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1 Peropoulos, F., Nikolopoulos, K., Spihourakis, G. and Assimakopoulos, V. (2013) Empirical heurisics for improving Inermien Demand Forecasing. Indusrial Managemen and Daa Sysems, 113 (5). pp ISSN Link o official URL (if available): Opus: Universiy of Bah Online Publicaion Sore hp://opus.bah.ac.uk/ This version is made available in accordance wih publisher policies. Please cie only he published version using he reference above. See hp://opus.bah.ac.uk/ for usage policies. Please scroll down o view he documen.

2 Empirical heurisics for improving Inermien Demand Forecasing Foios Peropoulos 1,*, Konsaninos Nikolopoulos 2, Georgios P. Spihourakis 1, Vassilios Assimakopoulos 1 1 Forecasing & Sraegy Uni, School of Elecrical and Compuer Engineering, Naional Technical Universiy of Ahens, Greece 2 The Business School, Bangor Universiy, Bangor, UK fois@fsu.gr; k.nikolopoulos@bangor.ac.uk; giorgos@fsu.gr; vassim@fsu.gr *corresponding auhor Absrac Purpose: Inermien demand appears sporadically, wih some ime periods no even displaying any demand a all. Even so, such paerns consiue considerable proporions of he oal sock in many indusrial seings. Forecasing inermien demand is a raher difficul ask bu of criical imporance for corresponding cos savings. The curren sudy examines he empirical oucomes of hree heurisics owards he modificaion of esablished inermien demand forecasing approaches. Design/mehodology/approach: Firs, opimizaion of he smoohing parameer used in Croson s approach is empirically explored, in conras o he use of an a priori fixed value as in earlier sudies. Furhermore, he effec of ineger rounding of he resuling forecass is considered. Lasly, we evaluae he performance of Thea model as an alernaive of SES esimaor for exrapolaing demand sizes and/or inervals. The proposed heurisics are implemened ino forecasing suppor sysem. Findings: The experimen is performed on 3,000 real inermien demand series from he auomoive indusry, while evaluaion is made boh in erms of bias and accuracy. Resuls indicae increased forecasing performance. Originaliy/Value: The curren research explores some very simple heurisics which have a posiive impac on he accuracy of inermien demand forecasing approaches. While, some of hese issues have been parially explored in he pas, he curren research focuses on a complee in-deph analysis of easy o employ modificaions o well esablished inermien demand approaches. By his, we enable he applicaion of such heurisics on an indusrial environmen, which may lead ino significan invenory and

3 producion cos reducions and oher benefis. Keywords: inermien demand; smoohing parameers; rounding; hea mehod; empirical invesigaion 1. Inroducion Demand and invenory forecass are required for virually all decision making siuaions regarding fuure evens, from shor erm forecass dealing wih invenories and scheduling o medium and long erm ones needed for sraegy and planning. Accurae forecass are of grea pracical imporance, linking invenory coss wih revenues, cusomer saisfacion, sock-ou coss and lead ime (for example Huang e al., 2011). Inermien demand paerns are characerized by infrequen demand arrivals coupled wih variable demand sizes, whenever demand occurs. Inermien demand iems may be engineering spare pars or oher iems wihin he range of producs offered by any organizaion and a any level of he supply chain. Sparse demand creaes significan problems in he manufacuring and supply environmen as far as forecasing is concerned. I is no only he variabiliy of he demand size, bu also he variabiliy of he inerval beween demands ha make inermien demand so difficul o forecas. If he fac ha slow moving iems may consiue up o 60% of he oal sock in any indusrial seing (Johnson e al., 2003) is also aken ino accoun, i becomes obvious ha small improvemens can insigae subsanial cos savings. The curren sudy examines hree empirical heurisics used wih esablished and commonly used forecasing approaches for inermien demand (Croson, 1972; Syneos and Boylan, 2001; Syneos, 2001). Firsly, opimizaion issues, regarding he opimal smoohing parameers used, are looked ino. Secondly, an inuiively aracive and pracically indispensable heurisic is invesigaed: rounding of he final forecass, as all demand sizes of SKUs are whole numbers. Lasly, an approach ha combines Croson mehod for inermien demand wih Thea model (Assimakopoulos and Nikolopoulos, 2000) is horoughly invesigaed. All hree heurisics resul in very promising resuls when applied o daa characerized by inermien demand paerns, where he presence of zero demand is eviden. Taking ino accoun ha such daa arise in many indusries, i comes wihou saying ha he

