccinc for shiga Prcfcctulc(Soumiya 2000)l fOr thc

cO■esponding land use.

VJues ofづ ,石,L崎 ル i and ost and idr admisttЫ e

violations arc shown in Tables 2 and 3 Thc linear mem―

bership functions for thc fuzzy sets associated with thc

thrcc ottectiVe filnctions are deined with acccptablc

minimum and maximum values ofthcm,as summarized in

Table 4.

Results and discussion

The FOM formulatcd with 4,082 variables and 4,919

constaints is solved by the silllplex mcthod. Here, ave

cxample solutions of palticultt intcrcst(rcfered to as

Solutions A E)are shOWn,Discussions are madc mainly to

demonstratc that(1)reduccd to the ECⅣ l by particular

manipulation of thc pttamctcrs,the FOⅣ I is totally inclu―

sivc and broad― spectrum,(2)the FONIs have the ability to

crcate management alternativcs in more practical and

flexible manncr.

Comparing FOM with ECM

Solution― A is produced for verincation of whcthcrthc FOM

gcncrates the samc noninfcrior solution as that (for

ε2=17,244 and ε3=200)generated by thc ECM in ourlast

study(N/1aeda et al.2006).For this,only crisp constraints are

considcrcd,reducing the values of all the admissible viola―

tions to null,asshownin Table 3.To rcflcctthe ε―constraints

in tlle solution,the equivalence relations ofそ :=そr=~ε2=~17,244 andそ:=イ =83=200 are intЮ duccd.

Thc values ofパ andぞF are COmputcd by the Zimmermann's

method(thcy Cnsurc a sufncicntlyヽ″ide rangc ofそ lto exploκ

its optilnal value).The nOninfe五 or solution obtained,ie,

optimal T― N load anocations to L卜4Us,is indeed in perfcct

agrccment with that obtained frolll the ECM.Another vcri―

■cation tcstお r Solution― B withそ:=イ =-82=~11,000andそ:=そ響=83=5,000 also shows thc same result of

の SPringer

'lable 5 Achier ecl valuessalisliLction lcvel

ol ol.r.jectir e functions ancl lllinirnLlrn

Solution : r (g/tla1') ;:(g/da),) :r(g/dal')

A

B

C

D

E

agreement,thus rcvcaling thatthe ECNIis only a rncmber of

the FONI family

ln Solution― A,thc second and third ottectiVCS(mini

mizations ofそ 2 and 73)are emphasized by giving smaller

valucs toそ :(=てr)andぞg(=そ r),as shown in Table 4,and

conscqucntly λ is decreascd (Table 5) Contrarily, in

Solution― B with rclatively large valucs ofこ」(=ζダ)and【 :

(=そ蟹),minimization ofそl is emphasized as a result of lcss

constrainedそ2 andそ3,lCading to enlargcment of λ.

Producing a variety of alternativcs for T― N load

allocation

Solutions C―E are differcnt three general solutions whcre

nonzcro values are givcn to the admissiblc violations in the

fuzzy constraints(scc Table 3).

Differcnce bet、veen Solutions C and D is in violation of

the constraints rclated to く〔Goa1 3>. It is assumed that

decision― makers aspire to more minimizeそ 3 0r maXilllizc

the 五ce yicld in Solution― D than in Solution― (〕, and

therefoК the viola● ons,″喀and偏 ,are taken smallcr in

Solution― D than in Solution―C(TablC 3).Resultantly,thc

sum of deviations fronl the optimunl standttd of T― N load

in paddy LMUsis rcduced fron1 3,061 g/day in Solution― Cto 2,674g/day in Solution― E),as shown in Table 5.

In Solution―E,thc constaints peltaining to the o● ec―

tives(Eq.36)arc fllZZy sinceィ ≠てμ (′ =1,2,3),and allthc constraints arc relaxed with the same admissible

violations as in Solution― C. 2ヽ point of diffcrence from

Solution― D as wcll as Solution― C is that thc value ofて r is

intentionally lowcred(そ蟹==200 in Tablc 4)by dcCiSiOn―

makcrs,who strongly dcsire to prioritize<Goa1 3>.As a

result,thc value of z3 iS CXtremely rcduced to 100 g/day,

which is a nliniinum in all the examplc solutions,as sho、 vn

in Tablc 5.

Two different solutions to optilllal a1locations of T― Nloads to LA/1Us and PSs,produccd by the ECM (SolutiOn―

A)and FOM(SolutiOn― C),are cOmparcd in Table 6.The

drawback of the ECⅣ l is that the resultant a1lowable loads

are pronc to be identiflcd with the lo、 vcr and upper lilnits of

the constraints even though they arc deterlllined rathcr

SubieCtiVCly with the crisp constraints.This can bc

25417

35371

2995ド

29171

29460

17.244

11.()()()

15.234

15、■97

14896

2()()

50()0

3061

2.674

100

()2C17

()951

()533

0537

0502

P`tdd)ヽVtltCl En、 iroll(2009,7:163-175

Table 6 Summary of optimal T-N load allocations to LMUs and PSs (Solutions A and C)

N PS

Solution-A Solr-rtiorr-C

T-N load (g/clay,) Nurnber ol cells Land use 'l -N loacl (g/tlar ) Nurnber ol cells Land use

180

104

126

1t19

211

284

554

616

425

307

35

23

1

578

1

66

(lit、

Cit、

til)ltlnti cloP icl(1

1)a(1(1、 ` licl(1

Ptt(ltl、 licl(l

Ptt(ld、 licI(l

[lpLin(l clく )p licl(1

tIPltind clol)liCl(1

25■ 17

159

237

104

111

189

284

353

379

616‐Ic)ttll T― N

215

|

516

)

102

280

1

219

t17

1o`td(g/dtt),

Cit、

Cit、

Cit、 .

