Research Article Mathematical Modeling for Water Quality ...downloads.hindawi.com/journals/jam/2013/731568.pdf · Research Article Mathematical Modeling for Water Quality Management

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 731568 14 pageshttpdxdoiorg1011552013731568

Research ArticleMathematical Modeling for Water Quality Managementunder Interval and Fuzzy Uncertainties

J Liu Y P Li and G H Huang

MOE Key Laboratory of Regional Energy Systems Optimization Resources and Environmental Research AcademyNorth China Electric Power University Beijing 102206 China

Correspondence should be addressed to Y P Li yongpingli33gmailcom

Received 4 October 2013 Accepted 29 November 2013

Academic Editor X S Qin

Copyright copy 2013 J Liu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this study an interval fuzzy credibility-constrained programming (IFCP) method is developed for river water qualitymanagement IFCP is derived from incorporating techniques of fuzzy credibility-constrained programming (FCP) and interval-parameter programming (IPP) within a general optimization framework IFCP is capable of tackling uncertainties presented asinterval numbers and possibility distributions as well as analyzing the reliability of satisfying (or the risk of violating) systemrsquosconstraints A real-world case for water quality management planning of the Xiangxi River in the Three Gorges Reservoir Region(which faces severe water quality problems due to pollution from point and nonpoint sources) is then conducted for demonstratingthe applicability of the developedmethodThe results demonstrate that high biological oxygendemand (BOD)discharge is observedat the Baishahe chemical plant and Gufu wastewater treatment plant For nonpoint sources crop farming generates large amountsof total phosphorus (TP) and total nitrogen (TN) The results are helpful for managers in not only making decisions of effluentdischarges from point and nonpoint sources but also gaining insight into the tradeoff between system benefit and environmentalrequirement

1 Introduction

Water is one of the most essential constituents for the humanlife which is crucial to various socioeconomic issues suchas industrial production agricultural activity environmen-tal protection and regional sustainability In recent yearsespecially in China degradation of water quality due topoint and nonpoint source pollutions has become one ofthe most pressing environmental concerns According tothe 2010 Report on Water Environmental Quality of Chinaapproximately 338 of the monitored river water (204 riverswith 409 monitoring stations) is in the worst two categoriesof water quality classification system (ie no longer fishableand of questionable agricultural value) around 538 of theassessed lakes and reservoirs are subject to different degrees ofeutrophication [1] Under such a circumstance water qualitymanagement is an essential task for preserving valuablewater resources and facilitating sustainable socioeconomicdevelopment in watershed systems [2] In fact water qualityplanning efforts are complicated with a variety of uncertain-ties which may be derived from the random characteristics

of natural processes (ie precipitation and climate change)and stream conditions (ie stream flow water supply andpointnonpoint source pollution) the errors in estimatedmodeling parameters and the vagueness of system objectivesand constraints In general the system objectives are oftenassociated with a number of socioeconomic and ecologicalfactors such as economic return environmental protectionand ecological sustainability while the constraints are relatedto pollutant discharges soil losses resources availabilitiesenvironmental requirements and policy regulations More-over these uncertainties may be further amplified by notonly interactions among various uncertain and dynamicimpact factors but also their associations with economicimplications of violated environmental requirements [3]

Fuzzy mathematic programming (FMP) based on fuzzysets theory can facilitate the analysis of system associatedwith uncertainties being derived from vagueness or fuzzi-ness [4] FMP method is suitable for situation when theuncertainties cannot be expressed as probability densityfunctions (PDFs) such that adoption of fuzzy membership

2 Journal of Applied Mathematics

functions becomes an attractive alternative [5] Previously anumber of FMP methods were developed for water qualitymanagement [5ndash15] For example Julien [6] investigated theapplication of a fuzzy possiblistic programming to addressimprecise parameters which were represented by possibilitydistributions in water quality decision-making problemsMujumdar and Sasikumar [8] formulated a fuzzy flexibleoptimization model for dealing with the systemrsquos fuzzy goaland constraints in a water quality management problemNie et al [12] proposed a fuzzy robust optimization modelfor water quality management of an agricultural systemto deal with uncertainties expressed as fuzzy membershipfunctions in both left- and right-hand-side coefficients (ofthe modelrsquos constraints) Maeda et al [13] employed a fuzzyflexible optimization model which involved fuzzy set theoryto express vagueness in constraints and objectives in riverwater quality management problems Liu et al [15] developeda two-stage fuzzy robust programming model for waterquality management to address fuzzy parameters which wererepresented by possibility distributions in the left- and right-hand sides of the constraints

Generally FMP methods can be classified into threecategories in view of the forms of uncertainties (i) fuzzyflexible programming (ii) robust programming and (iii)fuzzy possibilistic programming In detail fuzzy flexibleprogramming can deal with decision problems under fuzzygoal and constraints however it has difficulties in tacklingambiguous coefficients of the objective function and con-straints Robust programming improves upon fuzzy flexibleprogramming by allowing fuzzy parameters in the constraintsto be represented by possibility distributions However themain limitations of this method remain in its difficultiesin tackling uncertainties in a nonfuzzy decision space Infuzzy possibilistic programming fuzzy parameters that areregarded as possibility distributions are introduced into themodeling frameworks It can handle ambiguous coefficientsin the left- and right-hand sides of the constraints and in theobjective function

Fuzzy credibility-constrained programming (FCP) is acomputationally efficient fuzzy possibilistic programmingapproach that relies on mathematical concepts (ie theexpected value of a fuzzy number and the credibilitymeasure)and can support different kinds of fuzzy members such astriangular and trapezoidal forms as well as enabling thedecision maker to satisfy some constraints in at least somegiven confidence levels [16] When the credibility value of afuzzy event reaches 1 the fuzzy event will certainly occurwhen the credibility value of a fuzzy event reaches 0 thefuzzy event will not occur For example with respect towater quality management if the allowable total phosphorus(TP) discharges are 70 76 and 82 kgday and the amountof actual discharge may be 78 kgday then the credibilitydegree of the event wherein the total phosphorus dischargecan satisfy the riverrsquos self-purification capacity would be033 the credibility degree of its complement event (waterpollution occur) would be 067 No feature of fuzzy sets wouldbe missing by using credibility measure [16 17] Howeverthe main limitation of FCP lies within its deterministiccoefficients for the objective function leading to potential

losses of valuable uncertain information besides whenmanyuncertain parameters are expressed as fuzzy sets interactionsamong these uncertainties may lead to serious complexitiesparticularly for large-scale practical problems [18] In factin water quality management problems uncertainty is aninherent component of any economic analysis particularlythose (eg effluent trading programs) associated with envi-ronmental policy and project appraisal [19] For exampleone major characteristic of nonpoint source pollution thatdiffers from point source pollution is imperfect knowledgeabout pollutant loadings the crop productivity and costbenefit coefficients are easier to be presented as intervals thanby membership functions Interval-parameter programming(IPP) is an alternative for handling uncertainties in themodelrsquos left- andor right-hand sides as well as those that can-not be quantified as membership or distribution functionssince interval numbers are acceptable as its uncertain inputs[20]

Therefore the objective of this study is to develop aninexact fuzzy credibility-constrained programming (IFCP)method for water quality management through couplingfuzzy credibility-constrained programming (FCP) withinterval-parameter programming (IPP) The main advantageof IFCP is that it can effectively handle uncertain parametersexpressed as both fuzzy sets and interval values in theobjective function and constraints IFCP would not leadto serious complexities in its solution process and it isapplicable to large-scale practical problems Then thedeveloped IFCP method is applied to a real-world case ofwater quality management of the Xiangxi River which facessevere water quality problems due to point and nonpointsource pollution The results obtained can help decisionmakers to generate alternatives for industrial productionscale water supply cropped area livestock husbandry sizeand manurefertilizer application rate with consideration ofriver water quality management

Thepaperwill be organized as follows Section 2 describesthe development process of the IFCP Section 3 providesa case study of river water quality management Section 4presents result analysis and discussion Section 5 draws someconclusions and extensions

2 Methodology

When coefficients in the constraints are ambiguous and canbe expressed as possibility distributions the problem can beformulated as a fuzzy credibility-constrained programming(FCP) model as follows

Max119891 =119899

sum119895=1

119888119895119909119895 (1a)

subject to

Cr

119899

sum119895=1

119886119894119895119909119895le 119894

ge 120582119894 (1b)

119909119895ge 0 119894 = 1 2 119899 (1c)

Journal of Applied Mathematics 3

where 119909 = (1199091 1199092 119909

119899) is a vector of nonfuzzy deci-

sion variables 119888119895are benefit coefficients 119886

119894119895are technical

coefficients and 119894are right-hand-side coefficients Some or

all of these coefficients can be fuzzy numbers Cr is thecredibility measure which is firstly proposed and was widelyused in many research areas [16] Let 120585 be a fuzzy variablewith membership function 120583 and let 119903 be real numbers Thecredibility measure can be defined as follows [17]

Cr 120585 le 119903 =1

2(sup119909le119903

120583 (119909) + 1 minus sup119909gt119903

120583 (119909)) (2)

Noteworthy since Pos120585 le 119903 = sup119909le119903

120583(119909) and Nec120585 le119903 = 1 minus sup

119909gt119903120583(119909) the credibility measure can be defined

as follows

Cr 120585 le 119903 =1

2(Pos 120585 le 119903 +Nec 120585 le 119903) (3)

Similar to the probability measure

Cr 120585 le 119903 + Cr 120585 gt 119903 = 1 (4)

Consider a triangular fuzzy variable since it is the mostpopular possibility distribution the fuzzy variable 120585 fullydetermined by the triplet (119905 119905 119905) of crisp numbers with 119905 lt119905 lt 119905 whose membership function is given by

120583 (119903) =

119903 minus 119905

119905 minus 119905 if 119905 le 119903 le 119905

119903 minus 119905

119905 minus 119905 if 119905 le 119903 le 119905

0 otherwise

(5)

Based on this membership function credibility of 119903 le 120585 canbe expressed by

Cr 119903 le 120585 =

1 if 119903 le 119905

2119905 minus 119905 minus 119903

2 (119905 minus 119905) if 119905 le 119903 le 119905

119903 minus 119905

2 (119905 minus 119905) if 119905 le 119903 le 119905

0 if 119903 ge 119905

(6)

The inverse function of the credibilitymeasure is Crminus1(120582) = 119903when Cr(119903 le 120585) = 120582 Normally it is assumed that a significantcredibility level should be greater than 05 Therefore (6) canbe written as

Cr 119903 le 120585 =2119905 minus 119905 minus 119903

2 (119905 minus 119905)ge 120582 (7)

Then (7) can be transformed into a deterministic constraintas follows

119903 le 119905 + (1 minus 2120582) (119905 minus 119905) (8)

The fuzzy credibility-constrained programming (FCP)modelcan be formulated as follows

Max119891 =119899

sum119895=1

119888119895119909119895 (9a)

subject to

119899

sum119895=1

119886119894119895119909119895le 119887119894+ (1 minus 2120582

119894) (119887119894minus 119887119894) (9b)

119909119895ge 0 119894 = 1 2 119899 (9c)

Obviously model (9a) (9b) and (9c) can effectively dealwith uncertainties in the right-hand sides presented asfuzzy sets when coefficients in the left-hand sides and inthe objective function are deterministic However in real-world optimization problems uncertainties may exist inboth left- and right-hand sides (of the constraints) as wellas objective-function coefficients moreover the quality ofinformation that can be obtained is mostly not satisfactoryenough to be presented as fuzzy membership functions[2] For example economic return pollutant discharge andresources availability are easier to be expressed as intervalsthan membership functions [21] Since interval-parameterprogramming (IPP) is useful for addressing uncertaintiesexpressed as interval values in modeling parameters it canbe integrated into the FCP model to deal with uncertaintiespresented in fuzzy and interval formats Then an intervalfuzzy credibility-constrained programming (IFCP) can beformulated as follows

Max119891plusmn =119899

sum119895=1

119888plusmn119895119909plusmn119895 (10a)

subject to

119899

sum119895=1

119886plusmn119894119895119909plusmn119895le 119887119894+ (1 minus 2120582plusmn

119894) (119887119894minus 119887119894) (10b)

119909plusmn119895ge 0 119894 = 1 2 119899 (10c)

where the ldquominusrdquo and ldquo+rdquo superscripts represent the lower- andupper-bounds of interval parametersvariables respectivelyThen a two-step solution method is proposed for facilitatingcomputations of the IFCP model The first submodel can beformulated as follows

Max119891+ =1198961

sum119895=1

119888+119895119909+119895+119899

sum119895=1198961+1

119888+119895119909minus119895 (11a)


subject to

1198961

sum119895=1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816minus

Sign (119886minus119894119895) 119909+119895

+119899

sum119895=1198961+1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816+

Sign (119886+119894119895) 119909minus119895

le 119887119894+ (1 minus 2120582minus

119894) (119887119894minus 119887119894)

forall119894 = 1 2 119898

(11b)

119909+119895ge 0 119895 = 1 2 119896

1 (11c)

