A Robust Neutrosophic Modeling and Optimization Approach for Integrated Energy-Food-Water Security Nexus Management under Uncertainty

2020·0.99 MB·

by  Firoz Ahmad

water
Article
A Robust Neutrosophic Modeling and Optimization Approach
for Integrated Energy-Food-Water Security Nexus Management
under Uncertainty
FirozAhmad1,2,†,‡ ,ShafiqAhmad 3,*,‡ ,MazenZaindin4andAhmadYusufAdhami1,‡
1 DepartmentofStatisticsandOperationsResearch,AligarhMuslimUniversity,Aligarh202002,India;
[email protected](F.A.);[email protected](A.Y.A.)
2 IndianStatisticalInstitute,203B.T.Road,Kolkata700108,India
3 IndustrialEngineeringDepartment,CollegeofEngineering,KingSaudUniversity,P.O.Box800,
Riyadh11421,SaudiArabia
4 DepartmentofStatisticsandOperationsResearch,CollegeofScience,KingSaudUniversity,P.O.Box2455,
Riyadh11451,SaudiArabia;[email protected]
* Correspondence:ashafi[email protected]
† Currentaddress:IndianStatisticalInstitute,203B.T.Road,Kolkata700108,India.
‡ Theseauthorscontributedequallytothiswork.
Abstract: Naturalresourcesareaboonforhumanbeings,andtheirconservationforfutureuses
isindispensable.Mostimportantly,energy-food-watersecurity(EFWS)nexusmanagementisthe
utmostneedofourtime.Aneffectivemanagerialpolicyforthecurrentdistributionandconservation
tomeetfuturedemandisnecessaryandchallenging.Thus,thispaperinvestigatesaninterconnected
anddynamicEFWSnexusoptimizationmodelbyconsideringthesocio-economicandenvironmental
objectiveswiththeoptimalenergysupply,electricityconversion,foodproduction,waterresources
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1) allocation,andCO2emissionscontrolinthemulti-periodtimehorizons.Duetoreal-lifecomplexity,
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
variousparametersaretakenasintuitionisticfuzzynumbers. Anovelmethodcalledinteractive
Citation:Ahmad,F.;Ahmad,S.; neutrosophicprogrammingapproach(INPA)issuggestedtosolvetheEFWSnexusmodel.Toverify
Zaindin,M.;Adhami,A.Y.ARobust andvalidatetheproposedEFWSmodel,asyntheticcomputationalstudyisperformed.Theobtained
NeutrosophicModelingand solution results are compared with other optimization approaches, and the outcomes are also
OptimizationApproachfor evaluatedwithsignificantpracticalimplications.Thestudyrevealsthatthefoodproductionprocesses
IntegratedEnergy-Food-Water requiremorewaterresourcesthanelectricityproduction,althoughrecycledwaterhasnotbeenused
SecurityNexusManagementunder forfoodproductionpurposes.Theuseofacoal-firedplantisnotaprominentelectricityconversion
Uncertainty.Water2021,13,121.
source.However,naturalgaspowerplants’serviceisalsooptimallyexecutedwithamarginalrateof
https://doi.org/10.3390/w13020121
production.Finally,conclusionsandfutureresearchareaddressed.Thiscurrentstudyemphasizes
howtheproposedEFWSnexusmodelwouldbereliableandbeneficialinreal-worldapplicationsand
Received:28November2020
helppolicy-makersidentify,modify,andimplementtheoptimalEFWSnexuspolicyandstrategies
Accepted:6January2021
forthefutureconservationoftheseresources.
Published:7January2021
Publisher’sNote: MDPIstaysneu- Keywords: Energy-Food-Watersecuritynexusmanagement; robustmodelingandoptimization
tralwithregardtojurisdictionalclai- method; intuitionistic fuzzy parameters; interactive neutrosophic programming approach; opti-
msinpublishedmapsandinstitutio- malsocio-economicandenvironmentalprotectionpolicies
nalaffiliations.
1. Introduction
Copyright:©2021bytheauthors.Li-
Naturalresourcesarepreciousandessentialforhumanlives. Primarily,energy,food,
censee MDPI, Basel, Switzerland.
andwater(EFW)arethebasicneedsfordailylifeandhumansurvival. Arapidincrease
Thisarticleisanopenaccessarticle
inthepopulationsignificantlyenhancestheseresources’demand,causingdepreciation
distributedunderthetermsandcon-
ditionsoftheCreativeCommonsAt- anddepletionintheseresources’utilityrates. Asaresult,theacquisitionandconservation
tribution (CC BY) license (https:// oftheseresourcesbecomeindispensabletomeetfutureneeds. Theseresourcesareinter-
creativecommons.org/licenses/by/ relatedandaffecteachother,eitherdirectlyorindirectly.Thus,theinter-dependencenature
4.0/).
