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ThirdInternationalConferenceonProductionResearchThirdInternationalConferenceonProductionResearchThirdInternationalConferenceonProductionResearchThirdInternationalConferenceonProductionResearch– –– –
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EVALUATIONOFHOLTEVALUATIONOFHOLTEVALUATIONOFHOLTEVALUATIONOFHOLT----WINTERSMODELSINTHESOLIDWINTERSMODELSINTHESOLIDWINTERSMODELSINTHESOLIDWINTERSMODELSINTHESOLID
RESIDUAFORECASTING:ACASESTUDYINTHECITYOFRESIDUAFORECASTING:ACASESTUDYINTHECITYOFRESIDUAFORECASTING:ACASESTUDYINTHECITYOFRESIDUAFORECASTING:ACASESTUDYINTHECITYOF
TOLEDOTOLEDOTOLEDOTOLEDO– –– –PRPRPRPR
CiceroAparecidoBezerra
PontifíciaUniversidadeCatólicadoParaná–[email protected]
Abstract:Abstract:Abstract:Abstract:
The production (and treatment) of domestic solid residua has caught the society'sattention in a general way (specially, of the government) due to the great impact
caused in the environmental and social ambit. The problem is basically about the
disabilityofthecitiestofacethegrowingproductionofsolidresidua.Fromthisproblem,
the current studyaims toanalyze the Holt-Winters forecasting models, for the solid
residue production, having as a base, the data found in Toledo City – PR. For this
purpose, the methodology adoptedwas, the historical series decomposition of solid
residueproduction,ofthisregion,withintheperiodfrom1999tothefirstsemesterof
2003 (obtained through bibliographic and documental research), in order to identifypatternswhichcanbeprojectedmakinguseofHolt-Wintersmodels.Havingasabase,
the average error criterion, the results were satisfactorily adequate to the decision
processthatinvolvesthemeasuringofsolidresidueproductionforfutureperiodsupto
sixmonths.
Keywords:Keywords:Keywords:Keywords:solidresidua,Holt-Wintersmodel,forecasting.
1111 IntroductionIntroductionIntroductionIntroduction
Nowadays, it is noticed a social movement related to the growing generation of
garbage and to the palliative solutions presented by the government. If, on the one
hand,theresidueproductionaffectsboththeenvironmentalandsocialsetting,onthe
otherhand,thedemandforsubstructureofmanagementofthisactivityinthecitiesis
incessant.
The lack of adequate policies for the solid residue treatment, as well as the non-
existenceofacorrectmeasuringofgarbageproduction,inthecities,aresomeofthe
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factors that can becontributing to the saturationof the current treatmentmodeland
solid residue destination. This measurement should be accomplished in order to
foreseethefutureproduction,enablingproactiveactionsplannedandorganizedbythegovernment,inabletime.Inparticular,intheToledoCity–PR,thesepalliativesolutions
havealreadybeencausingthesystemcollapse,once,fromtheResolutionnumber307
of the National Council of Environment, the municipal administration prohibited the
companies,whichworkwithdebris collection, toplace thematerial at the municipal
landfill.Thisdeterminationtransferredtheresponsibilitytothecompanies,whichdon't
haveandadequateplaceforthedebris,reducingtheserviceperformingofthisnature.
This way, the current study is concerned: to analyze the precision of Holt-Wintersforecasting models, to the production of solid residua, having as a base, the data
collectedinthecityofToledo.Thisbeingthecase,itisexpectedtocontributeforthe
adequate anticipation of the measuring of this problem, through the formulation of
proactivepublicpoliciesrelatedtothegarbagedestinationandproduction.
2222 SolidresiduaSolidresiduaSolidresiduaSolidresidua
Conceptually, D’Almeida (2000) defines urban solid residue as the debris massgenerated because of the occurrence of human activities in urban agglomerate.
Figueiredo(1998)differsresiduefrompost-usedgoodsbythefactthatthelastones
represent a specific kind of residue, whoseorigin isn’t a direct consequence of the
consumption, but of the arbitrage of an average useful life established on the own
conceptionof the product.For the current study, solidresiduaare understood asall
solidmaterialinwhichitsownerattributesnomorevalueandhedesirestogetridofit,
attributingtothePublicAdministrationtheresponsibilityforitsfinalplacement.
AccordingtoIBGE(2000),Brazilproducesaround240thousandof tonsofgarbagea
day,numberwhichisinferiortothatoftheUnitedStates(607thousandtonsaday),but
superior to countries like Germany (85 thousand tons a day) and Sweden (10.4
thousandtonsaday).InBrazil,accordingtoZveibil(2001),theaverageoftheresidue
production for a cityof100,000 inhabitants, is0.55kilogramper inhabitant aday of
garbage.