119909minus119895ge 0 119895 = 119896

1+ 1 119899 (11d)

where 119888+119895(119895 = 1 2 119896

1) gt 0 119888+

119895(119895 = 119896

1+1 1198962+1 119899) lt

0 Sign(119886plusmn119894119895) = minus1 when 119886plusmn

119894119895lt 0 Sign(119886plusmn

119894119895) = 1 when 119886plusmn

119894119895gt

0 120582minus119894

is the lower bound of the credibility level value Theoptimal solutions of the first submodel would be 119909+

119895opt (119895 =1 2 119896

1) and 119909minus

119895opt (119895 = 1198961+1 1198962+1 119899) In the second

step the submodel corresponding to 119891minus can be formulated

Max119891minus =1198961

sum119895=1

119888minus119895119909minus119895+119899

sum119895=1198961+1

119888minus119895119909+119895 (12a)

subject to

1198961

sum119895=1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816+

Sign (119886+119894119895) 119909minus119895+119899

sum119895=1198961+1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816minus

Sign (119886minus119894119895) 119909+119895

le 119887119894+ (1 minus 2120582+

119894) (119887119894minus 119887119894) forall119894 = 1 2 119898

(12b)

0 le 119909minus119895le 119909+119895opt 119895 = 1 2 119896

1 (12c)

119909+119895ge 119909minus119895opt 119895 = 119896

1+ 1 119899 (12d)

The optimal solutions of model (11a) (11b) (11c) and (11d)would be 119909minus

119895opt (119895 = 1 2 1198961) and 119909+

119895opt (119895 = 1198961+ 1 1198962+

1 119899) can be obtained Through integrating the solutionsof the two submodels the solution for the objective-functionvalue and decision variables can be obtained as follows

119891plusmnopt = [119891minusopt 119891+

opt] (13a)

119909plusmn119895opt = [119909minus

119895opt 119909+

119895opt] (13b)

3 Case Study

TheXiangxi River (which ranges in longitude from 110∘251015840 to111∘061015840 E and in latitude from 30∘571015840 to 31∘341015840N) is locatedat 40 km upstream of the Three Gorges Reservoir [22] It

is 94 km long with a catchment area of 3099 km2 and itselevation generally ranges from 154m to 3000m It is locatedin the subtropical continental monsoon climate zone withan annual temperature of 156∘C (from 1961 to 2004) andthe long-term annual mean runoff depth of 688mm [23]Moreover it is one of the rainiest centers in the west ofHubei province with an average annual precipitation rangesfrom 900mm to 1200mmThe temporal precipitation of thisbasin is uneven which varies largely among different seasonsFor example more than 41 precipitation occurs in June toAugust and the rainfall in the spring autumn and winterseasons occupy 28 26 and 5 of the total precipitationper year respectively There are plenty of mineral resources(ie phosphate ore coal pyrite and granite) where thereserve of phosphorite is among the top three in Chinawhich reaches 357 million 119905 (ie tonne) [1] Relying on theseadvantages the number of phosphorus mining companiesand related chemical plants are increasing along the banksof the Xiangxi River Besides multiple crops such as ricemaize wheat citrus tea potato and vegetable are cultivatedin the catchment since the land-use patterns are diverse andthe tillable area land is approximately 2945 km2 In additionpig ox sheep and domestic fowl are the main live stocksin animal husbandry The main pattern is scattered livestockbreeding instead of large-scale standardized breeding

Currently water quality problems due to point andnonpoint source pollution discharges becomemore andmorechallenging in this catchment Main point sources includefive chemical plants (ie GF BSH PYK LCP and XJLY) sixphosphorus mining companies (ie XL XH XC GP JJWand SJS) and four wastewater treatment plants (WTPs) (ieGufu Nanyang Gaoyang and Xiakou) while four agricul-tural zones (AZ1 to AZ4) are the main nonpoint sourcesdue to the application of manurefertilizer These point andnonpoint sources scatter along a length of about 51 km riverstretch which is segmented into five reaches and the reachesare marked as I to V The main water quality problemsinclude (i) the immoderate discharge of high-concentrationphosphorus-containing wastewater and industrial soil wastes(ie chemical wastes slags and tailings) far exceed whatcan be decomposed by self-purification (according to thefield investigation 2389 119905 of phosphorus enters downstreamof the Xiangxi River) (ii) high potential for generating soilerosion and surface runoff due to the special geography andheavy rainfall (ie the average erosionmodulus reaches 6488t(km2sdota) in Xiangshan County in this catchment) (iii) largeamounts of nutrient pollutants (in terms of phosphorus andnitrogen) in livestock wastewater and wastes (from pig oxsheep and domestic fowl breeding) are drained into the riverby direct discharge or in rainfall In such a circumstancedecision makers should seek to develop a sound pollutioncontrol plan to ameliorate the current situation of the waterenvironment since it is infeasible and technical impossible toensure zero emission of pollutants

In this study the planning horizon is one year Moreoversince some crops should be grown in dry season while someother crops should be cultivated in wet season two periodsare chosen to cover the planning horizon The first period


is from June to October (ie dry season) and the secondperiod is from November to May of the next year (ie wetseason) The objective is to maximize the net system benefitsubject to the environmental requirements under uncertaintyover the planning horizon Policies in terms of the relatedhuman activities (ie industrial municipal and agriculturalactivities) and the pollutant discharges (from fifteen pointand four nonpoint sources) are critical for ensuring a max-imum system benefit and a safe water quality [24] Based onfield investigations and related literatures biological oxygendemand (BOD) total nitrogen (TN) and total phosphorus(TP) are selected as water quality indicators [1 25] Todevelop the local economy in a sustainable manner pollutantdischarge should be controlled by setting the thresholds forTP TN and BOD discharge in each reach [26] Howeverhuman-induced imprecision in acquiring these thresholds(ie lack of available data and biased judgment)make it morecomplicate

On the other hand uncertainties in the study systeminclude the following (a) cost of wastewater treatmentmanure and fertilizer purchase are associated with manyuncertain factors which are expressed as interval numbers(eg an interval of [30 35] RMByent is denoted as cost ofmanure purchase of AZ1 in dry season) (b) the BOD and TPtreatment efficiencies of wastewater in WTPs and chemicalplants are related to operating conditions of the treatmentfacilities which cannot be obtained as deterministic numbers(eg an interval of [089 092] is denoted as treatmentefficiencies of wastewater in Xiakou WTP) (c) nonpointsource losses of nitrogen and phosphorus from agriculturalzones fluctuate dynamically due to variability in soil erosion(corresponding to solid-phase nitrogen) and surface runoff(corresponding to dissolved nitrogen) (eg average soilloss from AZ1 planted with citrus in dry season would be[2049 2282] tha) and runoff from AZ1 planted with citrusin dry season would be [7807 9650]mm) (d) the amountof fertilizer and manure applications may vary with the soilfertility to meet the nutrient demands of each crop (ienitrogen and phosphorus) (e) energy and digestible proteindemands of human and animals are determined by cropsrsquoyield (eg yield of citrus planted in AZ1 during dry seasonwould be [103 126] tha) Therefore based on the IFCPmethod developed in Section 2 the study problem can beformulated as

Max119891plusmn =5

sum119894=1

2

sum119905=1

119871119905sdot BCplusmn119894119905sdot PLCplusmn119894119905+4

sum119904=1

2

sum119905=1

119871119905sdot BWplusmn119904119905sdotQWplusmn119904119905

+6

sum119901=1

2

sum119905=1

119871119905sdot BPplusmn119901119905sdot PLMplusmn

119901119905

+4

sum119903=1

BLplusmn119903sdotNLplusmn119903+4

sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot BAplusmn119895119896119905

sdot PAplusmn119895119896119905

minus4

sum119895=1

9

sum119896=1

2

sum119905=1

CMplusmn119895119905sdot AMplusmn119895119896119905

minus4

sum119895=1

9

sum119896=1

2

sum119905=1

CFplusmn119895119905sdot AFplusmn119895119896119905

minus5

sum119894=1

2

sum119905=1

119871119905sdot PLCplusmn119894119905sdotWCplusmn119894119905sdot CCplusmn119894119905

minus4

sum119904=1

2

sum119905=1

119871119905sdotQWplusmn119904119905sdot GTplusmn119904119905sdot CTplusmn119904119905

minus4

sum119895=1

9

sum119896=1

2

sum119905=1

PAplusmn119895119896119905

sdot IQplusmn119895119896119905

sdotWSPplusmn119905

minus5

sum119894=1

2

sum119905=1

119871119905sdot PLCplusmn119894119905sdot FWplusmn119894119905sdotWSPplusmn

119905

minus4

sum119904=1

2

sum119905=1

119871119905sdotQWplusmn119904119905sdotWSPplusmn

119905 (14a)

subject to

(1) wastewater treatment capacity constraints

QWplusmn119904119905sdot GTplusmn119904119905le TPCplusmn

119904119905(14b)

WCplusmn119894119905sdot PLCplusmn119894119905le TPDplusmn

119894119905 (14c)

(2) BOD discharge constraints

Cr PLCplusmn119894119905sdotWCplusmn119894119905sdot ICplusmn119894119905sdot (1 minus 120578plusmnBOD119894119905) le ABCplusmn

119894119905 ge 120582plusmn (14d)

Cr QWplusmn119904119905sdot GTplusmn119904119905sdot BMplusmn119904119905sdot (1 minus 1205781015840plusmnBOD119904119905) le ABWplusmn

119904119905 ge 120582plusmn

(14e)

(3) nitrogen discharge constraints

Cr

(119871119905sdot4

sum119903=1

AMLplusmn119903119905sdotNLplusmn119903+ 119871119905

sdotAMHplusmn119905sdot RPplusmn119905minus4

sum119895=1

9

sum119896=1

AMplusmn119895119896119905)

sdotMSplusmn119905sdot 120576plusmnNM + 119871

119905sdot RPplusmn119905sdot ACWplusmn

119905sdot DNRplusmn

119905

le ANLplusmn119905

ge 120582plusmn

(14f)

9

sum119896=1

(NSplusmn119895119896sdot SLplusmn119895119896119905

+ RFplusmn119895119896119905

sdot DNplusmn119895119896119905) sdot PAplusmn

119895119896119905

le MNLplusmn119895119905sdot TAplusmn119895119905

(14g)


(4) phosphorus discharge constraints

Cr PLCplusmn119894119905sdot [WCplusmn

119894119905sdot PCRplusmn

119894119905sdot (1 minus 120578plusmn

119879119875119894119905)

+ASCplusmn119894119905sdotSLRplusmn119894119905sdot PSCplusmn119894119905] le APCplusmn

119894119905 ge 120582plusmn

(14h)

Cr

(119871119905sdot4

sum119903=1

AMLplusmn119903119905sdotNLplusmn119903+ 119871119905sdot AMHplusmn

119905

sdotRPplusmn119905minus4

sum119895=1

9

sum119896=1

AMplusmn119895119896119905)

sdotMSplusmn119905sdot 120576plusmn119875119872

+ 119871119905sdot RPplusmn119905sdot ACWplusmn

119905sdot DPRplusmn

119905le APLplusmn

119905

ge 120582plusmn

(14i)

Cr QWplusmn119904119905sdot GTplusmn119904119905sdot PCMplusmn


119879119875119904119905) le APWplusmn

119904119905 ge 120582plusmn

(14j)

Cr PLMplusmn119901119905sdotWPMplusmn

119901119905sdotMWCplusmn


119901119905)

+PLMplusmn119901119905sdot ASMplusmn

119901119905sdot PCSplusmn119901119905sdot SLWplusmn

119901119905le APMplusmn

119901119905 ge 120582plusmn

(14k)

9

sum119896=1

(PSplusmn119895119896sdot SLplusmn119895119896119905

+ RFplusmn119895119896119905

sdot DPplusmn119895119896119905) sdot PAplusmn

119895119896119905le MPLplusmn

119895119905sdot TAplusmn119895119905

(14l)

(5) soil loss constraints

9

sum119896=1

SLplusmn119895119896119905

sdot PAplusmn119895119896119905

le MSLplusmn119895119905sdot TAplusmn119895119905 (14m)

(6) fertilizer and manure constraints

(1 minusNVFplusmn119905) sdot 120576plusmnNF sdot AF

plusmn

119895119896119905+ (1 minusNVMplusmn

119905)

sdot 120576plusmnNM sdot AMplusmn119895119896119905

minusNRplusmn119895119896119905

sdot PAplusmn119895119896119905

ge 0(14n)

120576plusmnPF sdot AFplusmn

119895119896119905+ 120576plusmnPM sdot AMplusmn

119895119896119905minus PRplusmn119895119896119905

sdot PAplusmn119895119896119905

ge 0 (14o)

9

sum119896=1

(120576plusmnNF sdot AFplusmn

119895119896119905+ 120576plusmnNM sdot AMplusmn

119895119896119905minusNRplusmn119895119896119905

sdot PAplusmn119895119896119905)


(14p)

9

sum119896=1

(120576plusmnPF sdot AFplusmn

119895119896119905+ 120576plusmn119875119872

sdot AMplusmn119895119896119905

minus PRplusmn119895119896119905

sdot PAplusmn119895119896119905)

le MPLplusmn119895119905sdot TAplusmn119895119905

(14q)