Water2021,13,121.https://doi.org/10.3390/w13020121 https://www.mdpi.com/journal/water
Water2021,13,121 2of27
ofEFWresourceshasbeenexploredasasortofcompositenetwork,calledenergy-food-
water security (EFWS) nexus management. According to the EIA report, [1] in the US,
approx. 89%oftheelectricitywasgeneratedbycoalandnatural-gas-firedpowerplants,
whereas a massive amount of water is consumed for pumping and cooling purposes.
Presently,about1.29billionpeoplethroughouttheworldarenotaccessingelectricity[2].
Tomeetsuchamassiveelectricitydemand,theaccumulationofwaterresourcesforcoal-
firedpowerplantsandthefuelextractionprocesswillbechallenging.Moreover,population
growthisexpectedtoincreaseconcerningfooddemand,about53%by2050[3]. Itcompels
theneedtorealizeandidentifytheinteractionsamongvariousresourcesrequirementsto
fulfillfuturedemandsaffectingtheEFWSnexussystem’scomposition.
According to [3]’s report, food production and supply purpose consume approx.
31%ofglobalenergy.AnincreaseinthemasspopulationresultsinhigherdemandforEFW
resourcesdrastically[3]. By2050,apredictiveanalysisindicatesthattheworldpopulation
willincreasebyabout49%[4].Furthermore,itisnoticedthatintegratedEFWSnexusmodel
arecomplexanddynamicduetonotonlytheinter-dependenciesofthesethreeresources
that influence each other at various level, but also the involvement of different factors
suchaseconomic,social,andenvironmental,etc. whichmadeitmorecomplicated. Nexus
managementisgainingmoreattentionduetoitsprimeconcernsaboutenergy,food,and
watersecurity. ItisexpectedthattheproposedEFWSnexusmodelwouldbewell-enough
toaddressandanswerthemostcriticalqueries,suchas“optimaldistributionofresources,”
“balancingthetrade-offbetweentotaleconomiccostandCO emissions”,“evaluatingand
2
designingthepoliciesandnecessaryinitiatives.” Quantitativeoptimizationmodelsare
powerfultoolsinidentifying,assessing,andsummarizingthecriticalfactorsandprovides
globallyacceptableoutcomestofacilitatethedecisionsandrelevantprocesses.
Uncertain parameters are often encountered in the EFWS decision-making model.
Unlikefuzzyandrandomparameters,uncertainparametersaredepictedastriangularor
trapezoidalintuitionisticfuzzynumbers. Afuzzyparameteronlydealswiththedegreeof
belongingness(acceptance)oftheelementintoafeasiblesolutionset. Itdoesnotconsider
thedegreeofnon-belongingness(rejections)oftheelementintothesamefeasiblesolution
set,anintegratedpartofthedecision-makingprocesses. Furthermore,uncertaintydue
torandomnessisindicatedwithrandomparameters. Sometimes,itmaynotbepossible
to have historical data for which the random parameters are estimated. According to
somespecifiedprobabilitydistributionfunction,theforecastingpatternandparameters
estimation of random variables is much dependent on the behavior and nature of the
historical data. An intuitionistic fuzzy parameter deals with the degree of belonging-
ness (acceptance) and degree of non-belongingness (rejections) of the element into the
same feasible solution set, simultaneously. For example, if the decision maker intends
toquantifythecostoftransportationwithsomeestimatedvalue,suchastransportation
costfromwaterresourceszonestoelectricityconversionplantis54$,thenthemostlikely
estimatedintervalwouldbe50–58$,alongwithsomehesitationdegreethatmaybegivenas
48–60$,whichensureslessviolationofriskswithdegreeofacceptanceandnon-acceptance.
Also, there is no scope for the historical data while dealing with intuitionistic fuzzy
parameters(Ahmadetal.[5]). Thus,theprimemotivebehindtheselectionofintuition-
isticfuzzyparametersistoavoidtheshortcomingsoffuzzyvaguerandomparameters.
ThereforethequantitativeandanalyticalstudyofEFWSnexusmanagementnetworkhas
theutmostneedforthetimeinbalancingtheequilibriumamongoptimalresourceuse,
socio-economicandenvironmentalobjectives,anddesigningfruitfulpolicies. Manypast
studiesconfronttheresearchdomainofEFWSnexus. Still,mostofthemareconfinedto
either individual resource management fields such as water sector, energy-conversion,
agricultural production, or two sectors such as water-energy nexus, particular prob-
lemssuchaswateruses,ortheoreticalandconceptualdevelopmentoftheEFWSnexus.
This study has unified all three resources and the socio-economic and environmental
objectivesatasingleplatform.
Water2021,13,121 3of27
Aninteractiveneutrosophicprogrammingapproach(INPA)isdevelopedtosolve
theproposedEFWSnexusmodel. ThediscussedINPAmanagesindeterminacyorneutral
thoughtswhilemakingdecisionsformultiobjectiveoptimizationproblems. Neutralityis
theregionofpropositions’valuesnegligence,whichexistsbetweentruthandfalsitydegree.