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Thesolidresidueproductionisrelatedtoitsfinaldestination.Themostusedalternative
in Brazil, according to IBGE (2000) is the open area landfills. This practice is very
criticizedfromthesocio-environmentalpointofview.Inthesameway,theincinerationoftheopenarealandfills,althoughbeinganalternativesourceofenergyproduction;it
alsohasastrongenvironmentalimpact(SCHOLZ,2000).TheNationalResearchof
BasicSanitation 2000(IBGE,2000)showsthat63.6%ofthecities usedopenareas
landfillsand32.2%adequatelandfills(13.8%sanitary,18.4%controlledlandfill),being
that5%didn’tinformthefinaldestinationofitsresidue.
Finally,accordingtoSoares-Baptista(2003),thematterofresiduemanagementisone
ofthegreatestchallengesofmunicipaladministrationofBrazil,seeingthattheNationalConstitutiondesignatesto themunicipalitiesthemainresponsiblyof themanagement
ofdomesticgarbage.
Afactorthatcancontributewiththematterrelatedtotheproductionofsolidresiduais
the use of demanding forecasting, because they enable the planning of resource
necessities through the future analysis (PELLEGRINI & FOGLIATTO, 2000). Such
models have started to be used (yet) in a limited and sector way at public
administration:fromthecomplementarylawnumber101from04/05/00,publishedatUnionOfficialDiary,section1,from05/05/2000,knownasFiscalResponsibilityLaw,
there is the obligation of revenue forecasting accompanied by the methodology of
calculus and used premises (VIEIRA, 2003). This way, it’s necessary to apply and
evaluatethesemethodologiesinthesolidresiduegeneration.
3333 ForecastingmodelsForecastingmodelsForecastingmodelsForecastingmodels
Thequantitative forecastingmethodsbasedonly on temporal seriesassume that itsfuturebehaviorcan'tbeforeseenthroughadeterministicfunction;howeveritcanbe
anticipatedthroughstochasticsprocedures.Amongthemostknownmodels,thesimple
andmovingaverage,exponentialsmoothingand,theBox-Jenkinsmethodologycanbe
quoted.Besidesthequotedmodels,itisimportanttoclarifythatthemultipleandsimple
regressiontechniquesareusedwithgreatsuccessinforecasting,butonlywhenthere
aremanysetsofdata.
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One of themost popular forecastingmodelwas developed in 1960,when the linear
modelofCharlesC.HoltwasextendedbyPeterR.Winterstocapturetheseasoning
directly, based on three smoothing equations (level, trend and seasoning) throughadditive and multiplicative equations, applied according to the series behavior. The
multiplicative model of Holt-Winters is composed of the following equations
(CARVALHO;LOIOLA&COELHO,2001):
))(1(11 −−
−
+−+=t t
st
t
t b L
S
Y L α α (1)
b t = β (Lt –Lt-1 )+(1 - β )b t-1 (2)
st
t
t
t S
L
Y S
−−+= )1( γ γ (3)
F t+m =(Lt +b t m )S t-s+m (4)
Where S represents the size of the seasoning, Lt represents the series level, bt
denotates the trend, St is the seasonal component and, Ft+m corresponds to the
forecasting for m periods ahead (CARVALHO; LOIOLA & COELHO, 2001). The
additive model of Holt-Winters is composed of the equations below (YAFFEE and
McGEE,2000):
Lt =α (Y t –S t-s )+(1–α )(Lt-1 +b t-1 ) (5)
b t = β (Lt –Lt-1 )+(1 - β )b t-1 (6)
S t =γ (Y t –Lt )+(1–γ )S t-s (7)
F t+m =Lt +b t m +S t-s+m (8)
Theses models bear the constants of smoothing α, β and γ. Theses constants are
valuesbetween0(zero)and1(one)establishedbytheanalyst.Fortheα,thehigher
thevalue,thefasterthemodelreactiontoarealvariationoftheobserveddatawillbe.
The same is applied to β,however it is related to the trend of theseries and, to γ
(relatedtotheseasonalfluctuation).Fromtheexponentialmethods,theHolt-Winters
modelsaretheonesthatbestrepresenttheseasoningandthedatatrend(KIRKHAM;
BOUSSABAINE&KIRKHAM,2002).