119871119905sdot4

sum119903=1


119905sdot RPplusmn119905minus4

sum119895=1

9

sum119896=1

AMplusmn119895119896119905

ge 0

(14r)

(7) energy and digestible protein constraints

4

sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905

sdotNECplusmn119896minus4

sum119903=1

ERLplusmn119903sdotNLplusmn119903

minus2

sum119905=1

ERHplusmn119905sdot RPplusmn119905ge 0

(14s)

4

sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905

sdot DPCplusmn119896

minus4

sum119903=1

DRLplusmn119903sdotNLplusmn119903minus2

sum119905=1

DRHplusmn119905sdot RPplusmn119905ge 0

(14t)

(8) production scale constraints

PLC119894min le PLCplusmn

119894119905le PLC

119894max (14u)

NL119903min le NLplusmn

119903le NL119903max (14v)

QW119904min le QWplusmn

119904119905le QW

119904max (14w)

PLM119901min le PLMplusmn

119901119905le PLM

119901max (14x)

(9) total yield of crops

4

sum119895=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905

ge MCYplusmn119896119905 (14y)

(10) planning area constraints

9

sum119896=1

PAplusmn119895119896119905

le TAplusmn119895119905 (14z)

(11) nonnegative constraints

PLCplusmn119894119905PAplusmn119895119896119905NLplusmn119903QWplusmn119904119905PLMplusmn

119901119905AMplusmn119895119896119905AFplusmn119895119896119905

ge 0(14aa)

where 119894 chemical plant 119894 = 1 Gufu (GF) 119894 = 2 Baishahe(BSH) 119894 = 3 Pingyikou (PYK) 119894 = 4 Liucaopo (LCP) 119894 = 5Xiangjinlianying (XJLY) 119895 agricultural zone 119895 = 1 4119896 main crop 119896 = 1 citrus 119896 = 2 tea 119896 = 3 wheat 119896 = 4potato 119896 = 5 rapeseed 119896 = 6 alpine rice 119896 = 7 secondrice 119896 = 8 maize 119896 = 9 vegetables 119901 phosphorus miningcompany 119901 = 1 Xinglong (XL) 119901 = 2 Xinghe (XH) 119901 = 3


Table 1 Net benefits from each production

Period119905 = 1 119905 = 2

Net benefits from chemical plant (RMByent)GF [7186 8345] [74375 8762]BSH [12915 14997] [13367 15748]PYK [7430 8628] [7690 9059]LCP [13242 15375] [13706 16144]XJLY [15244 17700] [15787 18585]

Net benefits from water supply (RMByenm3)Gufu [391 421] [434 477]Nanyang [290 312] [322 354]Gaoyang [353 380] [392 431]Xiakou [321 346] [357 393]

Net benefits from phosphorus mining company (RMByent)XL (Xinglong 119901 = 1) [150 173] [147 180]XH (Xinghe 119901 = 2) [126 145] [130 156]XC (Xingchang 119901 = 3) [135 155] [135 162]GP (Geping 119901 = 4) [144 166] [150 180]JJW (Jiangjiawan 119901 = 5) [137 158] [141 169]SJS (Shenjiashan 119901 = 6) [140 164] [145 175]

Xingchang (XC) 119901 = 4 Geping (GP) 119901 = 5 Jiangjiawan(JJW) 119901 = 6 Shenjiashan (SJS) 119903 livestock 119903 = 1 pig119903 = 2 ox 119903 = 3 sheep 119903 = 4 domestic fowls 119904 town119904 = 1 Gufu 119904 = 2 Nanyang 119904 = 3 Gaoyang 119904 = 4Xiakou 119905 time period 119905 = 1 dry season 119905 = 2 wet season119871119905 length of period (day) BCplusmn

119894119905 net benefit from chemical

plant 119894 during period 119905 (RMByent) PLCplusmn119894119905 production level

of chemical plant 119894 during period 119905 (tday) BWplusmn119904119905 benefit

from water supply to municipal uses (RMByenm3) QWplusmn119904119905

quantity of water supply to town 119904 in period 119905 (m3day)BPplusmn119901119905 average benefit for per unit phosphate ore (RMByent)

PLMplusmn119901119905 production level of phosphorus mining company 119901

during period 119905 (tday) BLplusmn119903 average benefit from livestock

119903 (RMByenunit) NLplusmn119903 number of livestock 119903 in the study

area (unit) CYplusmn119895119896119905 yield of crop 119896 planted in agricultural

zone 119895 during period 119905 (tha) BAplusmn119895119896119905 average benefit of

agricultural product (RMByent) PAplusmn119895119896119905 planting area of crop

119896 in agricultural zone 119895 during period 119905 (ha) CMplusmn119895119905 cost

of manure collectiondisposal in agricultural zone 119895 duringperiod 119905 (RMByent) CFplusmn

119895119905 cost of purchasing fertilizer in

agricultural zone 119895 during period 119905 (RMByent) AMplusmn119895119896119905 amount

of manure applied to agricultural zone 119895 with crop 119896 duringperiod 119905 (t) AFplusmn

119895119896119905 amount of fertilizer applied to agricultural

zone 119895 with crop 119896 during period 119905 (t) WCplusmn119894119905 wastewater

generation rate of chemical plant 119894 during period 119905 (m3t)CCplusmn119894119905 wastewater treatment cost of chemical plant 119894 during

period 119905 (RMByenm3) GTplusmn119904119905 wastewater discharge amount at

town 119904 during period 119905 (m3m3) CTplusmn119904119905 cost of municipal

wastewater treatment (RMByenm3) IQplusmn119895119896119905 irrigation quota for

crop 119896 in zone 119895 during period 119905 (m3ha) WSPplusmn119905 water

supply price (RMByenm3) FWplusmn119894119905 water consumption of per

unit production of chemical plant 119894 during period 119905 (m3t)ICplusmn119894119905 BOD concentration of raw wastewater from chemical

plant 119894 in period 119905 (kgm3) 120578plusmnBOD119894119905 BOD treatment efficiencyin chemical plant 119894 during period 119905 () ABCplusmn

119894119905 allowable

BOD discharge for chemical plant 119894 in period 119905 (kgday)BMplusmn119904119905 BOD concentration of municipal wastewater at town 119904

during period 119905 (kgm3) 1205781015840plusmnBOD119904119905 BOD treatment efficiencyof WTPs at town 119904 during period 119905 () ABWplusmn


BOD discharge for WTPs at town 119904 during period 119905 (kgday)NSplusmn119895119896 nitrogen content of soil in agricultural zone 119895 planted

with crop 119896 () SLplusmn119895119896119905 average soil loss from agricultural

zone 119895 planted with crop 119896 in period 119905 (tha) RFplusmn119895119896119905 runoff

from agricultural zone 119895 with crop 119896 in period 119905 (mm)DNplusmn119895119896119905 dissolved nitrogen concentration in the runoff from

agricultural zone 119895 planted with crop 119896 in period 119905 (mgL)MNLplusmn119895119905 maximum allowable nitrogen loss in agricultural

zone 119895 during period 119905 (tha) TAplusmn119895119905 tillable area of agricul-

tural zone 119895 during period 119905 (ha) AMLplusmn119903119905 amount of manure

generated by livestock 119903 [t(unitsdotdayminus1)] AMHplusmn119905 amount

of manure generated by humans [t(unitsdotdayminus1)] RPplusmn119905 total

rural population in the study area during period 119905 (unit)MSplusmn119905 manure loss rate in period 119905 () ACWplusmn

119905 wastewater

generation of per capita water consumption during period 119905[m3(unitsdotdayminus1)] DNRplusmn

0119905 dissolved nitrogen concentration

of rural wastewater during period 119905 (tm3) ANLplusmn119905 maximum

allowable nitrogen loss from rural life section in period119905 (t) PCRplusmn

119894119905 phosphorus concentration of raw wastewater

from chemical plant 119894 in period 119905 (kgm3) 120578plusmnTP119894119905 phosphorustreatment efficiency in chemical plant 119894 in period 119905 () ASCplusmn

119894119905

amount of slag discharged by chemical plant 119894 in period 119905(kgt) SLRplusmn

119894119905 slag loss rate due to rain wash in chemical

plant 119894 during period119905 () PSCplusmn119894119905 phosphorus content in

slag generated by chemical plant 119894 in period 119905 () APCplusmn119894119905

allowable phosphorus discharge for chemical plant 119894 in period119905 (kgday) PSplusmn

119895119896 phosphorus content of soil in agricultural

zone 119895 planted with crop 119896 () SLplusmn119895119896119905 average soil loss

from agricultural zone 119895 planted with crop 119896 in period 119905(tha) RFplusmn

119895119896119905 runoff from agricultural zone 119895 with crop 119896 in

period 119905 (mm) DPplusmn119895119896119905 dissolved phosphorus concentration

in the runoff from agricultural zone 119895 planted with crop 119896in period 119905 (mgL) MPLplusmn

119895119905 maximum allowable phosphorus

loss in agricultural zone 119895 during period 119905 (tha) DPRplusmn119905

dissolved phosphorus concentration of rural wastewater dur-ing period 119905 (tm3) APLplusmn


loss from rural life during period 119905 (t) PCMplusmn119904119905 phosphorus

concentration of municipal wastewater at town 119904 in period119905 (kgm3) 120578plusmnTP119904119905 phosphorus treatment efficiency of WTPat town 119904 in period 119905 () APWplusmn

119904119905 allowable phosphorus

discharge for WTP at town 119904 in period 119905 (kgday) WPMplusmn119901119905

wastewater generation from phosphorus mining company 119901in period 119905 (m3t) MWCplusmn

119901119905 phosphorus concentration of

wastewater from mining company 119901 in period 119905 (kgm3)120579plusmn119901119905 phosphorus treatment efficiency in mining company 119901


Table 2 Crop yields and net benefits

Agricultural zoneAZ1 AZ2 AZ3 AZ4

Yields (tha)Citrus [103 126] [98 119] [84 102] [92 112]Tea [01 02] [01 02] [03 06] [02 04]Wheat [14 21] [13 17] [12 25] [23 31]Potato [24 32] [20 28] [19 28] [28 38]Rapeseed [16 17] [13 15] [13 17] [17 18]Alpine rice [69 96] [67 79] [44 66] [57 70]Second rice [74 102] [73 85] [47 70] [62 74]Maize [29 39] [27 29] [25 26] [40 47]Vegetable [216 240] [104 134] [218 242] [262 291]

Net benefits (RMByent)Citrus [1082 1350] [1010 1260] [1126 1405] [1203 1502]Tea [18000 18900] [19125 20081] [20170 21178] [22500 23625]Wheat [1031 1242] [972 1169] [945 1134] [1010 1218]Potato [915 1001] [827 897] [893 969] [846 917]Rapeseed [2435 2934] [2191 2640] [2305 2778] [2490 3000]Alpine rice [1105 1214] [1047 1153] [1084 1178] [1149 1265]Second rice [1159 1274] [1098 1209] [1137 1236] [1205 1327]Maize [1295 1451] [1267 1422] [1230 1378] [1308 1467]Vegetable [2360 2691] [1613 1865] [1927 2197] [1845 1950]

() ASMplusmn119901119905 amount of slag discharged by mining company

119901 during period 119905 (kgt) PCSplusmn119901119905 phosphorus content in

generated slag () SLWplusmn119901119905 slag loss rate due to rain wash ()

APMplusmn119901119905 allowable phosphorus discharge formining company

119901 during period 119905 (kgday) MSLplusmn119895119905 maximum allowable soil

loss agricultural zone 119895 in period 119905 (tha) NVFplusmn119905 nitrogen

volatilizationdenitrification rate of fertilizer in period119905 ()NVMplusmn

119905 nitrogen volatilizationdenitrification rate of manure

in period 119905 () 120576plusmnNF nitrogen content of fertilizer () 120576plusmnPFphosphorus content of fertilizer () 120576plusmnNM nitrogen contentof manure () 120576plusmnPM phosphorus content of manure ()NRplusmn119895119896119905 nitrogen requirement of agricultural zone 119895 with crop