Thus, theneutrosophicdecisionsettheoryisamoregeneralizedandflexibleapproach
thanthefuzzyandintuitionisticfuzzysettheoryduetoindeterminacydegree. Therefore,
theproposedsolutionapproachcanalsoberegardedasacontributiontotheoptimization
techniquedomain.
Themainaimandobjectiveofthispaperaretoexploreandhighlightanintegrated
modeling and optimizing system for EFWS nexus management. A useful quantitative
andanalyticalmodelwouldestablishthetrade-offsamongobjectivesandconstraintsto
achievethesocio-economicdemandsandenvironmentalimpactswhileusingtheenergy,
food,andwaterresources. TheproposedEFWSmodelinherentlyestimatesthevarious
costsassociatedwiththelimitedEFWresources,socio-economicdemands,andCO emis-
2
sionsabatementoverthemulti-periodplanninghorizons. Itisalsocapableofdealingwith
moregeneralizeduncertaintyduetohesitationwhiledepictingtheparameters. Various
inter-connected components such as production, distribution, and consumption of the
EFWSnexusmodelarequantitativelyandanalyticallyexamined. TheproposedEFWS
nexusmodel’smainadvantagecanberealizedbyhavingitbeconsistentwithandcompe-
tentcomparedwithreal-lifemanagementproblems. Thecurrentstudyalsodemonstrates
howtheseresourcescanbeusedoptimallyandhowtheycanbeconservedtomeetfuture
demand. Practical implications are well addressed based on obtained solution results.
Post-optimalityanalysesareperformedtogenerateandidentifythemostpromisingsolu-
tionsunderdifferentriskfactors. Therefore,thepresentedstudycanhelpdecision-makers
designtheoptimalpoliciesandstrategiesforEFWSnexusmanagementunderuncertainty.
2. LiteratureReview
Over recent decades, optimal use of resources has made significant impacts in re-
search and development fields to identify, quantify, and analyze the decision-makers’
policiesandstrategieswhilemanagingtheEnergy-Water-Foodnexusmanagementsys-
tem. ThemosteffectiveandindispensablemodelingtextureoftheEFWSnetworkunder
various uncertainties was developed by many authors, practitioners, and researchers.
Bieberetal.[6]developedamethodologyandoutlookforresilientandsustainablesup-
plychainplanningintheenergy-water-foodnexusatthecity-regionlevel. Uenetal.[7]
proposedaholisticapproachforthesynergeticmanagementoftheWater-Energy-Food
nexusandappliedNSGA-IItosolvethemodel. Tsolasetal.[8]discussedanovelwater-
energy nexus optimization and presented a case study at the country and state level.
Namanyetal.[9]alsoaddressedaresourcesmanagement-orientedEnergy-water-Food
nexus optimization model with a real-life application. Gao et al. [10] introduced a wa-
ter–food–energynexusmanagementmodelforminimizingthecost-coalandproduction-
cost of agricultural products. Zhang et al. [11] attempted to develop an assessment-
optimizationmethodologyforWater-Energy-Foodnexussynergiesandconcludedwater
supplyasacriticalfactorinthemodelingapproach. Lietal.[12]exploredthesynergies
withintheWater-Energy-FoodnexussystemandpresentedasacasestudyShenzhencity
using the proposed model. Hamidov and Helming [13] presented a theoretical study
ontheever-increasingliteratureontheWEFnexusmanagementalongwithconcluding
remarks. Saifetal.[14]discussedanintegratedenergy-water-foodsupplychainoptimiza-
tionmodelwithacasestudy. Jietal.[15]addressedacrop-biomassproductionplanning
problemwithfood-energy-waternexusunderuncertainties. Thedevelopedmodelassists
thedecision-makersinadoptingadequatepoliciesandstrategies. Okolaetal.[16]designed
amultiobjectiveoptimizationframeworktohighlightthecomplexitylinkedwithFood-
Energy-WaterNexusandsuggestedtheoptimalproductionandresourceconsumption
policiesoverdifferentplanninghorizons.
Water2021,13,121 4of27
Manyresearchersdiscussedtheirworkbyincorporatingdifferentsortsorformsof
uncertaintyinthewater,energy,andfoodnexusoptimizationmodelasfarasuncertainties
areconcerned. Wesummarizedsomerecentdevelopmentsandadvancedstudiesinthe
uncertaintydomainintheEFWSnexusmodel. Lietal.[17]alsodevelopedanintegrated
agriculturalwater-energy-foodnexusmanagementmodelunderfuzzydemandsforthe
resourcesandappliedtoacasestudydata-set. Lietal.[18]suggestedawater-food-energy
nexusoptimizationoutlookforirrigatedagricultureunderdualstochasticuncertaintyof
available water resources and validated the model based on a real case study. Li et al.