4 curren research aims a improving seings and procedures by easy-o-implemen modificaions on convenional echniques. Furhermore, o our knowledge i is he firs ime ha he effeciveness of independenly selecing he smoohing values for he numeraor and he denominaor of each one of he series, via a widely used cos funcion (MSE), is being empirically invesigaed. Lasly, he full poenial of uilizing he Thea model in Croson s framework for he exrapolaion of is componens is analyzed and discussed. The remainder of he curren paper is srucured as follows. Firsly, a shor lieraure review on widely used inermien demand forecasing echniques is presened a Secion 2, followed by he experimenal srucure of our research and he descripion of he daa se used (Secion 3). Empirical resuls regarding he hree heurisics examined are presened and discussed in Secion 4. The implemenaion of he proposed hree heurisics is explored in Secion 5, hrough a dedicaed forecasing suppor sysem. Finally, some managerial implicaions are menioned (Secion 6), while conclusions are summed up and avenues for fuure work are proposed in Secion Inermien demand forecasing approaches Inermien demand daa or coun daa are frequenly observed in indusrial and invenory seings. Williams (1984) inroduced a number of classificaion rules in order for a spare par o be characerized as slow moving, inermien or lumpy. According o Teuner and Sani (2009), i is no an easy ask o forecas inermien demand, basically due o is erraic, and someimes lumpy, naure. Neverheless, i is very surprising ha so lile work has been done on forecasing inermien demand daa (Gooijer and Hyndman, 2005), wih several indusries and organizaions relying on he single exponenial smoohing (SES) mehod in order o forecas demand in a rouine sock conrol sysem (Brown, 1959). As firs shown by Croson (1972), he use of SES generally leads o inappropriae sock levels. As an alernaive, Croson proposed he decomposiion of he original inermien series ino wo separae series. The firs one includes all non-zero demand sizes, while he second series consiss of he respecive inervals beween wo consecuive non-zero demands. Each line is exrapolaed separaely, while he final forecas is simply calculaed as a raio of he wo. Assuming ẑ and pˆ o be he forecass of demand size and inerval, respecively, for period, Croson s forecas is given by:

5 z Yˆ ˆ pˆ In fac, laer research in his field is heavily based on his single research by Croson. Willemain e al. (1994) and Johnson and Boylan (1996) have underaken accuracy comparisons beween SES and Croson s mehod, demonsraing he superioriy of he laer, especially when he inerval beween demands exceeds 1.25 imes he updae period. Syneos and Boylan (2001) proved ha Croson s mehod is posiively biased. Towards he correcion of his behavior, hey proposed a modificaion of he original Croson s mehod (Syneos and Boylan, 2005), beer known as Syneos and Boylan Approximaion (SBA). This new esimaor is given by: Yˆ a zˆ 1 2 pˆ where a is he value of he exponenial smoohing consan used in he exrapolaion of he inervals series. Syneos, in his PhD Thesis (2001), proposed anoher unbiased esimaor, which can be obained as follows: ˆ a zˆ Y 1 2 a pˆ 2 Previous empirical sudies (Syneos and Boylan, 2001) have shown ha he biased behavior of Croson s mehod is more apparen in he case of daa wih high inermiency (i.e. many periods wih zero demand) when high smoohing values (a) are used. As a resul, Croson s mehod is no recommended o be used wih a values above This resul was verified by Teuner and Sani (2009), who analyzed he circumsances under which Croson s mehod and SBA approach end o be biased. According o heir findings, Croson s original mehod presens smaller bias if few demands are zero, whereas SBA modificaion has a smaller bias if many demands are zero. Moreover, i is argued ha forecass derived from Syneos mehod are ouperformed in erms of forecasing variance by he SB mehod (Syneos, 2001; Teuner and Sani, 2009). Teuner and Sani (2009) sugges he use of Syneos mehod as an alernaive o Croson and SBA mehods, bu heir findings are only based on simulaed daa. In all hree cases, he demand sizes and inervals are exrapolaed using SES, while forecass are updaed only in periods wih posiive demand. The esimae of he demand under he SES mehod is given by (Makridakis e al., 1998):