119896 during period 119905 (tha) PRplusmn119895119896119905 phosphorus requirement of

crop 119896 in agricultural zone 119895 during period 119905 (tha) NECplusmn119896

net energy content of crop 119896 (Mcalt) ERLplusmn119903 net energy

requirement of livestock 119903 (Mcalunit) ERHplusmn119905 net energy

requirement of human beings (Mcalunit) DPCplusmn119896 digestible

protein content of crop 119896 () DRLplusmn119903 digestible protein

requirement of livestock 119903 (tunit) DRHplusmn119905 digestible protein

requirement of human beings (tunit) MCYplusmn119895119896119905 minimum

crop production requirement for crop 119896 in period 119905 (t) TPCplusmn119904119905

capacity of wastewater treatment capacity (WTPs) (m3day)TPDplusmn119894119905 capacity of wastewater treatment capacity (chemical

plants) (m3day)Table 1 provides the net benefits of chemical plant prod-

uctions phosphorus mining company and municipal watersupply According to the cropsrsquo growth periods dry seasoncrops include wheat potato rapeseed and alpine rice secondrice maize and vegetable are identified as main crops during

the wet season citrus and tea grow over the entire yearTable 2 shows the net benefits of each crop in every AZ andyields of each crop To guarantee the stream water qualitywastewater treatment measures have to be adopted at eachpoint source Based on the local environmental regulationsa safe level of water quality must be guaranteed to protectaquatic life and maintain aerobic condition in the streamsystem [2] Thus the BOD and TP loading amount would becontrolled strictly However the imprecision of the allowableBOD and TP discharge could introduce uncertainties inthe water quality management For modeling purpose thevagueness of the allowable BOD and TP loading amount areencoded by triangular fuzzy membership functions Figure 1presents the fuzzy set with triangular membership functionThe minimum maximum and most likely values (119887 119887 and119887) that define these fuzzy sets are estimated according toprevious research regarding water quality monitoring andenvironmental capacities as tabulated in Table 3 Moreoverby setting acceptable interval credibility levels the constraintscan be at least basically satisfied and at best practicallysatisfied Lower-bound of the interval numbers would beno less than 05 and upper- bound of the interval numberswould be no more than 1 [16] Lower-bound of credibilitylevel represents a situationwhen the decisionmakers are opti-mistic about this study area which may imply a higher riskof violating the riverrsquos self-purification capacity Converselyupper-bound of credibility level corresponds to a situationwhen the decision makers prefer a conservative policy thatcould guarantee that the riverrsquos self-purification capacity besatisfied


Table 3 Allowable BOD and TP discharges

119887 119887 119887

119905 = 1 119905 = 2 119905 = 1 119905 = 2 119905 = 1 119905 = 2

Allowable BOD loading dischargefrom chemical plant (kgday)

GF 076 078 082 085 088 092BSH 4372 44297 444265 448605 45133 45424PYK 725 783 753 789 781 795LCP 9543 9723 9782 10283 10021 10843XJLY 4834 4943 49635 50645 5093 5186

Allowable BOD loading dischargefromWTP (kgday)

Gufu 145 143 150 148 155 153Nanyang 7 6 10 9 13 12Gaoyang 38 36 40 38 42 40Xiakou 28 30 30 32 32 34

Allowable TP loading dischargefrom chemical plant (kgday)


Allowable TP loading dischargefromWTP (kgday)


Allowable TP loading dischargefrom phosphorus mining company (kgday)

XL 15770 14331 17560 19245 19349 24159XH 8940 10419 9951 12053 10962 13687XC 10820 11074 12048 13825 13275 16576GP 14620 14003 16278 18199 17936 22395JJW 8980 10449 9997 12101 11013 13752SJS 8660 10219 9641 11740 10621 13261

Allowable TP and TN loading dischargefrom rural life (kgday)

TP 215 205 2445 244 274 283TN 1021 1060 1075 1070 1128 1180

4 Result Analysis

Table 4 shows the solutions for industrial production andwater supply during the two periods The results show thatsignificant variations in industrial production and watersupply exist among different chemical plants phosphorusmining companies and towns For chemical plants theproduction scale of BSH (ie [53623 65237] td (tonneday)during period 1 and [53078 62919] td during period 2)would be larger than the other chemical plants (especially

Table 4 Solutions for industrial productions and water supplies

Main point sources Period119905 = 1 119905 = 2

Chemical plant (tday)GF [2564 2741] [2113 2702]BSH [53623 65237] [53078 62919]PYK [8171 10209] [7598 8092]LCP [37924 40372] [35733 45348]XJLY [28087 31347] [32042 33781]

Phosphorus mining company (tday)XL [82478 126100] [70105 104489]XH [32600 42000] [35048 40000]XC [28687 40000] [26038 39368]GP [53499 77800] [65980 77800]JJW [42599 54700] [41257 54700]SJS [51053 68500] [60243 68500]

Water supply (m3day)Gufu [1172379 1485914] [1226489 2173914]Nanyang [76229 98764] [85973 110984]Gaoyang [278346 322587] [296232 322625]Xiakou [165671 208318] [159284 215609]

GF) because of its higher allowable BOD and TP dischargesand net benefits For water supply more water would bedelivered to Gufu (ie [1172379 1485914]m3day in period1 and [1226489 2173914]m3day in period 2) than thoseto the other towns due to its greater water demand higherallowable BOD discharge and higher economic return Forphosphorusmining company the production scale of XL (ie[82478 126100] td in period 1 and [70105 104489] td inperiod 2) would be larger than the other chemical plantsbecause of its higher allowable TP discharge and net benefit

The results for total crop areas and manurefertilizerapplications are listed in Table 5 The areas of citrus andtea would maintain the low levels over the planning hori-zon This may be attributed to their pollutant losses aswell as their low net energy and digestible protein con-tents (supplied for livestock) In period 1 wheat potatorapeseed and alpine rice should be cultivated The potatoarea (ie [14598 16203] ha) would account for the largestone of the entire croplands In period 2 wheat potatorapeseed and alpine rice would be harvested and secondrice maize and vegetable would be sown The vegetable area(ie [17452 19156] ha) would be the largest one amongall croplands The results indicate that the high levels ofarea planted with potato and vegetable are associated withtheir high crop yields good market price and low pollutantlosses In terms of manure and fertilizer application theirquantities would vary with crop areas The results indicatethat manure would be the main nitrogen and phosphorussource which satisfy the requirements of most crops due toits availability and low price to collect Solutions demonstratethat domestic fowl husbandry would reach the highest level


0b

Credibility

bb

0b bb

Crisp values Crisp values

Fuzz

y m

embe

rshi

pFuzzy set

10

05

Fuzz

y m

embe

rshi

p

10

05

Figure 1 Fuzzy set and credibility

014

803

4

142

188

4

948

010

604

0

102

140

8

675

018

937

1

172

224

1

114

4

013

724

2

122

152

6

727

0

30

60

90

120

GF BSH PYK LCP XJLY GF BSH PYK LCP XJLY

BOD

disc

harg

e (t)

Chemical plantt = 1 t = 2

Lower-boundUpper-bound

Figure 2 BOD discharges from chemical plants Note the five chemical plants are abbreviated as their initials in this figure GF Gufu 119894 = 1BSH Baishahe 119894 = 2 PYK Pingyikou 119894 = 3 LCP Liucaopo 119894 = 4 XJLY Xiangjinlianying 119894 = 5

(ie [3480 13256] times 103 unit) among all live stocks becauseit possesses more advantageous conditions than the otherlive stocks (ie higher allowable discharge higher revenueparameter and lower manure generation rate)

Figures 2 and 3 present the amounts of BOD dischargesfrom chemical plants andWTPs respectively The amount ofBOD discharge is associated with a number of factors (egproduction scale wastewater generation rate and wastewatertreatment facility) The BOD discharge from BSH would bemore than those from the other chemical plants whichwouldbe [8034 9371] t (ie tonne) in period 1 and [6040 7242] tin period 2The BOD discharge fromGF would be the lowestamong all chemical plants with [014 018] t in period 1 and[010 013] t in period 2 Among all WTPs Gufu wastewater

treatment plant would discharge the highest BOD level with[1762 2869] t in period 1 and [2672 3672] t in period 2 theBOD discharged from Nanyang WTP would be the lowest(ie [399 439] t in period 1 and [501 755] t in period 2)

Figure 4 shows TP discharges from point sources (iechemical plants WTPs and phosphorus mining companies)andnonpoint sources (ie crop farming and agricultural life)In Figure 4 symbol ldquoCPrdquo denotes chemical plant symbolldquoPMCrdquo denotes phosphorus mining company symbol ldquoCFrdquodenotes crop farming symbol ldquoALrdquo means agricultural lifeFor point sources the chemical plants would be the majorcontributor to water pollution The phosphorus pollutantscan be discharged from wastewater and solid wastes (iechemical wastes slags and tailings) The amount of TP


176

2

399

733

523

207

2

501

944

752

286

9

439

122

6

687

367

2

755

132

1

994

0

10

20

30

40

Gufu Nanyang Gaoyang Xiakou Gufu Nanyang Gaoyang Xiakou

BOD

disc

harg

e (t)

Wastewater treatment plantLower-boundUpper-bound

t = 1 t = 2

Figure 3 BOD discharges fromWTPs

991

5

147

647

3

471

8

143

0

876

7

125

766

9

496

9

198

6

211

85

261

949

5

808

4

197

1

192

06

341

154

56

760

7

672

7

0

50

100

150

200

250

CP WTP PMC CF AL CP WTP PMC CF AL

TP d

ischa

rge (

t)


t = 1 t = 2

Figure 4 Solutions of TP discharge Note CP chemical plant WTP wastewater treatment plant PMC phosphorus mining company CFcrop farming AL agricultural life

discharge from chemical plant would be [9915 21185] t inperiod 1 and [8767 19206] t in period 2 Since the wastew-ater should be sluiced strictly according to the integrateddischarge standards the amount of TP discharge fromWTPswould stay at a low level Most of the TP would be fromphosphorus-containing wastes (ie discharged directly andwashed by rainfall) For nonpoint sources the phosphoruspollutants from crop farming which can be generated thoughrunoff and soil erosion (the latter would be a larger pro-portion) would be more than that from agricultural lifeThe amount of TP discharge from crop farming would be

[4718 8084] t in period 1 and [7669 15456] t in period 2This is associated with its high soil loss rate low runoff andlow phosphorus concentration in the study area GenerallyTP discharge derives mainly from point sources particularlyfrom chemical plants The results also indicate that thenitrogen pollutants would be generated by nonpoint sources(ie mainly from crop farming) TN discharge from citrusand tea can be neglected (ie nearly equal to aero) Figure 5shows TN discharges from the other cropping areas TNdischarges fromwheat and potatowould be higher than thosefrom rapeseed and alpine rice in period 1 TNdischarges from


0

10

20

30

40

50

60

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

Wheat Potato Rapeseed Alpine rice Second rice Maize Vegetables

TN d

ischa

rge (

t)


t = 1 t = 2

Figure 5 TN discharges from agriculture activities

481

94

203

86

127

85

643

766

7

908

53

393

93

263

03

275

8 116

82

0

300

600

900

1200

CP MWS PMC LH CF

Net

syste

m b

enefi

t (RM

Byen)

106


Figure 6 Net benefits from industrial and agricultural activities Note CP chemical plant MWSmunicipal water supply PMC phosphorusmining company CF crop farming LH livestock husbandry

vegetable would be higher than those frommaize and secondrice in period 2 This difference may be attributed to theirplanting areas soil losses runoff and nitrogen concentration

Net system benefit can be obtained from different indus-trial and agricultural activities as shown in Figure 6 Chem-ical plants would be the major economic incoming source inthe study area and could generate the highest revenue (RMByen[48494 90653]times106)Municipal water supply and phospho-rus mining company would also make certain contributionto the economic development their net benefits would beRMByen [20386 39393] times 106 and RMByen [12785 26303] times106 respectively Livestock husbandrywould bring the lowestbenefit Such an industry-oriented pattern may be related

to the abundant mineral resources (particularly phosphorusore) which can generate high economic return Agriculturalactivities would make less contribution to the local economicdevelopment due to its topography which is not suitable forcultivation in large parts

5 Conclusions

In this study an interval fuzzy credibility-constrained pro-gramming (IFCP) has been advanced for water qualitymanagement under uncertainty This method integratesinterval-parameter programming (IPP) and fuzzy credibility-constrained programming (FCP) techniques within a general


Table 5 Solutions for agricultural production

Crops and live stocks Growth periodCrop area (ha)

Citrus Whole year [357 728]Tea Whole year [408 534]Wheat Dry season [11126 18942]Potato Dry season [14598 16203]Rapeseed Dry season [2965 3694]Alpine rice Dry season [2065 3932]Second rice Wet season [3512 3874]Maize Wet season [834 978]Vegetable Wet season [17452 19156]

Manure application(103 t)


Fertilizer application (t)Citrus Whole year [07 10]Tea Whole year [8043 8936]Wheat Dry season [8244 9173]Potato Dry season [10234 14542]Rapeseed Dry season [24 36]Alpine rice Dry season [396 605]Second rice Wet season [2927 3163]Maize Wet season [02 03]Vegetable Wet season [15204 16321]

Size of livestockhusbandry (103 unit)

Pig Whole year [0121 35462]Ox Whole year [0688 0813]Sheep Whole year [21430 34478]Domestic fowl Whole year [34800 132561]

optimization framework Generally the IFCP model hasadvantages in (1) handling uncertainties presented in termsof interval values and possibility distributions in the modeland (2) providing bases for determining optimalwater qualitymanagement plans with desired compromises between eco-nomic benefits and environmental capacity-violation risks