[19]addressedtheoptimization-assessmentofenergy,food,water,andlandframework
forbioenergyproductionininterval-valueduncertaintyanddemonstratedacasestudy
in northeast China. Yu et al. [20] explored a copula-based fuzzy interval-random pro-
grammingapproachunderjoint-riskandimplementedareal-worldcasestudyinHenan
Province, China. Elsayed et al. [21] studied a Nile Water-Food-Energy Nexus manage-
mentsystemanddepictedatrade-offsopportunityamongtheresourcesusepoliciesto
acasestudyinSudan. TanandZhang[22]suggestedarobustfractionalprogramming
approachtoaddresstheagriculturalwatermanagementmodelundertheriskuncertainty
andvalidatedusingsimulationsandcomparisonswithexistingalternatives. Hurford[23]
alsoidentifiedandassessedtheimpactsofwater-energy-foodsecurityindevelopingcoun-
tryandconcludingremarksaremadebasedonthestudy. GovindanandAl-Ansari[24]
discussedaflexiblecomputationalframeworkforintegratedenergy-water-foodsecurity
underriskfactorsandperformedacasestudyoftheagriculturalsectorintheStateofQatar.
Jietal.[25]developedaninexacthybridmodelforfood-energy-waternexusmanagement
underthemixed-characteruncertaintiesandpresentedpracticalimplicationsbasedonthe
casestudyinShandong,China. Yuetal.[26]propoundedaneffectivewater-energy-food
nexusplanningsystem,andamulti-levelintervalfuzzycredibility-constrainedprogram-
mingissuggestedtodealwiththeuncertaintiesandthenimplementedonareal-casestudy
inHenanProvince.Furthermore,Hangetal.[27]designedthelocalfood,energyandwater
productionsystembasedonthenexusconcept. Martinez-Hernandezetal.[28]developed
thenewsoftwaretoolfortechno-ecologicalsimulationoflocalfood-energy-watersystems.
Also,anintegratednexusmodelingnetworkisscarceintheliterature,whichcanintro-
duceandimplementtheenergy,food,andwaterresourcesefficiently. Inassociationwith
socio-economicobjectives,incorporatingenvironmentalprotectionsintoarobustmodeling
andoptimizationframeworkisalsotheneedforthecurrentstate-of-art. TheEFWSnexus
management perspective is necessary for the quantitative and qualitative study of the
complexinterconnectednetworkandevaluatingtheoutcomesforoptimalpoliciesand
developmentforthewholesystem.
3. DevelopmentofEnergy-Food-WaterSecurityNexusModel
TheproposedEFWSmodelisformulatedtooptimizeandmanagetheavailableenergy,
food,andwaternexusresourcesinintuitionisticfuzzyuncertaintyoverdifferentperiods.
Thevariouscostsassociatedwiththeoperations,production,supply,andCO emissions
2
abatementarecontemplatedoverthesocio-economicandenvironmentallyorientedobjec-
tive. ThequantityofCO emissionsisalsoachallengingtaskfromtheenvironmentalpoint
2
ofview. Thesevereimpactofsuchtoxicemissionsaffectsbothfloraandfauna. Thus,the
developedmodelintegratesandadherestoEFWSnexusmanagement’sessentialaspectsby
undertakingthevagueuncertaintyamongdifferentparameters. Theinter-connectedcom-
ponentintheproposedEFWSnexusmanagementmodelisshowninFigure1,whichalso
signifiestheresearchdomainofthisstudy. Theusefulnotionsanddescriptionsusedinthe
proposedEFWSmodelaredepictedinTable1. Theoptimalallocationoftheenergy-supply
quantityofcoalandnaturalgas,theavailablecapacityofthepowerplanttoproduceelec-
tricity,amountofgroundandsurfacewaterrequiredforfoodproduction,thevolumeof
ground,surface,andrecycledwaterneededforelectricityproduction,andsocio-economic
demandsforEFW(Energy,FoodandWater)turnoverinawell-specifiedproductionplan-
ningperiodsaretheprimecharacteristicfeaturesoftheproposedEFWSmodel. Thefirst
Water2021,13,121 5of27
objectiveisdevelopedtoreducethetotaleconomiccostcomprisingenergy-supplycostsfor
electricityproduction,costsofelectricity-production,costsofwater-delivery,costsoffood
production,andfinallyCO emissionsabatementcostsovertimehorizons. Thesecond
2
objectivemitigatestheCO emissionsproducedduringelectricityconversionandfood
2
production,respectively. Alltherelatedparametersaretakenastriangularintuitionistic
fuzzyandresolvedintotheircrispformusingarobustrankingfunction(SeeSection5.1).
Figure1.Inter-dependencerelationshipsofEnergy-Food-Watersecuritynexusmanagementsystem.