6 Yˆ ˆ ˆ i 1 Y ( Y Y ) (1 ) i0 Alhough a-smoohing values in he range [ ] are viewed as realisic (Croson, 1972; Willemain e al., 1994; Johnson and Boylan, 1996), mos of he empirical sudies invesigaing inermien demand assumed a consan value for a. The use of a smoohing consan is no a usual echnique in fas moving series, where an opimizaion procedure akes place owards he selecion of an opimal smoohing parameer which minimizes he in-sample MSE. Moreover, Snyder (2002) argued for he use of differen smoohing parameers for he demand sizes and inervals, as well as proposed he use of oher compaible models and mehods. Recen research on inermien demand forecasing has focused, oher han forecasing performance in erms of accuracy, on he variabiliy of inermien demand esimaes (Syneos and Boylan, 2010), he imporance of invenory obsolescence (Teuner e al., 2011) and invenory performance under differen ypes of informaion sharing (Ali e al., 2012). Anoher new research roue has been he emporal, non-overlapping aggregaion of inermien demand daa ino ime buckes so ha he resuling series are more likely o be non-inermien. This echnique has proven o be very promising in erms of forecasing accuracy (Nikolopoulos e al., 2011) and cusomer service levels (Babai e al., 2012). However, despie any research during he las 40 years, Croson s mehod is very ofen applied in pracice (Fildes e al., 2008) and incorporaed in commercial forecasing suppor sysems. As a resul, easy-o-apply modificaions on he original framework ha will lead in performance improvemen are considered as beneficial. Y i 3. Empirical Daa & Experimenal Srucure The empirical daabase used for he purposes of our research consiss of he individual monhly demand hisories of 3,000 SKUs from he auomoive indusry, over wo years (24 consecuive monhly demand observaions). The same daabase has been used in earlier sudies (Syneos and Boylan, 2005; Syneos e al., 2005). Deailed descripive saisics (o he second decimal place) on he demand daa series characerisics are presened in Table 1. I is worh menioning ha he daa in hand are considered as fas inermien, where he inermien demand inervals are a any ime less han 2, wih

7 a median value a The low degree of inermience in his daa se is coupled wih low demand sizes wih low degrees of variance. As a resul, he daa se is considered suiable as i conains series falling in all four classes of demand, according o Syneos e al. (2005): erraic, lumpy, smooh or inermien. A he same ime, he empirical daa are no o be relaed wih issues regarding variabiliy of he esimaes and invenory obsolescence. This empirical daase will no heavily affec he biased behavior of he Croson mehod, due o is low degree of inermiency. 3,000 Demand Sizes Demand Inervals Demand per period SKUs Mean SDev Mean SDev Mean SDev Min % ile Median % ile Max Table 1. Demand daa descripive saisics For he simulaion purposes of he curren research we held ou he las 11 observaions of each series, iniializing all mehods over he firs 13 periods. We performed a sliding simulaion (rolling evaluaion) over he ou-of-sample daa via producing one-sepahead forecass; hus we calculaed 11 one-sep-ahead errors for each series, for each of he forecasing mehods considered. The evaluaion of he resuls was performed by measuring he bias and accuracy of he examined mehods. Mean Error (ME) offers a way o deermine if an examined mehod is consisenly posiively or negaively biased, depending on he sign of he resuling value. ME can be calculaed across all series using he following equaion: where s Y and 1 mean( ME) n h n h s s Y Yˆ s1 1 s Yˆ are he acual and forecas values, respecively, of series s a ime period, n is he oal number of series considered and h is he number of ou-of-sample periods (horizon), hus n=3,000 and h=11. Accordingly, accuracy was calculaed using average values of Mean as well as Median Absolue Scaled Error (MASE and MdASE respecively, Hyndman and Koehler, 2006). These wo merics are widely applicable, scale independen and easy o inerpre: values of MASE greaer han one indicae ha

