Supply chain managemen of consumer goods based on linear forecasing models Parícia Ramos (paricia.ramos@inescporo.p) INESC TEC, ISCAP, Insiuo Poliécnico do Poro Rua Dr. Robero Frias, 378 4200-465, Poro, Porugal José Manuel Oliveira (jmo@inescporo.p) INESC TEC, Faculdade de Economia, Universidade do Poro Rua Dr. Robero Frias, 378 4200-465, Poro, Porugal Absrac In his work we apply linear forecasing models o a very broad collecion of reail sales of consumer goods from a Poruguese reailer. This allows us o draw conclusions for guidelines wihin his field, and also o conribue o general observaions relevan o he main field of forecasing. For each reail series he model wih he minimum value of he AIC for he in-sample period is seleced from all admissible models for furher evaluaion in he ou-of-sample. Boh one-sep and muliple-sep forecass are produced. The resuls show ha ARIMA models ouperform sae space models in ou-of-sample forecasing judged by MAPE. Keywords: Aggregae reail sales, Forecas accuracy, Sae space models, ARIMA models Inroducion Demand forecasing is one of he mos imporan issues ha is beyond all sraegic and planning decisions in any business organizaion. The imporance of accurae demand forecass in successful supply chain operaions and coordinaion has been recognized by many researchers (Wong and Guo, 2010; Arlo and Alain, 2010). A poor forecas would resul in eiher oo much or oo lile invenory, direcly affecing he profiabiliy of he supply chain and he compeiive posiion of he organizaion. Forecasing fuure sales is crucial o he planning and operaion of reail business a boh high and low levels. A he organizaional level, forecass of sales are needed as he essenial inpus o many decision aciviies in various funcional areas such as markeing, sales, producion/purchasing, as well as finance and accouning (Agrawal and Schorling, 1996; Chopra and Meindl, 2007). Reail sales ofen exhibi srong rend and seasonal variaions, presening challenges in developing effecive forecasing models. Hisorically, modeling and forecasing seasonal daa is one of he major research effors and many heoreical and heurisic mehods have been developed in he las several decades (Alon e al., 2001; Chu and Zhang, 2003; Zhang and Qi, 2005; Kuvulmaz e al., 2005, Pan e al., 2013). Exponenial smoohing and Auoregressive Inegraed Moving Average (ARIMA) models are he 1
mos widely-used approaches o ime series forecasing, and provide complemenary approaches o he problem. While exponenial smoohing mehods are based on a descripion of rend and seasonaliy in he daa, ARIMA models aim o describe he auocorrelaions in he daa. The ARIMA framework o forecasing originally developed by Box e al. (1994) involves an ieraive hree-sage process of model selecion, parameer esimaion and model checking. A saisical framework o exponenial smoohing mehods was recenly developed based on innovaions sae space models called ETS models (Hyndman e al., 2008a). Despie he invesigaor's effors, he several exising sudies have no led o a consensus abou he relaive forecasing performances of hese wo modeling frameworks when hey are applied o reail sales daa. The purpose of his work is o compare he forecasing performance of sae space models and ARIMA models when applied o a very broad collecion of reail sales of four differen caegories of consumer goods from he Poruguese reailer Jerónimo Marins. As far as we known i's he firs ime ETS models are esed for reail sales forecasing. The remainder of he paper is organized as follows. The nex secion describes he daases used in he sudy. Secion 3 discusses he mehodology used in he ime series modeling and forecasing. The empirical resuls obained in he research sudy are presened in Secion 4. The las secion offers he concluding remarks. Daa Jerónimo Marins is a Porugal-based inernaional group operaing in food disribuion, food manufacuring and services secors. Involving operaions in reail and wholesale formas, he Jerónimo Marins Group is he leader in food disribuion in Porugal, wih he brands Pingo Doce (leader in supermarkes) and Recheio (leader in cash & carry), in food sore chains in Poland (Biedronka) and in Colombia (Ara). The work presened in his paper