Table1.Notionsanddescriptions.
Indices Descriptions
e Associatedwiththeenergy-production
f Associatedwiththefoodproduction
j ∈ J Denotesthetypesofavailableenergysupplyandpowerconversionplant
t ∈ T Representstheeffectiveplanningtimeperiodssuchasatimehorizonforenergy, foodand
watersecuritymanagement
w Associatedwiththewater-resourcesuse
Decisionvariables
es Amountofavailableenergysupplyjintimeperiodt(PJ)
j,t
x Amountofelectricityconversionbyapowerplantousingenergysupplyjintimeperiodt(PJ)
j,t
f
gw Quantityofgroundwaterconsumedinfoodproductionprocessesintimeperiodt(bbl)
t
f
sw Quantityofsurfacewaterconsumedinfoodproductionprocessesintimeperiodt(bbl)
t
gwe Groundwaterquantitydeliveredtopower-conversionplantjintimeperiodt(bbl)
j,t
swe Surfacewaterquantitydeliveredtopower-conversionplantjintimeperiodt(bbl)
j,t
rwe Amountofrecycledwatersenttopower-conversionplantjintimeperiodt(bbl)
j,t
fo Totalquantityoffoodproductionintimeperiodt(ton)
t
Water2021,13,121 6of27
Table1.Cont.
Parameters
(cid:93)
esc Averagecostassociatedwithenergysupplyjintimeperiodt(million$/PJ)
j,t
f(cid:102)cj Fixedcostassociatedwithpowerconversionplantj(million$)
pc Averageoperationalcostincurredoverelectricityproductionbypowerconversionplantjin
(cid:103)j,t
timeperiodt(million$/PJ)
(cid:93)
f
cgw Totalcostrelatedtogroundwatersupplyforfoodproductionintimehorizont($/bbl)
t
(cid:94)
cgwe Totalcostrelatedtogroundwatersupplyforpowerconversionplantjintimehorizont($/bbl)
j,t
(cid:93)
f
csw Totalcostrelatedtosurfacewatersupplyforfoodproductionintimehorizont($/bbl)
t
(cid:93)
cswe Totalcostrelatedtosurfacewatersupplyforpowerconversionplantjintimehorizont($/bbl)
j,t
(cid:93)
crwe Totalcostrelatedtorecycledwatersupplyforpowerconversionplantjintimehorizont($/bbl)
j,t
c(cid:103)fot Unitcostassociatedwithfoodproductionintimeperiodt(million$/ton)
cea TotalcostofCO emissionabatementforelectricityproductionintimeperiodt($/kg)
(cid:103)t 2
cc UnitsofCO emissionperunitofelectricityproductionbypowerconversionplantjintime
(cid:103)j,t 2
periodt(millionkg/PJ)
c(cid:103)fat TotalcostofCO2emissionabatementforfoodproductionintimeperiodt($/ton)
f(cid:102)ft UnitsofCO2emissionperunitoffoodproductionintimeperiodt(millionkg/ton)
f(cid:103)ej,t Unitenergycarrierperunitofelectricityproductionfortransformationtechnique j intime
periodt(PJ/PJ)
av Totalavailableenergysupplyjintimeperiodt(PJ)
(cid:103)j,t
(cid:94)
f
aer Maximumunitofelectricityavailableforfoodproductionintimeperiodt(PJ)
t,max
(cid:94)
aerw Maximumunitofelectricityavailableforwatercollection,treatmentandsupplyintimeperiod
t,max
t(PJ)
e(cid:102)rf Demandofunitenergyforfoodproductionintimeperiodt(PJ/ton)
t
e(cid:102)rw Demandofunitenergyforwatercollection,treatmentandsupplyintimeperiodt(PJ)
t
d(cid:102)ee Socio-economicdemandforenergy(orelectricity)intimeperiodt(PJ)
t
d(cid:102)ef Socio-economicdemandforfoodintimeperiodt(ton)
t
w(cid:103)ft Unitsofwaterconsumptionperunitoffoodproductionintimeperiodt(bbl/ton)
sy Maximumquantityofavailablegroundwatersafeyieldintimeperiodt(bbl)
(cid:102)t
asw Maximumquantityofavailablesurfacewaterintimeperiodt(bbl)
(cid:103)t
arw Maximumquantityofavailablerecycledwaterintimeperiodt(bbl)
(cid:103)t
(cid:93)
tmcc MaximumpermissableCO emissionduringplanningperiods(millionton)
2
δ Lossfactorofwatersupplytothefoodsub-system
χ Unitwaterdemandperunitofelectricityproductionbypowerconversionplantj(bbl/GWh)
j
φ Lossfactorofwatersupplytothepowerconversionplantj
j
θ Average efficiency factor for CO emission abatement by power conversion plant j in time
j,t 2
periodt
3.1. ObjectiveFunctions
Thefirstobjectiveistominimizetotaleconomiccost,includingcostsofenergysupply
forelectricityproduction,electricityconversioncosts,costsofwatersupply,costsoffood
production,andcostsofCO emissionabatement. Thetotalsumofallcostsisdepicted
2
in(1).