8 forecass are worse, on average, han in-sample one sep forecass of he Naive mehod. The average values of MASE and MdASE are given by: 1 mean( MASE ) n h 1 mean( MdASE ) n n s1 n h s1 1 1 n 1 Y s k i1 Y median 1 n 1 where k is he number of in-sample periods, hus k=13. Three esimaors were used in erms of benchmarking: Naive, SES and Simple Moving Yˆ k s Y i2 s i s Yˆ Y Y s i s s i1 Y Average (SMA). Naive forecass are equal o he las acual demand, so: Yˆ 1 Y SES forecass were generaed using a consan level smoohing parameer, equal o Las, SMA a lengh of 13 periods was used because his was he esimaion procedure employed by he sofware manufacurer ha provided he empirical daa series used in his research (Syneos and Boylan, 2005). The esimae of he demand under he SMA(13) is given by: 13 1 Yˆ Y 13 i1 i s i1 4. Empirical Invesigaion & Discussion Bias and accuracy resuls for he hree benchmarks are repored in Table 2. Resuls for inermien demand mehods are also presened, when a smoohing consan in he range [ ] is seleced, he same for demand sizes and inervals. Overall, he mos unbiased mehod for he examined daase would be SBA mehod wih a=0.05, followed by Naïve and SES. Boh in Croson and SBA mehods, an increase in value of he a smoohing consan leads o a greaer absolue value of ME, resuling in more biased forecass. ME for Croson s mehod is negaive, denoing a posiively biasing behavior, whereas in he case of SBA he bias has a negaive direcion. In erms of accuracy, SBA scores he lowes values for MASE and MdASE, a a=0.15 and 0.2, respecively. I is worh noing ha, in conras o bias, increased values of a smoohing consan have a posiive effec on he measured (via MdASE) accuracy for boh Croson

9 and SBA mehods. Furhermore, we should also noe he overall good performance of SES mehod, which scored he same accuracy level wih SBA a a=0.05. Empirical resuls of Table 2 indicae ha Syneos mehod is no suiable for he examined daa, being ouperformed from Croson and SBA in erms of boh bias and accuracy. As a resul, furher analysis of he curren research is basically based on he performance of he laer mehods. Mehod BIAS ACCURACY ME MASE MdASE Naive E-04 SMA(13) E-04 SES(0.05) E-04 Croson(0.05) E-04 Croson(0.1) E-04 Croson(0.15) E-04 Croson(0.2) E-04 SBA(0.05) E-04 SBA(0.1) E-04 SBA(0.15) E-04 SBA(0.2) E-04 Syneos(0.05) E-04 Table 2. Resuls of benchmarks and sandard inermien demand mehods 4.1. Opimizing a smoohing parameer The firs heurisic of our research examines he opimizaion of a smoohing parameer, raher han using a consan value for demand sizes and inervals across all series. A linear opimizaion procedure akes place, where all values in he range [ ] are examined separaely using a sep of 0.01, and he one minimizing he in-sample MSE is seleced as he opimal, resuling in differen a values for each series. The value of he in-sample MSE is given by: 1 MSE k k Y Yˆ 1 This linear opimizaion is a common pracice for fas moving series, where smoohing parameers are seleced in order o bes fi he in-sample forecas model. In his case, 2