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To determine the model precision, standard measures obtained by the difference
betweenforeseenandobservedvaluesareusedasdefinedbelow:
Whereetistheerrorintperiod,YtistheobserveddatumandFtistheforecastingfort
periods. If observations and forecasting for n periods exist, it will be possible to
determinestandardstatisticalmeasures,forthesetofperiods:
∑=
=
n
t
t e
n
ME
1
1 (10)
∑=
=
n
t
t e
n MAE
1
1 (11)
∑=
=
n
t
t e
n MSE
1
21 (12)
The valuesupplied by the average error (10) tends to be small, once negative and
positiveerrorsfoundduringtheperiodstendtonullifyeachother.Itsmeritistoinform
whether the forecasting was systemically above or below the observed. Both the
absolute average error (11) and the square average error (12) turn the errors into
positive ones to then calculate the average, what seems to provide more accurate
information concerning the error's amplitude. These statistics deal with forecasting
measures whose size depends on the data scale, however they don't ease the
comparisonamongdifferenttemporalseriesanddifferenttimeintervals(MAKRIDAKIS;
WHEELWRIGHT &HYNDMAN, 1998). Such situation is overcome by error relative
measures,fromthefollowingequations:
t t t F Y e −= (9)
100
−
=
t
t t
t
Y
F Y PE (13)
∑=
=
n
t
t PE
n MPE
1
1 (14)
∑=
=
n
t
t PE
n MAPE
1
1 (15)
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Howeverthesamewaytheaverageerror(10),thepercentileoftheaverageerror(14)
tendstobesmall.Yetthemeanabsolutepercentileerror(15)issignificantonlyifthe
scalehasasignificantorigin.
4444 CasestudyCasestudyCasestudyCasestudy
ToledoCity islocated inthewestofParanáState,andithasapopulationofaround
100 thousand inhabitants. Toledo's economy is basically turned to the agriculture
consortedwith the agro industry. Asbasicproducts therearesoybeans,wheatand
corn,developedin5,282thousandruralproperties.Toledoconcentratesthecountry's
secondbiggestswineherdandthefirstofParanáState.ItisimportanttostandoutthatToledoistheheadquarterofthegreatestpoultryfrigorificofParanáStateandit isthe
biggest swine slaughterhouse of Latin America (PREFEITURA MUNICIPAL DE
TOLEDO,2001).
ToledoCitygenerateseveryyear,around15,000,000kilosofdomesticgarbage,whose
averagecompositionisformedby30%ofrecycledmaterial,50%oforganicmatterand
20%ofsewage.Table1,showsthemonthlyproductioninkilos,ofsolidresiduainthe
cityofToledo:
Table1:SolidresiduamonthlyquantityproducedinkilosinToledoCityTable1:SolidresiduamonthlyquantityproducedinkilosinToledoCityTable1:SolidresiduamonthlyquantityproducedinkilosinToledoCityTable1:SolidresiduamonthlyquantityproducedinkilosinToledoCityMonth 1999 2000 2001 2002 2003
January 1,034,590 1,175,145 1,222,530 1,279,880 1,406,290
February 935,595 1,041,930 978,840 1,011,700 1,142,125
March 1,076,358 1,059,045 1,101,360 1,056,470 1,177,625
April 1,049,761 999,280 1,023,415 1,181,270 1,151,565
May 1,025,905 1,140,370 1,108,820 1,190,180 1,223,095
June 1,066,205 1,127,735 1,090,295 1,114,640 1,023,630
July 1,074,140 1,006,765 1,047,860 1,162,950 -
August 951,865 1,048,395 1,037,110 1,181,160 -
September 952,505 1,015,745 968,165 1,081,560 -
October 932,270 1,011,100 1,130,220 1,167,850 -
November 905,475 1,020,440 1,085,635 1,128,700 -
December 1,106,065 1,186,485 1,210,060 1,315,830 -
Althoughtheannualaverageofresidueproduction,inToledoCity,iswithintheaverage
(for a city of 100,000 inhabitants, the average is 0.55 kilos per inhabitant a day of
garbage,accordingtoZveibil,2001),themonthlyvariationisbigbecauseofseveral
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conditioningelements.Monthlyvariationscanbeidentifiedupto35%inthemonthly
weight of the collected residua. Another problem is the final destination, mostly
depositedinopenareas,witheventualearthcovering.
4.1 Preliminary analysis
It'spossibletoverifythattherewasanaverageincreaseintheproductioninthesolid
residue,between1999and2003,of17.63%beingthatthebiggestincreaseoccurredin
the biennial 2001/2002, with a percentile of 8.74%. The monthly average is
1,101,088.48 kilos with standard deflection of 13,517.94 kilos, ranging between
905,407and 1,407,015 kilos, being that the year2003 showed the biggestmonthlyvariation(standarddeflectionof116,037.63kilos). In2002,thesmallestvariationwas
shown(68,853.22kilos).