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J T T J T T (cid:18) (cid:93) (cid:93)(cid:19)
MinO1 =∑ ∑esj,t×e(cid:93)scj,t+ ∑ f(cid:102)cj+ ∑ ∑ xj,t×p(cid:103)cj,t+ ∑ gwtf ×cgwtf +swtf ×cswtf
j=1t=1 t=1 j=1t=1 t=1
+ ∑J ∑T (cid:16)gwte×c(cid:93)gwte+swte×c(cid:103)swte+rwej,t×c(cid:93)rwej,t(cid:17)+ ∑T fot×c(cid:103)fot (1)
j=1t=1 t=1
J T
+ ∑ ∑ xj,t×c(cid:93)eaj,t×c(cid:103)cj,t+ fot×c(cid:103)fat× f(cid:102)ft
j=1t=1
ThesecondobjectivefunctionrepresentstheminimizationofCO emissionproduced
2
duringtheelectricityconversionandfoodproduction,respectively. Thus,theminimization
ofnetCO emissionispresentedin(2).
2
J T (cid:16) (cid:17)
∑ ∑
MinO2 = θj,t×c(cid:103)cj,t+ f(cid:102)ft (2)
j=1t=1
3.2. Constraints
Theconstraint(3)representsthattheproduced-electricitybyeachpowerconversion
plantineachtimeperiodmustnotbegreaterthantheenergy-supplyconversionquantity.
xjt× f(cid:103)ej,t ≤ esj,t ∀j,t (3)
Theconstraint(4)ensuresthatthedeliveredfossilfuelssuchascoalandnaturalgas
mustbelessthantheirmaximumavailablequantityoverthetimeperiods.
es ≤ av ∀j,t (4)
j,t (cid:103)j,t
Theconstraint(5)representsthatthesuppliedenergyforfoodproductionmustbe
lessthanitsmaximumpermissibleelectricitysupplyineachtimeperiod.
fo ×erf ≤ aerf ∀j,t (5)
t (cid:101)t (cid:102)t,max
Theconstraint(6)ensuresthattherequiredquantityofelectricityforwatercollection,
treatmentandsupplypurposemustbelessthanitsmaximumallocatedamount.
(cid:32) (cid:33)
J (cid:16) (cid:17)
erw× gwf +swf + ∑ gwe +swe +rwe ≤ aerw ∀j,t (6)
(cid:101)t t t j,t j,t j,t (cid:102)t,max
j=1
Theconstraint(7)ensuresthattheproducedelectricityfrompowerconversionplants
mustmeetthesocio-economicdemandsofelectricityafterdeliveringforfoodproduction
andwatercollection,treatmentandsupplypurpose.
(cid:32) (cid:33)
∑J xj,t−e(cid:101)rtf × fot−e(cid:101)rwt × gwtf +swtf + ∑J (cid:16)gwej,t+swej,t+rwej,t(cid:17) ≥ d(cid:101)eet ∀j,t (7)
j=1 j=1
Theconstraint(8)representsthatnetwatersupplyfromallsourcesmustmeetthe
waterdemandforthefoodproductionineachtimeperiod.
(cid:16) (cid:17)
(1−δ)× gwtf +swtf ≥ fot×w(cid:102)ft ∀j,t (8)
Theconstraint(9)representsthatnetwatersupplyfromallsourcesmustmeetthe
waterrequirementforelectricityproduction.
(cid:0)1−φ (cid:1)×(cid:16)gwe +swe +rwe (cid:17) ≥ χ ×x ∀j,t (9)
j j,t j,t j,t j j,t
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Theconstraint(10)ensuresthatthedeliverdgroundwaterquantitymustbelessthan
orequalstothemaximumavailablegroundwaterquantityineachtimeperiods.
J
gwf + ∑gwe ≤ sy ∀j,t (10)
t j,t (cid:102)t
j=1
Theconstraint(11)ensuresthatthesuppliedsurfacewaterquantitymustbelessthan
orequalstothemaximumavailablesurfacewaterquantityineachtimeperiods.
J
swf + ∑swe ≤ asw ∀j,t (11)
t j,t (cid:103)t
j=1
Theconstraint(12)ensuresthatthedeliverdrecycledwaterquantitymustbelessthan
orequalstothemaximumavailablerecycledwaterquantityineachtimeperiods.
J
∑rwe ≤ arw ∀j,t (12)
j,t (cid:103)t
j=1
The constraint (13) ensures that the produced food must meet the socio-economic
demandforfood.
fo ≥ d(cid:102)ef ∀j,t (13)
t t
The constraint (14) ensures that the produced CO quantity must be less than the
2
maximumpermissibleCO emissionsduringthetimeperiods.