The developed model has been applied to a real-worldcase of planning water quality management in the XiangxiRiver of the Three Gorges Reservoir Region The objectiveis to maximize the net system benefit subject to the envi-ronmental requirements under uncertainty over the planning

horizon Pollutant discharges generated by various point andnonpoint sources were considered simultaneously Intervalsolutions for production activities (ie industrial municipaland agricultural) and pollutant discharges (ie BOD TP andTN) under interval credibility levels have been generatedby solving two deterministic submodels The detailed resultsof related production scales and pollutant discharges canhelp identify desired water quality management schemes fordeveloping the local economy in a sustainable manner Someuseful suggestions for the local economy development ina sustainable manner could be summarized (i) advancingwastewater treatment technologies (eg tertiary treatmentand depth processing technologies) to further improve pol-lutant removal efficiency (ii) controlling the generation ofphosphorus-containing wastes (from chemical plants andphosphorus mining companies) strictly in the productionprocess and taking effective treatments and disposal mea-sures to reach the goal of achieving TP abatement (iii) takingcontrol practices on soil erosion for reducing the transport ofnitrogen and phosphorus pollutants to the river

Although reasonable solutions and desired managementpolicies have been obtained through the IFCP managementmodel there are still some extensive research works tobe done For example the proposed IFCP method candeal with uncertainties expressed as fuzzy sets and intervalnumbers however the main limitations of the IFCP methodremain in its difficulties in tackling uncertainties expressed asprobabilistic distributions (stochastic uncertainties) Undersuch a circumstance stochastic mathematical programmingmethod is a suitable option to be introduced into the pro-posed IFCP method Moreover decision support regardingpollution management could be further provided by incor-porating certain water quality simulation models into IFCPframework which can effectively reflect dynamic interactionsbetween pollutant loading and water quality

Acknowledgments

This research was supported by the Natural Sciences Founda-tion of China (Grant nos 51379075 51225904 and 51109077)the 111 Project (B14008) and the Program for InnovativeResearch Team in University (IRT1127) The authors aregrateful to the editors and the anonymous reviewers for theirinsightful comments and suggestions

References

[1] P Du Y P Li and G H Huang ldquoInexact chance-constrainedwaste-load allocation model for water quality management ofXiangxihe Riverrdquo Journal of Environmental Engineering vol 139pp 1178ndash1197 2013

[2] Y P Li and G H Huang ldquoTwo-stage planning for sustainablewater-quality management under uncertaintyrdquo Journal of Envi-ronmental Management vol 90 no 8 pp 2402ndash2413 2009

[3] Y P Li W Li and G H Huang ldquoTwo-stage inexact-probabilistic programming model for water quality manage-mentrdquo Environmental Engineering Science vol 29 no 7 pp 713ndash725 2012


[4] Y P Li and G H Huang ldquoFuzzy-stochastic-based violationanalysis method for planning water resources managementsystems with uncertain informationrdquo Information Sciences vol179 no 24 pp 4261ndash4276 2009

[5] X S Qin G H Huang G M Zeng A Chakma and Y FHuang ldquoAn interval-parameter fuzzy nonlinear optimizationmodel for stream water quality management under uncer-taintyrdquo European Journal of Operational Research vol 180 no3 pp 1331ndash1357 2007

[6] B Julien ldquoWater quality management with imprecise informa-tionrdquo European Journal of Operational Research vol 76 no 1pp 15ndash27 1994

[7] C-S Lee and C-G Wen ldquoFuzzy goal programming approachfor water quality management in a river basinrdquo Fuzzy Sets andSystems vol 89 no 2 pp 181ndash192 1997

[8] P P Mujumdar and K Sasikumar ldquoA fuzzy risk approach forseasonal water quality management of a river systemrdquo WaterResources Research vol 38 no 1 pp 51ndash59 2002

[9] H-W Chen and N-B Chang ldquoDecision support for allocationof watershed pollution load using grey fuzzy multiobjectiveprogrammingrdquo Journal of the American Water Resources Asso-ciation vol 42 no 3 pp 725ndash745 2006

[10] S Karmakar and P P Mujumdar ldquoGrey fuzzy optimizationmodel for water quality management of a river systemrdquoAdvances in Water Resources vol 29 no 7 pp 1088ndash1105 2006

[11] Y P Li G H Huang S L Nie and D W Mo ldquoInterval-parameter robust quadratic programming for water qualitymanagement under uncertaintyrdquo Engineering Optimization vol40 no 7 pp 613ndash635 2008

[12] X H Nie G H Huang D Wang and H L Li ldquoRobustoptimisation for inexact water quality management underuncertaintyrdquo Civil Engineering and Environmental Systems vol25 no 2 pp 167ndash184 2008

[13] S Maeda T Kawachi K Unami J Takeuchi T Izumi and SChono ldquoFuzzy optimization model for integrated managementof total nitrogen loads from distributed point and nonpointsources in watershedrdquo Paddy andWater Environment vol 7 no3 pp 163ndash175 2009

[14] T Y Xu andX SQin ldquoSolvingwater qualitymanagement prob-lem through combined genetic algorithmand fuzzy simulationrdquoJournal of Environmental Informatics vol 22 no 1 pp 39ndash482013

[15] M Liu G X Nie M Hu R F Liao and Y S Shen ldquoAninterval-parameter fuzzy robust nonlinear programmingmodelfor water quality managementrdquo Journal of Water Resource andProtection vol 5 pp 12ndash16 2013

[16] Y M Zhang and G H Huang ldquoOptimal water resourceplanning under fixed budget by interval-parameter credibilityconstrained programmingrdquo Engineering Optimization vol 43no 8 pp 879ndash889 2011

[17] M S Pishvaee S A Torabi and J Razmi ldquoCredibility-basedfuzzy mathematical programming model for green logisticsdesign under uncertaintyrdquo Computers and Industrial Engineer-ing vol 62 no 2 pp 624ndash632 2012

[18] Y P Li G H Huang and S L Nie ldquoOptimization of regionaleconomic and environmental systems under fuzzy and randomuncertaintiesrdquo Journal of Environmental Management vol 92no 8 pp 2010ndash2020 2011

[19] R Brouwer and C De Blois ldquoIntegrated modelling of riskand uncertainty underlying the cost and effectiveness of waterquality measuresrdquo Environmental Modelling and Software vol23 no 7 pp 922ndash937 2008

[20] Y R Fan and G H Huang ldquoA robust two-step methodfor solving interval linear programming problems within anenvironmental management contextrdquo Journal of EnvironmentalInformatics vol 19 no 1 pp 1ndash9 2012

[21] Y Xu G H Huang and T Y Xu ldquoInexact management mod-eling for urban water supply systemsrdquo Journal of EnvironmentalInformatics vol 20 no 1 pp 34ndash43 2012

[22] Y Chen S Y Cheng L Liu et al ldquoAssessing the effects of landuse changes on non-point source pollution reduction for theThree Gorges Watershed using the SWAT modelrdquo Journal ofEnvironmental Informatics vol 22 no 1 pp 13ndash26 2013

[23] L He G Q Wang and C Zhang ldquoApplication of looselycoupled watershed model and channel model in Yellow RiverChinardquo Journal of Environmental Informatics vol 19 no 1 pp30ndash37 2012

[24] Y P Li andGHHuang ldquoA recourse-based nonlinear program-ming model for stream water quality managementrdquo StochasticEnvironmental Research and Risk Assessment vol 26 no 2 pp207ndash223 2012

[25] L Ye Q-H Cai R-Q Liu and M Cao ldquoThe influence oftopography and land use on water quality of Xiangxi River inthree Gorges Reservoir regionrdquo Environmental Geology vol 58no 5 pp 937ndash942 2009

[26] F A Deviney Jr D E Brown and K C Rice ldquoEvaluation ofbayesian estimation of a hidden continuous-time markov chainmodel with application to threshold violation in water-qualityindicatorsrdquo Journal of Environmental Informatics vol 19 no 2pp 70ndash78 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


functions becomes an attractive alternative [5] Previously anumber of FMP methods were developed for water qualitymanagement [5ndash15] For example Julien [6] investigated theapplication of a fuzzy possiblistic programming to addressimprecise parameters which were represented by possibilitydistributions in water quality decision-making problemsMujumdar and Sasikumar [8] formulated a fuzzy flexibleoptimization model for dealing with the systemrsquos fuzzy goaland constraints in a water quality management problemNie et al [12] proposed a fuzzy robust optimization modelfor water quality management of an agricultural systemto deal with uncertainties expressed as fuzzy membershipfunctions in both left- and right-hand-side coefficients (ofthe modelrsquos constraints) Maeda et al [13] employed a fuzzyflexible optimization model which involved fuzzy set theoryto express vagueness in constraints and objectives in riverwater quality management problems Liu et al [15] developeda two-stage fuzzy robust programming model for waterquality management to address fuzzy parameters which wererepresented by possibility distributions in the left- and right-hand sides of the constraints

Generally FMP methods can be classified into threecategories in view of the forms of uncertainties (i) fuzzyflexible programming (ii) robust programming and (iii)fuzzy possibilistic programming In detail fuzzy flexibleprogramming can deal with decision problems under fuzzygoal and constraints however it has difficulties in tacklingambiguous coefficients of the objective function and con-straints Robust programming improves upon fuzzy flexibleprogramming by allowing fuzzy parameters in the constraintsto be represented by possibility distributions However themain limitations of this method remain in its difficultiesin tackling uncertainties in a nonfuzzy decision space Infuzzy possibilistic programming fuzzy parameters that areregarded as possibility distributions are introduced into themodeling frameworks It can handle ambiguous coefficientsin the left- and right-hand sides of the constraints and in theobjective function

Fuzzy credibility-constrained programming (FCP) is acomputationally efficient fuzzy possibilistic programmingapproach that relies on mathematical concepts (ie theexpected value of a fuzzy number and the credibilitymeasure)and can support different kinds of fuzzy members such astriangular and trapezoidal forms as well as enabling thedecision maker to satisfy some constraints in at least somegiven confidence levels [16] When the credibility value of afuzzy event reaches 1 the fuzzy event will certainly occurwhen the credibility value of a fuzzy event reaches 0 thefuzzy event will not occur For example with respect towater quality management if the allowable total phosphorus(TP) discharges are 70 76 and 82 kgday and the amountof actual discharge may be 78 kgday then the credibilitydegree of the event wherein the total phosphorus dischargecan satisfy the riverrsquos self-purification capacity would be033 the credibility degree of its complement event (waterpollution occur) would be 067 No feature of fuzzy sets wouldbe missing by using credibility measure [16 17] Howeverthe main limitation of FCP lies within its deterministiccoefficients for the objective function leading to potential

losses of valuable uncertain information besides whenmanyuncertain parameters are expressed as fuzzy sets interactionsamong these uncertainties may lead to serious complexitiesparticularly for large-scale practical problems [18] In factin water quality management problems uncertainty is aninherent component of any economic analysis particularlythose (eg effluent trading programs) associated with envi-ronmental policy and project appraisal [19] For exampleone major characteristic of nonpoint source pollution thatdiffers from point source pollution is imperfect knowledgeabout pollutant loadings the crop productivity and costbenefit coefficients are easier to be presented as intervals thanby membership functions Interval-parameter programming(IPP) is an alternative for handling uncertainties in themodelrsquos left- andor right-hand sides as well as those that can-not be quantified as membership or distribution functionssince interval numbers are acceptable as its uncertain inputs[20]

Therefore the objective of this study is to develop aninexact fuzzy credibility-constrained programming (IFCP)method for water quality management through couplingfuzzy credibility-constrained programming (FCP) withinterval-parameter programming (IPP) The main advantageof IFCP is that it can effectively handle uncertain parametersexpressed as both fuzzy sets and interval values in theobjective function and constraints IFCP would not leadto serious complexities in its solution process and it isapplicable to large-scale practical problems Then thedeveloped IFCP method is applied to a real-world case ofwater quality management of the Xiangxi River which facessevere water quality problems due to point and nonpointsource pollution The results obtained can help decisionmakers to generate alternatives for industrial productionscale water supply cropped area livestock husbandry sizeand manurefertilizer application rate with consideration ofriver water quality management

Thepaperwill be organized as follows Section 2 describesthe development process of the IFCP Section 3 providesa case study of river water quality management Section 4presents result analysis and discussion Section 5 draws someconclusions and extensions

2 Methodology

When coefficients in the constraints are ambiguous and canbe expressed as possibility distributions the problem can beformulated as a fuzzy credibility-constrained programming(FCP) model as follows

Max119891 =119899

sum119895=1

119888119895119909119895 (1a)

subject to

Cr

119899

sum119895=1

119886119894119895119909119895le 119894

ge 120582119894 (1b)

119909119895ge 0 119894 = 1 2 119899 (1c)


where 119909 = (1199091 1199092 119909






Cr 120585 le 119903 =1

2(sup119909le119903

120583 (119909) + 1 minus sup119909gt119903

120583 (119909)) (2)




as follows

Cr 120585 le 119903 =1

2(Pos 120585 le 119903 +Nec 120585 le 119903) (3)


Cr 120585 le 119903 + Cr 120585 gt 119903 = 1 (4)


120583 (119903) =

119903 minus 119905

119905 minus 119905 if 119905 le 119903 le 119905

119903 minus 119905

119905 minus 119905 if 119905 le 119903 le 119905

0 otherwise

(5)