According to the methodology proposed by Makridakis; Wheelwright & Hyndman
(1998),itisimportanttoanalyzethetemporalseriesinordertofindoutpatternsinthe
data, specially, related to the seasoning, trend and cycle. To discover patterns,
Libonati; Ribeiro Filho; Carvalho & Lemis (2004), suggest the additive classical
decomposition, in which a certain data set is composed by the addition of theseasoningtrendandresidue.Thefirststepfor theadditiveclassicaldecomposition is
the determinationof data trend, through the use of centeredmoving average of 12
periods,whoseresultisseeninChart1:
Chart1:TrendChart1:TrendChart1:TrendChart1:Trend
Removingthecomponentthatdeterminesthetrendoftheobserveddata,itispossible
toestimatetheseasonalcomponentthroughtheaverageofeachmonthlyobservation,
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presumingthattheyareconstantfromyeartoyear.Thisway,theseasonalindexescan
beobservedinChart2:
Chart2:SeasonalindexesChart2:SeasonalindexesChart2:SeasonalindexesChart2:Seasonalindexes
Finally, subtracting both the trend and seasonal indexes from the observed data,
irregularseriesareobtainedaccordingtoChart3:
Chart3:ResidueChart3:ResidueChart3:ResidueChart3:Residue
Throughtheclassicaldecompositionoftheobserveddata,itispossibletoconclude
that these data show well defined trends and seasoning, enabling the Holt-Winters
models application, considering its capacity of representing these standards
(KIRKHAM;BOUSSABAINE&KIRKHAM,2002).
4.2 Holt-Winters model application
FortheapplicationofHolt-Winters(additiveandmultiplicative)models,theMicrosoft
Excel2000®programwasused.TheobservationsfromJanuaryof2000toDecember
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of2002wereusedastestdataforthemodels,reservingthefirstsixmonthsoftheyear
2003 for the comparisons of the forecasting generated by the models. Due to the
observationofseasoningcycles,themodelswerebegunfromtheforthmonth.Forthedeterminationofvaluesof thesofteningconstantsα,βandγthatminimizedthemean
absolutepercentileerror–MAPE(15)oftheforecasting,Solversupplementwasused,
present in theMicrosoftExcel2000®program.Thisway, for theadditivemodel, the
valuesfrom0.0064(α),1(β)and0.1590(γ)werefound.Forthemultiplicativemodel,
the values were 0.0755 (α), 1 (β) and 0.2045 (γ). The forecasting results, of both
models,canbeobservedinChart4:
Chart4:HoltChart4:HoltChart4:HoltChart4:Holt----WintersmodelbehaviorfordatatestingWintersmodelbehaviorfordatatestingWintersmodelbehaviorfordatatestingWintersmodelbehaviorfordatatesting
For theadditivemodel considering thewhole period (4years) the found errorswere
1.72forthemeanerror–ME(14)and6.04%forthemeanabsolutepercentileerror–MAPE(15).Forthemultiplicativemodelthevalueswere0.04and7.25%respectively.
It is also observed that the values generated by the models tend to adjust to the
observeddatafromthelastyearofthetest.
4.3 Comparison of the results
According toMakridakis;Wheelwright;Hyndman (1998), theprecision is determined
whileamodelmanagedtoreproducedatathatisalreadyknown.Forthisreasonthedata of the year 2003 were reserved, for the precision comparison of generated
forecasting,whatcanbeseeninChart5:
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Chart5:BehaviorofHoltChart5:BehaviorofHoltChart5:BehaviorofHoltChart5:BehaviorofHolt----WintersmodelfortheknowndataWintersmodelfortheknowndataWintersmodelfortheknowndataWintersmodelfortheknowndata
Table2showstheresultsobtainedbytheadditiveandmultiplicativemodel:
Table2:AdditiveandmultiplicativeHoltTable2:AdditiveandmultiplicativeHoltTable2:AdditiveandmultiplicativeHoltTable2:AdditiveandmultiplicativeHolt----WintersforecastingvaluesWintersforecastingvaluesWintersforecastingvaluesWintersforecastingvalues
Additive Multiplicative
PeriodObserved
dataForeseen
dataError
%Abs.error
Foreseen
dataError
%Abs.error
Jan/03 1,406,290 1,177,873 228,416 16.24 1,159,309 246,980 17.56
Feb/03 1,142,125 1,131,182 10,942 0.96 1,169,871 -27,746 2.43
Mar/03 1,177,625 1,155,617 22,007 1.87 1,240,644 -63,019 5.35
Apr/03 1,151,565 1,218,342 -66,777 5.80 1,192,540 -40,975 3.56
May/03 1,223,095 1,222,148 946 0.08 1,200,943 22,151 1.81
Jun/03 1,023,630 1,183,176 -160,106 15.64 1,257,239 -233,609 22.82
Consideringthewholetestperiod(1semester),thevaluesfortheaverageerrorswere-
0.38and-2.46,fortheadditiveandmultiplicativemodels,respectively.Thepercentile
absoluteaverageerrorswere6.76%(additive)and8.92%(multiplicative).