was developed using 67 ime series of sales of consumer goods of a Pingo Doce supermarke of around 1500 m 2 beween January 2007 and July 2012 (67 monhs). Figure 1 shows he ime plo of he number of differen producs sold per day in his supermarke during ha period. I can be seen ha he number of differen producs sold per day is increasing and i has an annual seasonal behavior. Figure 1 Number of differen producs sold per day beween January 2007 and July 2012. To illusrae he broad collecion of ime series analyzed in his work, Figure 2 shows he ime plo of monhly sales of six producs sold beween January 2007 and July 2012 (67 observaions). All hese series are obviously non-saionary exhibiing srong rend 2
and/or seasonal paerns providing a good esing ground for comparing he wo forecasing mehods. Forecasing models Figure 2 Time series of consumer goods sold by Pingo Doce.. ETS models Exponenial smoohing mehods have been used wih success o generae easily reliable forecass for a wide range of ime series since he 1950s (Gardner, 1985; Gardner, 2006). In hese mehods forecass are calculaed using weighed averages where he weighs decrease exponenially as observaions come from furher in he pas. The mos common represenaion of hese mehods is he componen form. Componen form represenaions of exponenial smoohing mehods comprise a forecas equaion and a smoohing equaion for each of he componens included in he mehod. The componens ha may be included are he level componen, he rend componen and he seasonal componen. By considering all he combinaions of he rend and seasonal componens, fifeen exponenial smoohing mehods are possible. Each mehod is usually labeled by a pair of leers (T,S) defining he ype of Trend and Seasonal Trend = N,A,A,M,M and componens. The possibiliies for each componen are: { d d} Seasonal { N,A,M} y y y =. For illusraion, denoing he ime series by 1, 2,, n and he forecas of y + h, based on all of he daa up o ime, by y ˆ + h, he componen form for 3
he mehod (A,A) (addiive Hol-Winers mehod) is (Hyndman and Ahanasopoulos, 2013): yˆ = l + hb + s + h + m+ h m ( ) ( 1 α )( ) l = α y s + l + b m 1 1 ( ) ( 1 β ) b = β l l + b * * 1 1 ( ) ( γ ) s = γ y l b + 1 s, 1 1 m where m denoes he period of he seasonaliy, l denoes an esimae of he level of he series a ime, b denoes an esimae of he rend of he series a ime, s denoes an esimae of he seasonaliy of he series a ime and y ˆ+ h denoes he poin forecas for h periods ahead where h + m = ( h 1)mod m + 1. The iniial saes l0, b0, s1 m, s0 and he smoohing parameers *,, α β γ are esimaed from he observed daa. The smoohing * parameers α, β, γ are consrained beween 0 and 1 so ha he equaions can be inerpreed as weighed averages. Deails abou all he oher mehods may be found in (Hyndman and Ahanasopoulos, 2013). To be able o generae forecas inervals and oher properies, Hyndman e al. (2008a) (amongs ohers) developed a saisical framework for all exponenial smoohing mehods. In his saisical framework each sochasic model, referred as an innovaions sae space model, consiss of a measuremen equaion ha describes he observed daa, and sae equaions ha describe how he unobserved componens or saes (level, rend, seasonal) change over ime. For each exponenial smoohing mehod, Hyndman e al. (2008a) described wo possible innovaions sae space models, one corresponding o a model wih addiive random errors and oher corresponding o a model wih muliplicaive random errors, giving a oal of 30 poenial models. To disinguish he models wih addiive and muliplicaive errors, an exra leer E was added: he riple of leers (E,T,S) refers o he hree componens: Error, Trend and Seasonaliy. The noaion ETS(,,) helps in remembering he order in which he componens are specified. For illusraion, he equaions of he model ETS(A,A,A) (addiive Hol-Winers mehod wih addiive errors) are (Hyndman and Ahanasopoulos, 2013): y = l 1 + b 1 + s m + ε (5) l = l 1 + b 1 + αε (6) b = b + βε (7) s 1 = s + γε m and he equaions of he model ETS(M,A,A) (addiive Hol-Winers mehod wih muliplicaive errors) are: (9) y = ( l 1 + b 1 + s m)(1 + ε ) l = l + b + α( l + b + s ) ε (10) 1 1 1 1 m b = b + β( l + b + s ) ε 1 1 1 m s = s + γ ( l + b + s ) ε m 1 1 m (1) (2) (3) (4) (8) (11) (12) 4