Cr 119903 le 120585 =

1 if 119903 le 119905

2119905 minus 119905 minus 119903

2 (119905 minus 119905) if 119905 le 119903 le 119905

119903 minus 119905

2 (119905 minus 119905) if 119905 le 119903 le 119905

0 if 119903 ge 119905

(6)


Cr 119903 le 120585 =2119905 minus 119905 minus 119903

2 (119905 minus 119905)ge 120582 (7)


119903 le 119905 + (1 minus 2120582) (119905 minus 119905) (8)


Max119891 =119899

sum119895=1

119888119895119909119895 (9a)

subject to

119899

sum119895=1

119886119894119895119909119895le 119887119894+ (1 minus 2120582

119894) (119887119894minus 119887119894) (9b)

119909119895ge 0 119894 = 1 2 119899 (9c)


Max119891plusmn =119899

sum119895=1

119888plusmn119895119909plusmn119895 (10a)

subject to

119899

sum119895=1


119894) (119887119894minus 119887119894) (10b)

119909plusmn119895ge 0 119894 = 1 2 119899 (10c)


Max119891+ =1198961

sum119895=1

119888+119895119909+119895+119899

sum119895=1198961+1

119888+119895119909minus119895 (11a)


subject to

1198961

sum119895=1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816minus

Sign (119886minus119894119895) 119909+119895

+119899

sum119895=1198961+1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816+

Sign (119886+119894119895) 119909minus119895

le 119887119894+ (1 minus 2120582minus

119894) (119887119894minus 119887119894)

forall119894 = 1 2 119898

(11b)

119909+119895ge 0 119895 = 1 2 119896

1 (11c)

119909minus119895ge 0 119895 = 119896

1+ 1 119899 (11d)

where 119888+119895(119895 = 1 2 119896

1) gt 0 119888+

119895(119895 = 119896

1+1 1198962+1 119899) lt


119894119895lt 0 Sign(119886plusmn

119894119895) = 1 when 119886plusmn

119894119895gt

0 120582minus119894


119895opt (119895 =1 2 119896

1) and 119909minus

119895opt (119895 = 1198961+1 1198962+1 119899) In the second


Max119891minus =1198961

sum119895=1

119888minus119895119909minus119895+119899

sum119895=1198961+1

119888minus119895119909+119895 (12a)

subject to

1198961

sum119895=1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816+

Sign (119886+119894119895) 119909minus119895+119899

sum119895=1198961+1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816minus

Sign (119886minus119894119895) 119909+119895

le 119887119894+ (1 minus 2120582+

119894) (119887119894minus 119887119894) forall119894 = 1 2 119898

(12b)

0 le 119909minus119895le 119909+119895opt 119895 = 1 2 119896

1 (12c)

119909+119895ge 119909minus119895opt 119895 = 119896

1+ 1 119899 (12d)


119895opt (119895 = 1 2 1198961) and 119909+

119895opt (119895 = 1198961+ 1 1198962+



opt] (13a)

119909plusmn119895opt = [119909minus

119895opt 119909+

119895opt] (13b)

3 Case Study








Max119891plusmn =5

sum119894=1

2

sum119905=1


sum119904=1

2

sum119905=1


+6

sum119901=1

2

sum119905=1


119901119905

+4

sum119903=1


sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot BAplusmn119895119896119905

sdot PAplusmn119895119896119905

minus4

sum119895=1

9

sum119896=1

2

sum119905=1


minus4

sum119895=1

9

sum119896=1

2

sum119905=1


minus5

sum119894=1

2

sum119905=1


minus4

sum119904=1

2

sum119905=1


minus4

sum119895=1

9

sum119896=1

2

sum119905=1

PAplusmn119895119896119905

sdot IQplusmn119895119896119905

sdotWSPplusmn119905

minus5

sum119894=1

2

sum119905=1


119905

minus4

sum119904=1

2

sum119905=1


119905 (14a)

subject to



119904119905(14b)


119894119905 (14c)



119894119905 ge 120582plusmn (14d)


119904119905 ge 120582plusmn

(14e)


Cr

(119871119905sdot4

sum119903=1



sum119895=1

9

sum119896=1

AMplusmn119895119896119905)




119905

le ANLplusmn119905

ge 120582plusmn

(14f)

9

sum119896=1


+ RFplusmn119895119896119905


119895119896119905


(14g)






119879119875119894119905)


119894119905 ge 120582plusmn

(14h)

Cr

(119871119905sdot4

sum119903=1


119905


sum119895=1

9

sum119896=1

AMplusmn119895119896119905)




119905le APLplusmn

119905

ge 120582plusmn

(14i)



119879119875119904119905) le APWplusmn

119904119905 ge 120582plusmn

(14j)




119901119905)




119901119905 ge 120582plusmn

(14k)

9

sum119896=1


+ RFplusmn119895119896119905


119895119896119905le MPLplusmn

119895119905sdot TAplusmn119895119905

(14l)


9

sum119896=1

SLplusmn119895119896119905

sdot PAplusmn119895119896119905




plusmn

119895119896119905+ (1 minusNVMplusmn

119905)



sdot PAplusmn119895119896119905

ge 0(14n)



119895119896119905minus PRplusmn119895119896119905

sdot PAplusmn119895119896119905

ge 0 (14o)

9

sum119896=1



119895119896119905minusNRplusmn119895119896119905

sdot PAplusmn119895119896119905)


(14p)

9

sum119896=1


119895119896119905+ 120576plusmn119875119872

sdot AMplusmn119895119896119905


sdot PAplusmn119895119896119905)


(14q)

119871119905sdot4

sum119903=1



sum119895=1

9

sum119896=1

AMplusmn119895119896119905

ge 0

(14r)


4

sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905


sum119903=1


minus2

sum119905=1


(14s)

4

sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905


minus4

sum119903=1


sum119905=1


(14t)



119894119905le PLC

119894max (14u)


119903le NL119903max (14v)


119904119905le QW

119904max (14w)


119901119905le PLM

119901max (14x)


4

sum119895=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905



9

sum119896=1

PAplusmn119895119896119905

le TAplusmn119895119905 (14z)




ge 0(14aa)




Period119905 = 1 119905 = 2













































119905 wastewater







119894119905




















































119887 119887 119887

119905 = 1 119905 = 2 119905 = 1 119905 = 2 119905 = 1 119905 = 2










XL 15770 14331 17560 19245 19349 24159XH 8940 10419 9951 12053 10962 13687XC 10820 11074 12048 13825 13275 16576GP 14620 14003 16278 18199 17936 22395JJW 8980 10449 9997 12101 11013 13752SJS 8660 10219 9641 11740 10621 13261


TP 215 205 2445 244 274 283TN 1021 1060 1075 1070 1128 1180

4 Result Analysis










0b

Credibility

bb

0b bb


Fuzz

y m

embe

rshi

pFuzzy set

10

05

Fuzz

y m

embe

rshi

p

10

05


014

803

4

142

188

4

948

010

604

0

102

140

8

675

018

937

1

172

224

1

114

4

013

724

2

122

152

6

727

0

30

60

90

120


BOD

disc

harg

e (t)









176

2

399

733

523

207

2

501

944

752

286

9

439

122

6

687

367

2

755

132

1

994

0

10

20

30

40


BOD

disc

harg

e (t)


t = 1 t = 2


991

5

147

647

3

471

8

143

0

876

7

125

766

9

496

9

198

6

211

85

261

949

5

808

4

197

1

192

06

341

154

56

760

7

672

7

0

50

100

150

200

250


TP d

ischa

rge (

t)


t = 1 t = 2





0

10

20

30

40

50

60

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4


TN d

ischa

rge (

t)


t = 1 t = 2


481

94

203

86

127

85

643

766

7

908

53

393

93

263

03

275

8 116

82

0

300

600

900

1200

CP MWS PMC LH CF

Net

syste

m b

enefi

t (RM

Byen)

106






5 Conclusions















Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 119909 = (1199091 1199092 119909






Cr 120585 le 119903 =1

2(sup119909le119903

120583 (119909) + 1 minus sup119909gt119903

120583 (119909)) (2)




as follows

Cr 120585 le 119903 =1

2(Pos 120585 le 119903 +Nec 120585 le 119903) (3)


Cr 120585 le 119903 + Cr 120585 gt 119903 = 1 (4)


120583 (119903) =

119903 minus 119905

119905 minus 119905 if 119905 le 119903 le 119905

119903 minus 119905

119905 minus 119905 if 119905 le 119903 le 119905

0 otherwise

(5)


Cr 119903 le 120585 =

1 if 119903 le 119905

2119905 minus 119905 minus 119903

2 (119905 minus 119905) if 119905 le 119903 le 119905

119903 minus 119905

2 (119905 minus 119905) if 119905 le 119903 le 119905

0 if 119903 ge 119905

(6)


Cr 119903 le 120585 =2119905 minus 119905 minus 119903

2 (119905 minus 119905)ge 120582 (7)


119903 le 119905 + (1 minus 2120582) (119905 minus 119905) (8)


Max119891 =119899

sum119895=1

119888119895119909119895 (9a)

subject to

119899

sum119895=1

119886119894119895119909119895le 119887119894+ (1 minus 2120582

119894) (119887119894minus 119887119894) (9b)

119909119895ge 0 119894 = 1 2 119899 (9c)


Max119891plusmn =119899

sum119895=1

119888plusmn119895119909plusmn119895 (10a)

subject to

119899

sum119895=1


119894) (119887119894minus 119887119894) (10b)

119909plusmn119895ge 0 119894 = 1 2 119899 (10c)


Max119891+ =1198961

sum119895=1

119888+119895119909+119895+119899

sum119895=1198961+1

119888+119895119909minus119895 (11a)


subject to

1198961

sum119895=1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816minus

Sign (119886minus119894119895) 119909+119895

+119899

sum119895=1198961+1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816+

Sign (119886+119894119895) 119909minus119895

le 119887119894+ (1 minus 2120582minus

119894) (119887119894minus 119887119894)

forall119894 = 1 2 119898

(11b)

119909+119895ge 0 119895 = 1 2 119896

1 (11c)

119909minus119895ge 0 119895 = 119896

1+ 1 119899 (11d)

where 119888+119895(119895 = 1 2 119896

1) gt 0 119888+

119895(119895 = 119896

1+1 1198962+1 119899) lt


119894119895lt 0 Sign(119886plusmn

119894119895) = 1 when 119886plusmn

119894119895gt

0 120582minus119894


119895opt (119895 =1 2 119896

1) and 119909minus

119895opt (119895 = 1198961+1 1198962+1 119899) In the second


Max119891minus =1198961

sum119895=1

119888minus119895119909minus119895+119899

sum119895=1198961+1

119888minus119895119909+119895 (12a)

subject to

1198961

sum119895=1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816+

Sign (119886+119894119895) 119909minus119895+119899

sum119895=1198961+1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816minus

Sign (119886minus119894119895) 119909+119895

le 119887119894+ (1 minus 2120582+

119894) (119887119894minus 119887119894) forall119894 = 1 2 119898

(12b)

0 le 119909minus119895le 119909+119895opt 119895 = 1 2 119896

1 (12c)

119909+119895ge 119909minus119895opt 119895 = 119896

1+ 1 119899 (12d)


119895opt (119895 = 1 2 1198961) and 119909+

119895opt (119895 = 1198961+ 1 1198962+



opt] (13a)

119909plusmn119895opt = [119909minus

119895opt 119909+

119895opt] (13b)

3 Case Study








Max119891plusmn =5

sum119894=1

2

sum119905=1


sum119904=1

2

sum119905=1


+6

sum119901=1

2

sum119905=1


119901119905

+4

sum119903=1


sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot BAplusmn119895119896119905

sdot PAplusmn119895119896119905

minus4

sum119895=1

9

sum119896=1

2

sum119905=1


minus4

sum119895=1

9

sum119896=1

2

sum119905=1


minus5

sum119894=1

2

sum119905=1


minus4

sum119904=1

2

sum119905=1


minus4

sum119895=1

9

sum119896=1

2

sum119905=1

PAplusmn119895119896119905

sdot IQplusmn119895119896119905

sdotWSPplusmn119905

minus5

sum119894=1

2

sum119905=1


119905

minus4

sum119904=1

2

sum119905=1


119905 (14a)

subject to



119904119905(14b)


119894119905 (14c)



119894119905 ge 120582plusmn (14d)


119904119905 ge 120582plusmn

(14e)


Cr

(119871119905sdot4

sum119903=1



sum119895=1

9

sum119896=1

AMplusmn119895119896119905)




119905

le ANLplusmn119905

ge 120582plusmn

(14f)

9

sum119896=1


+ RFplusmn119895119896119905


119895119896119905


(14g)






119879119875119894119905)


119894119905 ge 120582plusmn

(14h)

Cr

(119871119905sdot4

sum119903=1


119905


sum119895=1

9

sum119896=1

AMplusmn119895119896119905)




119905le APLplusmn

119905

ge 120582plusmn

(14i)



119879119875119904119905) le APWplusmn

119904119905 ge 120582plusmn

(14j)




119901119905)




