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ThirdInternationalConferenceonProductionResearchThirdInternationalConferenceonProductionResearchThirdInternationalConferenceonProductionResearchThirdInternationalConferenceonProductionResearch– –– –

Americas’Americas’Americas’Americas’Region2006Region2006Region2006Region2006(ICPR(ICPR(ICPR(ICPR----AM06)AM06)AM06)AM06)

IFPRIFPRIFPRIFPR– –– –ABEPROABEPROABEPROABEPRO----PUCPRPUCPRPUCPRPUCPR----PPGEPSPPGEPSPPGEPSPPGEPS

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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