2
J T T
∑ ∑ (cid:0) (cid:1) ∑ (cid:93)
xj,t×c(cid:103)cj,t 1−θj,t + fot×(cid:102)fft ≤ tmcc ∀j,t (14)
j=1t=1 t=1
Theconstraint(15)depictsthenon-negativityrestrictionsoverallthedecisionvariables.
es ,x ,gwf,swf,gwe ,swe ,rwe , fo ≥0 ∀j,t (15)
j,t j,t t t j,t j,t j,t t
where (·) representtheintuitionisticfuzzyparametersinvolvedintheEFWSoptimiza-
(cid:101)
tionmodel.
Allthediscussedmultiobjectiveoptimizationmethodsinthesupplementarymaterials
arebasedonfuzzydecisionsettheory. Inthefuzzyoptimizationtechnique,eachobjective
function’smembershipfunctionsaremaximizedtoachievetheoptimalglobalsolution.
Sometimes,theneutralthoughtsorindeterminacydegreeinevitablyexistswhilemaking
decisions. Thefuzzysetcannotbeappliedtotackletheneutralideasduetotheabsenceof
anindeterminacydegree. Hencefuzzyprogrammingisalsonotapplicableforthiscase.
Thebetterrepresentativeofsuchanindeterminatesituationcanbehandledbyneutrosophic
decisionsettheoryefficiently. Thus,theproposedINPAcontemplatesdifferentaspects
ofeachobjectives’marginalevaluationsbyhavingthetruth,indeterminacy,andafalsity
degreeatatime. Thein-depthuncertaintyquantificationtechniqueunderneutrosophicset
theorymakesitmorepowerfulandpromisinginyieldingabettersolutionthanthefuzzy
settheory.
3.2.1. ProposedInterativeNeutrosophicProgrammingApproach(INPA)
The real-life complexity most often creates the indeterminacy situation or neutral
thoughtswhilemakingoptimaldecisions. Apartfromtheacceptanceandrejectionde-
greesinthedecision-makingprocess,theindeterminacydegreealsohasmuchimportance.
Thustocovertheneutralthoughtsorindeterminacydegreeoftheelementintothefeasible
solutionset,Smarandache[29]investigatedaneutrosophicset. Thename“neutrosophic”
istheadvancecombinationoftwoexplicitterms,namely;“neutre”extractedfromFrench
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means,neutral,and“sophia”adoptedfromGreekmeans,skill/wisdom,thatunanimously
providethedefinition“knowledgeofneutralthoughts”(seeSmarandache[29],Ahmadand
Adhami[30],Ahmadetal.[31],AdhamiandAhmad[32]). TheNSconsidersthreesorts
ofmembershipfunctions,suchastruth(degreeofbelongingness),indeterminacy(degree
ofbelongingnessuptosomeextent),andafalsity(degreeofnon-belongingness)degrees
into the feasible solution set. The idea of independent, neutral thoughts differs from
the NS with all the uncertain decision sets such as FS and IFS. The updated literature
worksolelyhighlightsthatmanypractitionersorresearchershavetakenthedeepinter-
estintheneutrosophicresearchfield(see, AhmadandAdhami[33], Ahmadetal.[34],
Ahmadetal.[5]). TheNSresearchdomainwouldgetexposureinthefutureandassistin
dealingwithindeterminacyorneutralthoughtsinthedecision-makingprocess. Thisstudy
also fetches the novel ideas of neutrosophic optimization techniques based on the NS.
Anovelinteractiveneutrosophicprogrammingapproachisdevelopedtosolvethemulti-
objectiveEFWSmodelunderintuitionisticfuzzyparameters. Themarginalevaluationof
eachobjectivefunctionisquantifiedbythetruth,indeterminacy,andfalsitymembership
functionsundertheneutrosophicdecisionset. Thus,NSplaysavitalroleinoptimizing
multiobjective optimization problems by incorporating, executing, and implementing
neutralthoughts.