where β * = α β, 0 α 1 < <, 0 < β < α, 0 < γ < 1 α (13) 2 and ε is a zero mean Gaussian whie noise process wih variance σ. Equaions (5) and (9) are he measuremen equaion and Equaions (6)-(8) and (10)-(12) are he sae equaions. The erm innovaions comes from he fac ha all equaions in his ype of specificaion use he same random error process ε. The measuremen equaion shows he relaionship beween he observaions and he unobserved saes. The ransiion equaion shows he evoluion of he sae hrough ime. I should be emphasized ha hese models generae opimal forecass for all exponenial smoohing mehods and provide an easy way o obain maximum likelihood esimaes of he model parameers (for more deails see Hyndman and Khandakar (2008b)). ARIMA models ARIMA is one of he mos versaile linear models for forecasing seasonal and nonseasonal ime series. I has enjoyed grea success in boh academic research and indusrial applicaions during he las hree decades. The class of ARIMA models is broad. I can represen many differen ypes of sochasic seasonal and non-seasonal ime series such as pure auoregressive (AR), pure moving average (MA) and mixed AR and MA processes (Brockwell and Davis, 1991). The heory of ARIMA models has been developed by many researchers and is wide applicaion was due o he work by Box and Jenkins (1994) who developed a sysemaic and pracical model building mehod. The muliplicaive seasonal ARIMA model, denoed as ARIMA ( p, d, q) ( P, D, Q) m, has he following form (Wei, 2005): where φ ( B) Φ ( B m )(1 B) d (1 B m ) D y = c + θ ( B) Θ ( B m ) ε (14) p P q Q φ ( B) = 1 φ B φ B p 1 θ ( B) = 1+ θ B + + θ B q 1 p m m Pm P Φ P( B ) = 1 Φ1B ΦPB q m m Qm q Θ Q( B ) = 1+ Θ 1B + + ΘQB and m is he seasonal frequency, B is he backward shif operaor, d is he degree of regular differencing, and D is he degree of seasonal differencing, φ ( B ) and θ ( B) are he regular auoregressive and moving average polynomials of orders p and q respecively, Φ m ( B ) and ( m Θ B ) are he seasonal auoregressive and moving average P Q polynomials of orders P and Q respecively, c = µ (1 φ1 φ P )(1 Φ1 Φ P ) d m D where µ is he mean of (1 B) (1 B ) y process and ε is a zero mean Gaussian 2 whie noise process wih variance σ. The roos of he polynomials φ ( B ), Φ ( B m ), θ ( B ) and ( m Θ B ) should lie ouside a uni circle o ensure causaliy and inveribiliy q Q (Shumway and Soffer, 2011). For d + D 2, c = 0 is usually assumed because a quadraic or a higher order rend in he forecas funcion is paricularly dangerous. p p q P 5
Empirical sudy Esimaion resuls In order o use ETS models for forecasing he values of iniial saes and smoohing parameers need o be known. I is easy o compue he likelihood of ETS models and so maximum likelihood esimaes are usually preferred. A grea advanage of he ETS saisical framework is ha informaion crieria can be used for model selecion, namely he Akaike s Informaion Crieria (AIC). For ETS models, AIC is defined as (Hyndman and Ahanasopoulos, 2013): AIC = 2log( L) + 2k (15) where L is he likelihood of he model and k is he oal number of parameers and iniial saes ha have been esimaed. Some of he combinaions of (Error, Trend, Seasonal) can lead o numerical difficulies. Specifically, he models ha can cause such insabiliies are: ETS(M,M,A), ETS(M,Md,A), ETS(A,N,M), ETS(A,A,M), ETS(A,Ad,M), ETS(A,M,N), ETS(A,M,A), ETS(A,M,M), ETS(A,Md,N), ETS(A,Md,A), and ETS(A,Md,M) (Hyndman and Ahanasopoulos, 2013). Usually hese paricular combinaions are no considered when selecing a model. The ime series analysis was carried using he saisical sofware R programming language and he specialized package forecas (Hyndman and Khandakar, 2008b; R Developmen Core Team, 2013). For each ime series of monhly sales all admissible ETS models were applied using he in-sample daa beween January 2007 and July 2011 (firs 55 observaions). The iniial saes and he parameers were esimaed by maximizing he likelihood of each model. The ETS model wih he minimum value of he AIC was seleced o produce forecass and forecas inervals on he ou-of-sample period (Augus 2011 o July 2012, las 12 observaions). The main ask in ARIMA forecasing is selecing an appropriae