119901119905 ge 120582plusmn

(14k)

9

sum119896=1


+ RFplusmn119895119896119905


119895119896119905le MPLplusmn

119895119905sdot TAplusmn119895119905

(14l)


9

sum119896=1

SLplusmn119895119896119905

sdot PAplusmn119895119896119905




plusmn

119895119896119905+ (1 minusNVMplusmn

119905)



sdot PAplusmn119895119896119905

ge 0(14n)



119895119896119905minus PRplusmn119895119896119905

sdot PAplusmn119895119896119905

ge 0 (14o)

9

sum119896=1



119895119896119905minusNRplusmn119895119896119905

sdot PAplusmn119895119896119905)


(14p)

9

sum119896=1


119895119896119905+ 120576plusmn119875119872

sdot AMplusmn119895119896119905


sdot PAplusmn119895119896119905)


(14q)

119871119905sdot4

sum119903=1



sum119895=1

9

sum119896=1

AMplusmn119895119896119905

ge 0

(14r)


4

sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905


sum119903=1


minus2

sum119905=1


(14s)

4

sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905


minus4

sum119903=1


sum119905=1


(14t)



119894119905le PLC

119894max (14u)


119903le NL119903max (14v)


119904119905le QW

119904max (14w)


119901119905le PLM

119901max (14x)


4

sum119895=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905



9

sum119896=1

PAplusmn119895119896119905

le TAplusmn119895119905 (14z)




ge 0(14aa)




Period119905 = 1 119905 = 2













































119905 wastewater







119894119905




















































119887 119887 119887

119905 = 1 119905 = 2 119905 = 1 119905 = 2 119905 = 1 119905 = 2










XL 15770 14331 17560 19245 19349 24159XH 8940 10419 9951 12053 10962 13687XC 10820 11074 12048 13825 13275 16576GP 14620 14003 16278 18199 17936 22395JJW 8980 10449 9997 12101 11013 13752SJS 8660 10219 9641 11740 10621 13261


TP 215 205 2445 244 274 283TN 1021 1060 1075 1070 1128 1180

4 Result Analysis










0b

Credibility

bb

0b bb


Fuzz

y m

embe

rshi

pFuzzy set

10

05

Fuzz

y m

embe

rshi

p

10

05


014

803

4

142

188

4

948

010

604

0

102

140

8

675

018

937

1

172

224

1

114

4

013

724

2

122

152

6

727

0

30

60

90

120


BOD

disc

harg

e (t)









176

2

399

733

523

207

2

501

944

752

286

9

439

122

6

687

367

2

755

132

1

994

0

10

20

30

40


BOD

disc

harg

e (t)


t = 1 t = 2


991

5

147

647

3

471

8

143

0

876

7

125

766

9

496

9

198

6

211

85

261

949

5

808

4

197

1

192

06

341

154

56

760

7

672

7

0

50

100

150

200

250


TP d

ischa

rge (

t)


t = 1 t = 2





0

10

20

30

40

50

60

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4


TN d

ischa

rge (

t)


t = 1 t = 2


481

94

203

86

127

85

643

766

7

908

53

393

93

263

03

275

8 116

82

0

300

600

900

1200

CP MWS PMC LH CF

Net

syste

m b

enefi

t (RM

Byen)

106






5 Conclusions















Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










subject to

1198961

sum119895=1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816minus

Sign (119886minus119894119895) 119909+119895

+119899

sum119895=1198961+1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816+

Sign (119886+119894119895) 119909minus119895

le 119887119894+ (1 minus 2120582minus

119894) (119887119894minus 119887119894)

forall119894 = 1 2 119898

(11b)

119909+119895ge 0 119895 = 1 2 119896

1 (11c)

119909minus119895ge 0 119895 = 119896

1+ 1 119899 (11d)

where 119888+119895(119895 = 1 2 119896

1) gt 0 119888+

119895(119895 = 119896

1+1 1198962+1 119899) lt


119894119895lt 0 Sign(119886plusmn

119894119895) = 1 when 119886plusmn

119894119895gt

0 120582minus119894


119895opt (119895 =1 2 119896

1) and 119909minus

119895opt (119895 = 1198961+1 1198962+1 119899) In the second


Max119891minus =1198961

sum119895=1

119888minus119895119909minus119895+119899

sum119895=1198961+1

119888minus119895119909+119895 (12a)

subject to

1198961

sum119895=1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816+

Sign (119886+119894119895) 119909minus119895+119899

sum119895=1198961+1

1003816100381610038161003816100381611988611989411989510038161003816100381610038161003816minus

Sign (119886minus119894119895) 119909+119895

le 119887119894+ (1 minus 2120582+

119894) (119887119894minus 119887119894) forall119894 = 1 2 119898

(12b)

0 le 119909minus119895le 119909+119895opt 119895 = 1 2 119896

1 (12c)

119909+119895ge 119909minus119895opt 119895 = 119896

1+ 1 119899 (12d)


119895opt (119895 = 1 2 1198961) and 119909+

119895opt (119895 = 1198961+ 1 1198962+



opt] (13a)

119909plusmn119895opt = [119909minus

119895opt 119909+

119895opt] (13b)

3 Case Study








Max119891plusmn =5

sum119894=1

2

sum119905=1


sum119904=1

2

sum119905=1


+6

sum119901=1

2

sum119905=1


119901119905

+4

sum119903=1


sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot BAplusmn119895119896119905

sdot PAplusmn119895119896119905

minus4

sum119895=1

9

sum119896=1

2

sum119905=1


minus4

sum119895=1

9

sum119896=1

2

sum119905=1


minus5

sum119894=1

2

sum119905=1


minus4

sum119904=1

2

sum119905=1


minus4

sum119895=1

9

sum119896=1

2

sum119905=1

PAplusmn119895119896119905

sdot IQplusmn119895119896119905

sdotWSPplusmn119905

minus5

sum119894=1

2

sum119905=1


119905

minus4

sum119904=1

2

sum119905=1


119905 (14a)

subject to



119904119905(14b)


119894119905 (14c)



119894119905 ge 120582plusmn (14d)


119904119905 ge 120582plusmn

(14e)


Cr

(119871119905sdot4

sum119903=1



sum119895=1

9

sum119896=1

AMplusmn119895119896119905)




119905

le ANLplusmn119905

ge 120582plusmn

(14f)

9

sum119896=1


+ RFplusmn119895119896119905


119895119896119905


(14g)






119879119875119894119905)


119894119905 ge 120582plusmn

(14h)

Cr

(119871119905sdot4

sum119903=1


119905


sum119895=1

9

sum119896=1

AMplusmn119895119896119905)




119905le APLplusmn

119905

ge 120582plusmn

(14i)



119879119875119904119905) le APWplusmn

119904119905 ge 120582plusmn

(14j)




119901119905)




119901119905 ge 120582plusmn

(14k)

9

sum119896=1


+ RFplusmn119895119896119905


119895119896119905le MPLplusmn

119895119905sdot TAplusmn119895119905

(14l)


9

sum119896=1

SLplusmn119895119896119905

sdot PAplusmn119895119896119905




plusmn

119895119896119905+ (1 minusNVMplusmn

119905)



sdot PAplusmn119895119896119905

ge 0(14n)



119895119896119905minus PRplusmn119895119896119905

sdot PAplusmn119895119896119905

ge 0 (14o)

9

sum119896=1



119895119896119905minusNRplusmn119895119896119905

sdot PAplusmn119895119896119905)


(14p)

9

sum119896=1


119895119896119905+ 120576plusmn119875119872

sdot AMplusmn119895119896119905


sdot PAplusmn119895119896119905)


(14q)

119871119905sdot4

sum119903=1



sum119895=1

9

sum119896=1

AMplusmn119895119896119905

ge 0

(14r)


4

sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905


sum119903=1


minus2

sum119905=1


(14s)

4

sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905


minus4

sum119903=1


sum119905=1


(14t)



119894119905le PLC

119894max (14u)


119903le NL119903max (14v)


119904119905le QW

119904max (14w)


119901119905le PLM

119901max (14x)


4

sum119895=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905



9

sum119896=1

PAplusmn119895119896119905

le TAplusmn119895119905 (14z)




ge 0(14aa)




Period119905 = 1 119905 = 2













































119905 wastewater







119894119905




















































119887 119887 119887

119905 = 1 119905 = 2 119905 = 1 119905 = 2 119905 = 1 119905 = 2










XL 15770 14331 17560 19245 19349 24159XH 8940 10419 9951 12053 10962 13687XC 10820 11074 12048 13825 13275 16576GP 14620 14003 16278 18199 17936 22395JJW 8980 10449 9997 12101 11013 13752SJS 8660 10219 9641 11740 10621 13261


TP 215 205 2445 244 274 283TN 1021 1060 1075 1070 1128 1180

4 Result Analysis










0b

Credibility

bb

0b bb


Fuzz

y m

embe

rshi

pFuzzy set

10

05

Fuzz

y m

embe

rshi

p

10

05


014

803

4

142

188

4

948

010

604

0

102

140

8

675

018

937

1

172

224

1

114

4

013

724

2

122

152

6

727

0

30

60

90

120


BOD

disc

harg

e (t)









176

2

399

733

523

207

2

501

944

752

286

9

439

122

6

687

367

2

755

132

1

994

0

10

20

30

40


BOD

disc

harg

e (t)


t = 1 t = 2


991

5

147

647

3

471

8

143

0

876

7

125

766

9

496

9

198

6

211

85

261

949

5

808

4

197

1

192

06

341

154

56

760

7

672

7

0

50

100

150

200

250


TP d

ischa

rge (

t)


t = 1 t = 2





0

10

20

30

40

50

60

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4


TN d

ischa

rge (

t)


t = 1 t = 2


481

94

203

86

127

85

643

766

7

908

53

393

93

263

03

275

8 116

82

0

300

600

900

1200

CP MWS PMC LH CF

Net

syste

m b

enefi

t (RM

Byen)

106






5 Conclusions















Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Max119891plusmn =5

sum119894=1

2

sum119905=1


sum119904=1

2

sum119905=1


+6

sum119901=1

2

sum119905=1


119901119905

+4

sum119903=1


sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot BAplusmn119895119896119905

sdot PAplusmn119895119896119905

minus4

sum119895=1

9

sum119896=1

2

sum119905=1


minus4

sum119895=1

9

sum119896=1

2

sum119905=1


minus5

sum119894=1

2

sum119905=1


minus4

sum119904=1

2

sum119905=1


minus4

sum119895=1

9

sum119896=1

2

sum119905=1

PAplusmn119895119896119905

sdot IQplusmn119895119896119905

sdotWSPplusmn119905

minus5

sum119894=1

2

sum119905=1


119905

minus4

sum119904=1

2

sum119905=1


119905 (14a)

subject to



119904119905(14b)


119894119905 (14c)



119894119905 ge 120582plusmn (14d)


119904119905 ge 120582plusmn

(14e)


Cr

(119871119905sdot4

sum119903=1



sum119895=1

9

sum119896=1

AMplusmn119895119896119905)




119905

le ANLplusmn119905

ge 120582plusmn

(14f)

9

sum119896=1


+ RFplusmn119895119896119905


119895119896119905


(14g)






119879119875119894119905)


119894119905 ge 120582plusmn

(14h)

Cr

(119871119905sdot4

sum119903=1


119905


sum119895=1

9

sum119896=1

AMplusmn119895119896119905)




119905le APLplusmn

119905

ge 120582plusmn

(14i)



119879119875119904119905) le APWplusmn

119904119905 ge 120582plusmn

(14j)




119901119905)




119901119905 ge 120582plusmn

(14k)

9

sum119896=1


+ RFplusmn119895119896119905


119895119896119905le MPLplusmn

119895119905sdot TAplusmn119895119905

(14l)


9

sum119896=1

SLplusmn119895119896119905

sdot PAplusmn119895119896119905




plusmn

119895119896119905+ (1 minusNVMplusmn

119905)



sdot PAplusmn119895119896119905

ge 0(14n)



119895119896119905minus PRplusmn119895119896119905

sdot PAplusmn119895119896119905

ge 0 (14o)

9

sum119896=1



119895119896119905minusNRplusmn119895119896119905

sdot PAplusmn119895119896119905)


(14p)

9

sum119896=1


119895119896119905+ 120576plusmn119875119872

sdot AMplusmn119895119896119905


sdot PAplusmn119895119896119905)


(14q)

119871119905sdot4

sum119903=1



sum119895=1

9

sum119896=1

AMplusmn119895119896119905

ge 0

(14r)


4

sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905


sum119903=1


minus2

sum119905=1


(14s)

4

sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905


minus4

sum119903=1


sum119905=1


(14t)



119894119905le PLC

119894max (14u)


119903le NL119903max (14v)


119904119905le QW

119904max (14w)


119901119905le PLM

119901max (14x)


4

sum119895=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905



9

sum119896=1

PAplusmn119895119896119905

le TAplusmn119895119905 (14z)




ge 0(14aa)