Definition 1. Ahmad et al. [5] Let there be a universal discourse Y such that y ∈ Y, then a
neutrosophicset AinYcanbedepictedbytruthµ (y),indeterminacyσ (y)andafalsityν (y)
A A A
membershipfunctionsinthefollowingform:
A = {< y, µ (y), σ (y), ν (y) > |y ∈Y}
A A A
where µ (y), σ (y) and ν (y) are real standard or non-standard subsets belong to ]0−,1+[,
A A A
alsogivenas,µ (y) :Y →]0−,1+[,σ (y) :Y →]0−,1+[,andν (y) :Y →]0−,1+[. Thereis
A A A
norestrictiononthesumofµ (y), σ (y)andν (y),sowehave
A A A
0− ≤ supµ (y)+σ (y)+ supν (y) ≤3+
A A A
BellmanandZadeh[35]firstpropoundedtheideaofafuzzydecisionset. Afterthat,
itiswidelyadoptedbymanyresearchers. Thefuzzydecisionconceptcomprisesfuzzy
decision(D),fuzzygoal(G),andfuzzyconstraints(C),respectively.Herewerecallthemost
extensivelyusedfuzzydecisionsetwiththeaidofthefollowingmathematicalexpressions:
D =O∩C
Consequently,wealsodepicttheneutrosophicdecisionsetD ,whichcontemplate
N
overneutrosophicobjectivesandconstraintsasfollows:
D = (∩K O )(∩I C) = (x, µ (x), σ (x), ν (x))
N k=1 k i=1 i D D D
where
(cid:26) (cid:27)
µ (x),µ (x),...,µ (x)
µ (x) = min O1 O2 Ok ∀ x ∈ X
D µ (x),µ (x),...,µ (x)
C1 C2 Ci
(cid:26) (cid:27)
σ (x),σ (x),...,σ (x)
σ (x) = max O1 O2 Ok ∀ x ∈ X
D σ (x),σ (x),...,σ (x)
C1 C2 Ci
(cid:26) (cid:27)
ν (x),ν (x),...,ν (x)
ν (x) = max O1 O2 Ok ∀ x ∈ X
D ν (x),ν (x),...,ν (x)
C1 C2 Ci
where µ (x),σ (x) and ν (x) are the truth, indeterminacy and a falsity membership
D D D
functionsofneutrosophicdecisionsetD respectively.
N
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Theboundsfork-thobjectivefunctionundertheneutrosophicenvironmentcanbe
obtainedasfollows:
Uµ =U , Lµ = L fortruthmembership
k k k k
Uσ = Lµ+s , Lσ = Lµ forindeterminacymembership
k k k k k
Uν =Uµ, Lν = Lµ+t for falsitymembership
k k k k k
wheres andt ∈ (0,1)arepredeterminedrealnumbersprescribedbydecision-makers.
k k
The linear-type truth µ (O (x)), indeterminacy σ (O (x)) and a falsity ν (O (x))
k k k k k k
membershipfunctionsunderneutrosophicenvironmentcanbefurnishedasfollows:
µ (O (x)) =  1Ukµ−Ok(x) iiff OLµk(≤x)O≤(Lxµk) ≤Uµ (16)
k k  0Ukµ−Lµk if Ok(x) ≥k Uµ k
k k
 1 if O (x) ≤ Lσ
σ (O (x)) =  Ukσ−Ok(x) if Lσk≤O (xk) ≤Uσ (17)
k k Uσ−Lσ k k k
 0 k k if O (x) ≥Uσ
k k
 0 if O (x) ≤ Lν
ν (O (x)) =  Ok(x)−Lνk if Lνk≤O (xk) ≤Uν (18)
k k Uν−Lν k k k
 1 k k if O (x) ≥Uν
k k
IfforanymembershipL =U ,thenthevalueofthesemembershipwillbeequalto1.
k k
IntroducingtheideaofBellmanandZadeh[35],wemaximizetheoverallachievement
functiontoreachtheoptimalsolutionofeachobjectives. Themathematicalexpressionfor
achievementfunctionisdefinedasfollows:
Max mink=1,2,3,...,K µk(Ok(x))
Min maxk=1,2,3,...,K σk(Ok(x))
Min maxk=1,2,3,...,K νk(Ok(x)) (19)
subjectto
alltheconstraintsof(24)
Also,assumethatµ (O (x)) ≥ α,σ (O (x)) ≤ βandν (O (x)) ≤ γ,forallk.
k k k k k k
Withtheaidofauxiliaryparametersα, βandγ,theproblem(19)canbetransformed
intothefollowingproblem(20):
(INPA) Maxψ(x) = η(α−β−γ)+(1−η)∑K (µ (O (x))−σ (O (x))−ν (O (x)))
k=1 k k k k k k
subjectto
µ (O (x)) ≥ α,
k k
σ (O (x)) ≤ β,
k k (20)
ν (O (x)) ≤ γ,
k k
α ≥ β, 0≤ α+β+γ ≤3,
α,β,γ ∈ [0,1]
alltheconstraintsof(24)
where η is the compensation co-efficient between the overall satisfaction level and the
sumofindividualmarginalevaluationofeachobjectivefunctioninneutrosophicenvi-
ronment. Thus,thedevelopmentofproposedINPA(20)hasanewachievementfunction
whichisrepresentedbyaconvexcombinationofdifferencesamongtheboundsfortruth,
indeterminacy,andfalsitydegreesofobjectivefunction(α−β−γ),andthesumofdif-
ferencesamongtheseachievementdegrees(µ (O (x))−σ (O (x))−ν (O (x)))tomake
k k k k k k
suregeneratinganestablishedbalancedcompromisesolution.