model order, ha is he values of p, q, P, Q, d and D (he seasonal period is 12, m = 12 ). We use he auomaic model selecion algorihm ha was proposed by Hyndman and Khandakar (2008b). We sar by choosing he values of d and D by applying uni-roo ess. I is recommended ha seasonal differencing be done firs because someimes he resuling series will be saionary and here will be no need for a furher regular differencing. D = 0 or D = 1 depending on he OCSB es (Osborn e al., 1988). Once he value of D is seleced, d is chosen by applying successive KPSS uni-roo ess (Kwiakowski e al., 1992). Once d and D are known, he orders of p, q, P and Q are seleced via Akaike s Informaion Crieria: AIC = 2log( L) + 2( p + q + P + Q + k + 1) (16) 2 where k = 2 if c 0 and 1 oherwise (he oher parameer being σ ), and L is he maximized likelihood of he model fied o he differenced daa (1 B) d (1 B m ) D y. Raher han considering every possible combinaion of p, q, P and Q, he algorihm uses a sepwise search o raverse he model space: (a) The bes model (wih smalles AIC) is seleced from he following four: ARIMA (2, d,2) (1, D,1) 12 ARIMA (0, d,0) (0, D,0) 12, ARIMA (1, d,0) (1, D,0) 12, ARIMA (0, d,1) (0, D,1) 12. 6
If d + D 1, hese models are fied wih c 0, oherwise c = 0. This is called he curren model. (b) Thireen variaions on he curren model are considered where: one of p, q, P and Q is allowed o vary from he curren model by ± 1; p and q boh vary from he curren model by ± 1; P and Q boh vary from he curren model by ± 1; he consan c is included if he curren model has c = 0 and excluded oherwise. Whenever a model wih a lower AIC is found, i becomes he new curren model and he procedure is repeaed. This process finishes when we canno find a model close o he curren model wih a lower AIC. For each ime series of monhly sales he sep-wise algorihm described above was applied using he in-sample daa beween January 2007 and July 2011 (firs 55 observaions) o find an appropriae ARIMA model. The parameers of he models are esimaed by maximizing he likelihood. The seleced model was used o produce forecass and forecas inervals on he ou-of-sample period (Augus 2011 o July 2012, las 12 observaions). Forecas evaluaion resuls For each reail series boh seleced models (ETS and ARIMA) were used o forecas on he ou-of-sample period from Augus 2011 o July 2012 (12 observaions). Boh onesep and muliple-sep forecass were produced. Using each model fied for he insample period, poin forecass of he nex 12 monhs (one-sep forecass) and he forecas accuracy measures based on he errors obained were compued. The values of he average MAPE (mean absolue percenage error) of one-sep forecass obained are presened in Table 1. Supposing T is he oal number of observaions, N is he insample size and h is he sep-ahead, muli-sep forecass were obained using he following algorihm: For h = 1 o T N For i = 1 o T N h + 1 Selec he observaion a ime N + h + i 1 as ou-of-sample Use he observaions unil ime N + i 1 o esimae he model Compue he h -sep error on he forecas for ime N + h + i 1 Compue he forecas accuracy measures based on he errors obained for sepahead h Compue he mean of he forecas accuracy measures In our case sudy T = 67 and N = 55. I should be emphasized ha in muli-sep forecass he model is esimaed recursively in each sep i using he observaions unil ime N + i 1. Boh one-sep and muli-sep forecass are imporan in faciliaing a shor and long planning and decision making. They simulae he real-world forecasing environmen in which daa need o be projeced for shor and long periods (Alon e al., 2001). The values of he average MAPE of muli-sep forecass obained are also presened in Table 1. The resuls from Table 1 show ha ARIMA models ouperform sae space models on boh one-sep and muli-sep forecass judged by MAPE. ARIMA consisenly forecass more accuraely han ETS on one-sep forecass and on all seps-ahead of muli-sep forecass. 7