Period119905 = 1 119905 = 2













































119905 wastewater







119894119905




















































119887 119887 119887

119905 = 1 119905 = 2 119905 = 1 119905 = 2 119905 = 1 119905 = 2










XL 15770 14331 17560 19245 19349 24159XH 8940 10419 9951 12053 10962 13687XC 10820 11074 12048 13825 13275 16576GP 14620 14003 16278 18199 17936 22395JJW 8980 10449 9997 12101 11013 13752SJS 8660 10219 9641 11740 10621 13261


TP 215 205 2445 244 274 283TN 1021 1060 1075 1070 1128 1180

4 Result Analysis










0b

Credibility

bb

0b bb


Fuzz

y m

embe

rshi

pFuzzy set

10

05

Fuzz

y m

embe

rshi

p

10

05


014

803

4

142

188

4

948

010

604

0

102

140

8

675

018

937

1

172

224

1

114

4

013

724

2

122

152

6

727

0

30

60

90

120


BOD

disc

harg

e (t)









176

2

399

733

523

207

2

501

944

752

286

9

439

122

6

687

367

2

755

132

1

994

0

10

20

30

40


BOD

disc

harg

e (t)


t = 1 t = 2


991

5

147

647

3

471

8

143

0

876

7

125

766

9

496

9

198

6

211

85

261

949

5

808

4

197

1

192

06

341

154

56

760

7

672

7

0

50

100

150

200

250


TP d

ischa

rge (

t)


t = 1 t = 2





0

10

20

30

40

50

60

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4


TN d

ischa

rge (

t)


t = 1 t = 2


481

94

203

86

127

85

643

766

7

908

53

393

93

263

03

275

8 116

82

0

300

600

900

1200

CP MWS PMC LH CF

Net

syste

m b

enefi

t (RM

Byen)

106






5 Conclusions















Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119879119875119894119905)


119894119905 ge 120582plusmn

(14h)

Cr

(119871119905sdot4

sum119903=1


119905


sum119895=1

9

sum119896=1

AMplusmn119895119896119905)




119905le APLplusmn

119905

ge 120582plusmn

(14i)



119879119875119904119905) le APWplusmn

119904119905 ge 120582plusmn

(14j)




119901119905)




119901119905 ge 120582plusmn

(14k)

9

sum119896=1


+ RFplusmn119895119896119905


119895119896119905le MPLplusmn

119895119905sdot TAplusmn119895119905

(14l)


9

sum119896=1

SLplusmn119895119896119905

sdot PAplusmn119895119896119905




plusmn

119895119896119905+ (1 minusNVMplusmn

119905)



sdot PAplusmn119895119896119905

ge 0(14n)



119895119896119905minus PRplusmn119895119896119905

sdot PAplusmn119895119896119905

ge 0 (14o)

9

sum119896=1



119895119896119905minusNRplusmn119895119896119905

sdot PAplusmn119895119896119905)


(14p)

9

sum119896=1


119895119896119905+ 120576plusmn119875119872

sdot AMplusmn119895119896119905


sdot PAplusmn119895119896119905)


(14q)

119871119905sdot4

sum119903=1



sum119895=1

9

sum119896=1

AMplusmn119895119896119905

ge 0

(14r)


4

sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905


sum119903=1


minus2

sum119905=1


(14s)

4

sum119895=1

9

sum119896=1

2

sum119905=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905


minus4

sum119903=1


sum119905=1


(14t)



119894119905le PLC

119894max (14u)


119903le NL119903max (14v)


119904119905le QW

119904max (14w)


119901119905le PLM

119901max (14x)


4

sum119895=1

CYplusmn119895119896119905

sdot PAplusmn119895119896119905



9

sum119896=1

PAplusmn119895119896119905

le TAplusmn119895119905 (14z)




ge 0(14aa)




Period119905 = 1 119905 = 2













































119905 wastewater







119894119905




















































119887 119887 119887

119905 = 1 119905 = 2 119905 = 1 119905 = 2 119905 = 1 119905 = 2










XL 15770 14331 17560 19245 19349 24159XH 8940 10419 9951 12053 10962 13687XC 10820 11074 12048 13825 13275 16576GP 14620 14003 16278 18199 17936 22395JJW 8980 10449 9997 12101 11013 13752SJS 8660 10219 9641 11740 10621 13261


TP 215 205 2445 244 274 283TN 1021 1060 1075 1070 1128 1180

4 Result Analysis










0b

Credibility

bb

0b bb


Fuzz

y m

embe

rshi

pFuzzy set

10

05

Fuzz

y m

embe

rshi

p

10

05


014

803

4

142

188

4

948

010

604

0

102

140

8

675

018

937

1

172

224

1

114

4

013

724

2

122

152

6

727

0

30

60

90

120


BOD

disc

harg

e (t)









176

2

399

733

523

207

2

501

944

752

286

9

439

122

6

687

367

2

755

132

1

994

0

10

20

30

40


BOD

disc

harg

e (t)


t = 1 t = 2


991

5

147

647

3

471

8

143

0

876

7

125

766

9

496

9

198

6

211

85

261

949

5

808

4

197

1

192

06

341

154

56

760

7

672

7

0

50

100

150

200

250


TP d

ischa

rge (

t)


t = 1 t = 2





0

10

20

30

40

50

60

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4


TN d

ischa

rge (

t)


t = 1 t = 2


481

94

203

86

127

85

643

766

7

908

53

393

93

263

03

275

8 116

82

0

300

600

900

1200

CP MWS PMC LH CF

Net

syste

m b

enefi

t (RM

Byen)

106






5 Conclusions















Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Period119905 = 1 119905 = 2













































119905 wastewater







119894119905




















































119887 119887 119887

119905 = 1 119905 = 2 119905 = 1 119905 = 2 119905 = 1 119905 = 2










XL 15770 14331 17560 19245 19349 24159XH 8940 10419 9951 12053 10962 13687XC 10820 11074 12048 13825 13275 16576GP 14620 14003 16278 18199 17936 22395JJW 8980 10449 9997 12101 11013 13752SJS 8660 10219 9641 11740 10621 13261


TP 215 205 2445 244 274 283TN 1021 1060 1075 1070 1128 1180

4 Result Analysis










0b

Credibility

bb

0b bb


Fuzz

y m

embe

rshi

pFuzzy set

10

05

Fuzz

y m

embe

rshi

p

10

05


014

803

4

142

188

4

948

010

604

0

102

140

8

675

018

937

1

172

224

1

114

4

013

724

2

122

152

6

727

0

30

60

90

120


BOD

disc

harg

e (t)









176

2

399

733

523

207

2

501

944

752

286

9

439

122

6

687

367

2

755

132

1

994

0

10

20

30

40


BOD

disc

harg

e (t)


t = 1 t = 2


991

5

147

647

3

471

8

143

0

876

7

125

766

9

496

9

198

6

211

85

261

949

5

808

4

197

1

192

06

341

154

56

760

7

672

7

0

50

100

150

200

250


TP d

ischa

rge (

t)


t = 1 t = 2





0

10

20

30

40

50

60

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4


TN d

ischa

rge (

t)


t = 1 t = 2


481

94

203

86

127

85

643

766

7

908

53

393

93

263

03

275

8 116

82

0

300

600

900

1200

CP MWS PMC LH CF

Net

syste

m b

enefi

t (RM

Byen)

106






5 Conclusions















Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































119887 119887 119887

119905 = 1 119905 = 2 119905 = 1 119905 = 2 119905 = 1 119905 = 2










XL 15770 14331 17560 19245 19349 24159XH 8940 10419 9951 12053 10962 13687XC 10820 11074 12048 13825 13275 16576GP 14620 14003 16278 18199 17936 22395JJW 8980 10449 9997 12101 11013 13752SJS 8660 10219 9641 11740 10621 13261


TP 215 205 2445 244 274 283TN 1021 1060 1075 1070 1128 1180

4 Result Analysis










0b

Credibility

bb

0b bb


Fuzz

y m

embe

rshi

pFuzzy set

10

05

Fuzz

y m

embe

rshi

p

10

05


014

803

4

142

188

4

948

010

604

0

102

140

8

675

018

937

1

172

224

1

114

4

013

724

2

122

152

6

727

0

30

60

90

120


BOD

disc

harg

e (t)









176

2

399

733

523

207

2

501

944

752

286

9

439

122

6

687

367

2

755

132

1

994

0

10

20

30

40


BOD

disc

harg

e (t)


t = 1 t = 2


991

5

147

647

3

471

8

143

0

876

7

125

766

9

496

9

198

6

211

85

261

949

5

808

4

197

1

192

06

341

154

56

760

7

672

7

0

50

100

150

200

250


TP d

ischa

rge (

t)


t = 1 t = 2





0

10

20

30

40

50

60

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4


TN d

ischa

rge (

t)


t = 1 t = 2


481

94

203

86

127

85

643

766

7

908

53

393

93

263

03

275

8 116

82

0

300

600

900

1200

CP MWS PMC LH CF

Net

syste

m b

enefi

t (RM

Byen)

106






5 Conclusions















Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119887 119887 119887

119905 = 1 119905 = 2 119905 = 1 119905 = 2 119905 = 1 119905 = 2










XL 15770 14331 17560 19245 19349 24159XH 8940 10419 9951 12053 10962 13687XC 10820 11074 12048 13825 13275 16576GP 14620 14003 16278 18199 17936 22395JJW 8980 10449 9997 12101 11013 13752SJS 8660 10219 9641 11740 10621 13261


TP 215 205 2445 244 274 283TN 1021 1060 1075 1070 1128 1180

4 Result Analysis










0b

Credibility

bb

0b bb


Fuzz

y m

embe

rshi

pFuzzy set

10

05

Fuzz

y m

embe

rshi

p

10

05


014

803

4

142

188

4

948

010

604

0

102

140

8

675

018

937

1

172

224

1

114

4

013

724

2

122

152

6

727

0

30

60

90

120


BOD

disc

harg

e (t)









176

2

399

733

523

207

2

501

944

752

286

9

439

122

6

687

367

2

755

132

1

994

0

10

20

30

40


BOD

disc

harg

e (t)


t = 1 t = 2


991

5

147

647

3

471

8

143

0

876

7

125

766

9

496

9

198

6

211

85

261

949

5

808

4

197

1

192

06

341

154

56

760

7

672

7

0

50

100

150

200

250


TP d

ischa

rge (

t)


t = 1 t = 2





0

10

20

30

40

50

60

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4


TN d

ischa

rge (

t)


t = 1 t = 2


481

94

203

86

127

85

643

766

7

908

53

393

93

263

03

275

8 116

82

0

300

600

900

1200

CP MWS PMC LH CF

Net

syste

m b

enefi

t (RM

Byen)

106






5 Conclusions















Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0b

Credibility

bb

0b bb


Fuzz

y m

embe

rshi

pFuzzy set

10

05

Fuzz

y m

embe

rshi

p

10

05


014

803

4

142

188

4

948

010

604

0

102

140

8

675

018

937

1

172

224

1

114

4

013

724

2

122

152

6

727

0

30

60

90

120


BOD

disc

harg

e (t)









176

2

399

733

523

207

2

501

944

752

286

9

439

122

6

687

367

2

755

132

1

994

0

10

20

30

40


BOD

disc

harg

e (t)


t = 1 t = 2


991

5

147

647

3

471

8

143

0

876

7

125

766

9

496

9

198

6

211

85

261

949

5

808

4

197

1

192

06

341

154

56

760

7

672

7

0

50

100

150

200

250


TP d

ischa

rge (

t)


t = 1 t = 2





0

10

20

30

40

50

60

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4


TN d

ischa

rge (

t)


t = 1 t = 2


481

94

203

86

127

85

643

766

7

908

53

393

93

263

03

275

8 116

82

0

300

600

900

1200

CP MWS PMC LH CF

Net

syste

m b

enefi

t (RM

Byen)

106






5 Conclusions















Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










176

2

399

733

523

207

2

501

944

752

286

9

439

122

6

687

367

2

755

132

1

994

0

10

20

30

40


BOD

disc

harg

e (t)


t = 1 t = 2


991

5

147

647

3

471

8

143

0

876

7

125

766

9

496

9

198

6

211

85

261

949

5

808

4

197

1

192

06

341

154

56

760

7

672

7

0

50

100

150

200

250


TP d

ischa

rge (

t)


t = 1 t = 2





0

10

20

30

40

50

60

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4


TN d

ischa

rge (

t)


t = 1 t = 2


481

94

203

86

127

85

643

766

7

908

53

393

93

263

03

275

8 116

82

0

300

600

900

1200

CP MWS PMC LH CF

Net

syste

m b

enefi

t (RM

Byen)

106






5 Conclusions















Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

10

20

30

40

50

60

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4

AZ1

AZ2

AZ3

AZ4


TN d

ischa

rge (

t)


t = 1 t = 2


481

94

203

86

127

85

643

766

7

908

53

393

93

263

03

275

8 116

82

0

300

600

900

1200

CP MWS PMC LH CF

Net

syste

m b

enefi

t (RM

Byen)

106






5 Conclusions















Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Mathematical Modeling for Water Quality ...downloads.hindawi.com/journals/jam/2013/731568.pdf · Research Article Mathematical Modeling for Water Quality Management