Li plttn(l clol)liCl(|

Ptid(1、 :licl(1

1).ld(1、 :「 icld

P`tcl(l、 :1う cl(1

Pをt(ld、 flcl(1

ヒipltlnd cl olっ licld

2995ヽTottl T-N loacl (s/cLrr t

PS

Solution.,\ Solutior C

PS number 1-N conccntitttion{3/nl｀ ) T-N loacl (g/da1 ) PS number I― N conccntrtui()n t8/nlr) T-N loacl (g/da1')

I

2

3r-ll5t'lotal 'l'-N load (g/cla1 )

150

739

15 ()

100

10()

135

166t14

150

240

160

17244

133

653

133

884

884

119

11、 748

133

212

141

15234

I

|

)

TottJ T― N!oad(g/day)

recognized in quite limited T-N load values for Solution-A(Table 6). On the contrary, in Solution-C, three out of fourdifferent T-N load values given to paddy LMUs, 18.9, 35.3,37.9 glday, are not explicitly given values as lower orupper limits. This tendency can also be seen for LMUs ofother land use types and for the result from Solution-D as

well. This suggests that subjectivity or intentionality indetermination of the input parameters is likely to success-

fully be removed in optimal solutions of the FOM.The FOM includes no weights or parameters associated

with objective functions, while the ECM contains 12 &nd s3.

Preference of decision-makers to each objective is incor-porated in the FOM through proper deflnition of themembership functions. Normally the values of z'i and z7'

can be determined in a logical way, but alternately be

directly specilied by decision-makers in accordance withtheir attitude toward accomplishments of the objectives,like in Solutions A, B, and E,. Thus, the fuzzy optimizationmodel developed can describe decision-makers' aspirationfor goals in a more flexible manner than the Lorf\zzye-constraint model.

The distribution map of T-N loads allocated to LMUs inSolution-C is illustrated in Fig. 7. From a comparison ofFigs. 4 and l, it can readily be seen that a pattern of load

allocation is significantly contingent on that of land use

practice. The flow length (Fig.6) has also a strong effecton the spatial variation of the allocated load amount, due to

an assumption that effluent loads decay exponentially withthe increasing flow length. Additionally, the loads allocated

to sandy LMUs are likely to be increased. This is because

if the soil texture of a LMU is sandy, then the baseflow

with the greater purification efflciency represented by 2D

accounts for 90Vo of the total discharge from that LMU.

Conclusions

An intra-watershed management model based on fuzzylinear programing has been presented which is capable ofoptimally allocating total allowable T-N loads to PSs and

NPSs in a watershed. The method presented takes advan-

tage of the GIS combined with the DEM, in order to deal

with NPSs on a grid basis. Effluent limitation standard fortotal T-N load at the outlet of watershed is imposed as aprime constraint for river water quality conservation. T-Ntranspol'ts along with direct runoff, baseflow and river floware defined to make them fit the grid-based optimization.Vagueness in setting up parameters in constraints is

2 SPringer

Padd、 ヽヽ「t、tcr En、 il on(2009)7:163-175

Fig. 7 Optimization result of T-N loads allocated to LMUs (Solu-tion-C)

appropriately expressed in the framework of fuzzy opti-mization. The demonstrative application of the model tothe real watershed reveals that our last nonfuzzy model,

ECM, is placed as a particular case of the FOM developed

in this study. It is also demonstrated that the FOM can

provide optimal or satisfactory solutions to the T-N load

allocation problem with more appropriate representation

of decision-makers' attitude to the accomplishment ofmanagement goals. Subject to the optimal solutions,

overloading pollutant dischargers may be required to take

effective actions to reduce their current loads up to their

own allocated limits. Reduction of use of nitrogen fertil-izer, cropland retirement in farmland area, or enhancement

of wastewater treatment capacity in industrial plants and

sewage works could effectively be conducted if the allo-

cation limits are used as the management targets.

Further studies are necessary for making the proposed

model less sensitive to the uncertainty of the input param-

eters like river discharge and self-puriflcation coefflcient. Astochastic approach could be adopted to treat such param-

eter uncertainty in the fizzy optimization model.

Acknowledgments The authors are grateful to the Department ofLake Biwa and the Environment, Shiga Prefectural Government,

japan. lor ol'ltling the CilS source clntit

c;ualitr' clulu iu the stutlr alca.

ancl the l)S-relittecl uater'

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