Improvemens on one-sep forecass are of 51%. On muli-sep forecass improvemens increase wih he increasing of he sep unil sep-ahead 6 and hen decrease unil sep-ahead 12. The improvemens are 0%, 12%, 27%, 34%, 37%, 40%, 36%, 31%, 29%, 20%, 17% and 13% respecively. In boh models he value of MAPE ends o increase wih he increasing of he sep unil sep-ahead 6 and hen ends o decrease unil sep-ahead 12. The resuls of our analysis also show ha muli-sep forecass are more accurae han one-sep forecass (wih excepion o seps-ahead 4, 5 and 6 in he ARIMA model). This is no surprising since muli-sep forecass incorporae informaion ha is more updaed. Producing esimaes of uncerainy is an essenial aspec of forecasing which is ofen ignored. We also evaluaed he performance of boh forecasing mehodologies in producing forecas inervals ha provide coverages which are close o he nominal raes. Table 2 and Table 3 show he mean percenage of imes ha he nominal 95% and 80% forecas inervals conain he rue observaions for boh one-sep and muliple-sep forecass, respecively. The resuls indicae ha boh ETS and ARIMA produce coverage probabiliies ha are very close o he nominal raes. ETS produces beer coverage probabiliies for boh 80% and 90% forecas inervals on boh one-sep and muliple-sep forecass. I can also be observed ha boh mehods slighly underesimae he coverage probabiliies for he nominal 80% forecas inervals. Conclusions Accurae reail sales forecasing can have a grea impac on effecive managemen of reail operaions. Reail sales ime series ofen exhibi srong rend and seasonal variaions presening challenges in developing effecive forecasing models. How o effecively model hese series and how o improve he qualiy of forecass are sill ousanding quesions. Despie he invesigaor's effors, he several exising sudies have no led o a consensus abou he relaive forecasing performances of ETS and ARIMA modeling frameworks when hey are applied o reail sales daa. The purpose of his work was o compare he forecasing performance of sae space models and ARIMA models when applied o a very broad collecion of reail sales of four differen caegories of consumer goods from he Poruguese reailer Jerónimo Marins. For each ime series of monhly sales all admissible ETS models were applied using he in-sample period. The ETS model wih he minimum value of he AIC was seleced o produce forecass and forecas inervals on he ou-of-sample period. The auomaic model selecion algorihm proposed by Hyndman and Khandakar (2008b) was used o selec an appropriae ARIMA model for each ime series of monhly sales. The model seleced by he sep-wise algorihm for each ime series was hen used o produce forecass and forecas inervals on he ou-of-sample period. Boh one-sep and muliple-sep forecass were produced. The resuls indicae ha ARIMA models ouperform sae space models in ou-of-sample forecasing judged by MAPE. On boh modeling approaches muli-sep forecass are generally beer han one-sep forecass which is no surprising because muli-sep forecass incorporae informaion ha is more updaed. The performance of boh forecasing mehodologies in producing forecas inervals ha provide coverages which are close o he nominal raes was also evaluaed. The resuls indicae ha boh ETS and ARIMA produce coverage probabiliies ha are very close o he nominal raes. However, ETS produces beer coverage probabiliies for boh 80% and 90% forecas inervals on boh one-sep and muliple-sep forecass. We could also observe ha boh mehods slighly underesimae he coverage probabiliies for he nominal 80% forecas inervals. 8
Model One-sep forecass Table 1 MAPE (%) for ou-of-sample period forecass (January 2007 o July 2012). Sep-ahead of muli-sep forecass 1 2 3 4 5 6 7 8 9 10 11 12 ARIMA 64.83 44.31 53.20 60.96 67.74 67.42 68.26 59.38 50.65 49.29 41.78 44.35 45.76 ETS 132.93 44.53 60.69 83.59 102.31 107.51 113.90 92.81 73.32 69.21 52.40 53.67 52.56 Table 2 Forecas 80% inerval coverage for ou-of-sample period forecass (January 2007 o July 2012). Model One-sep forecass Sep-ahead of muli-sep forecass 1 2 3 4 5 6 7 8 9 10 11 12 ARIMA 75 74 75 75 74 73 75 75 77 76 77 76 66 ETS 80 75 78 78 78 80 80 82 84 82 81 86 78 Table 3 Forecas 95% inerval coverage for ou-of-sample period forecass (January 2007 o July 2012). Model One-sep forecass Sep-ahead of muli-sep forecass 1 2 3 4 5 6 7 8 9 10 11 12 ARIMA 92 89 91 91 90 89 92 91 90 90 89 89 90 ETS 94 90 91 90 90 91 93 94 93 93 92 95 97 9
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