5555 LimitationsandrecommendationsLimitationsandrecommendationsLimitationsandrecommendationsLimitationsandrecommendations
Someconsiderationsmustbetaken:
• Considering the research priority about solid residua production, naturally,
greatereffortsmustbededicatedtothosestudiesthatdealwiththecausative
variables of the problem.This way, the goal of this article isn't to substitute
studiesaboutthecausesrelatedtotheproblem,buttoenableafuturemeasure
that is adequate enough, for the actions to be accomplished in the present,
dealingwiththepointsthatcanleadtotheforeseensituation.
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• This situation (implemented actionsat this moment) is particularly important,
evidencinganecessityofcomplementamongthecausalstudiesandstochastic
natureones,presentedinthisstudy.
• Theforecastingperiodhorizonlastedsixmonths.AlthoughtheUnitedNations
Conference About Environment and Development (CONFERÊNCIA DAS
NAÇÕESUNIDASSOBREMEIOAMBIENTEEDESENVOLVIMENTO,1996)
suggested longer time spaces for the adequate decision involving public
departments,itisbelievedthat,evenduringthisperiodoftime,theuseofthese
forecastingmodelscanaidthedecisiveprocess.
Forthecurrentstudytherewasn'tmonthlydataavailabilitythatcouldexplaintheseries
behavior. It's recommended, in the presence of these data, the application of
regressionmodels,knownaseconometricsmodels,whichassumethataYvariable
canbeforeseen,incaseofaXexplanatoryvariableisavailable.Besides,theuseof
moreanalyticalmodelsissuggested,amongtheonestherearethemodelsbasedon
themethodologyofBox-Jenkins.
6666 ConclusionConclusionConclusionConclusion
From the presentedmodels, the additivemodel obtained the best performance, not
onlyforthetestdata,butalsoforthedatausedforthefinalforecasting,havingasthe
base the error criteria. However it is noticed that both models didn't capture the
standardswhicharepresentinthedataserieswithaccuracy,mainlythemultiplicative
model.Evenso,it isbelievedthatthepresentedvaluessupplyasufficientlyadequate
margin for decision aid, according to Toledo's Environmental Department. It is
importanttoobservethattheprecisionisn'ttheonlycriterionforthereliabilityofanymodel: the stimulus for the action in the organization is what will determine the
forecastingsuccess(MAKRIDAKIS,WHEELWRIGHT&HYNDMAN,1998).
Oncethedomesticsolidresiduaproductioncanbemeasuredwitharelativelyadequate
safetymarginwithinaperiodofsixmonths,itisthepublicadministrationtasktoplan
(andperform)proactivemeasures inthemanagementofthisproblem.Themeasures
to be adopted aren't new, there are abundant reports showing its efficacy and
efficiency, since they are implemented inable time: selective collection with formal,
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semiformal,andinformalwebsandconsequentrecycling.Cesa&Conto(2003)stand
outthatdifferentfieldsofknowledgecan(andmust)contributetotheminimizationof
the problems which are faced by the urban population when handling the residua,among them: civil, operations and material engineering; architecture; marketing;
administration;psychology.
This study evaluate the Holt-Winters forecasting models concerning the domestic
productionofgarbageinToledoCity,inordertocontributetothemanagementof this
problem.However,accordingtoMakridakis;Wheelwright&Hyndman(1998),Pellegrini
& Fogliatto (2000), and Vieira (2003), allied to this knowledge, the involved
professionals'perceptionintheanalyzedmatter,whoseexperienceacquiredthroughthe direct observation of this phenomenonand in the perceptionof random factors,
mustbetakenintoconsiderationforcorrectmeasuringofthesituationand,mainlyin
thesensitizationoftheauthoritiestoeffectivelyimplantpublicpolicieswhichareableto
minimizetheproblemscausedbythesolidresiduaproduction.
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