10 he opimizaion procedure is applied direcly and exclusively o he decomposed series, i.e. he demands sizes and inervals, which may lead o differen opimal a values, as suggesed by Snyder (2002). Mehod BIAS ACCURACY ME MASE MdASE Croson( opimal a) E-04 SBA( opimal a) E-04 Syneos( opimal a) E-04 Table 3. Opimizing a smoohing value The resuls of his empirical heurisic for each inermien demand mehod (Croson, SBA and Syneos) are presened in Table 3. In erms of accuracy, he resuls are almos idenical wih hose of he implemenaions of he mehods where consan a smoohing value is se equal o 0.05 across all series. However, here is a significan reducion of he value of ME meric in he case of SBA mehod. The calculaed bias drops o 0.021, which means a 25% error improvemen. There is however no significan evidence ha opimizaion benefis Croson and Syneos mehods. Demands Inervals a-value (numeraor) (denominaor) % % 13 in-sample % % observaions % % % % 18 in-sample % % observaions % % % % 23 in-sample % % observaions % % Table 4. Disribuions of opimal a-values Table 4 presens he disribuions of he opimal a-values for boh demand and inervals in hree insances of he rolling procedure. In more deail, he number of ime series selecing as opimal a-values in he ranges [ ], [ ] and [ ] along wih he relevan

11 percenages are demonsraed. The hree insances considered in his analysis were compleed by 13, 18 and 23 in-sample observaions respecively. A close observaion of Table 4 makes i clear ha smaller values of a are generally seleced (inside he range [ ]), especially in he case of shor available hisory. As more observaions become available, opimizaion enables a selecive choice of greaer a-values as well, for up o 17% of he ime series The effec of rounding When forecasing SKUs, providing decimal forecas values does no make much sense. This simple idea leads us o round he produced forecass so ha he reporing values would be whole numbers. Table 5 presens he resuls of he rounding effec, when indicaive implemenaions of inermien demand mehods are used. In comparison wih Table 2, he bias measured in all cases is almos a he same levels (if no even lower). Furhermore, here are noable improvemens in erms of accuracy, as compued via MASE, where he calculaed forecass are approximaely 2% more accurae for all mehods esed. Mehod BIAS ACCURACY ME MASE MdASE Croson(0.1, Round) E-04 SBA(0.1, Round) E-04 SBA(0.2, Round) E-04 Syneos(0.05, Round) E-04 Table 5. Rounding SKUs forecass 4.3. Combining Croson mehod wih Thea model The use of SES mehod in order o exrapolae he decomposed Croson s series has been criicized in many sudies (see for example Snyder, 2002). We consider he use of an alernaive, modern forecasing echnique, he Thea model (he winner of M3 forecasing compeiion, Makridakis and Hibon, 2000), inroduced by Assimakopoulos and Nikolopoulos (2000). Thea mehod decomposes he original series in wo (or more) separae series (he so-called hea lines), whose primary qualiaive characerisic is he beer approximaion of he long-erm behavior of he daa or he augmenaion of shor-erm feaures, depending on he value of he Thea coefficien.

12 These hea lines are exrapolaed separaely. A he curren sudy, we implemen he Classic Thea model, as a hree seps procedure: 1. Each ime-series is decomposed ino wo Thea lines, he linear regression line (which is referred also as Thea Line (Θ=0)) and he Thea Line (Θ=2), which is calculaed as follows: TheaLine( 2) 2Y LRL Where Y refers o he -h acual observaion of he raw daa, while LRL denoes he -h observaion of he linear regression line, expressing he linear relaionship beween raw daa and ime. 2. The linear regression line is exrapolaed in he usual way while he second line is exrapolaed via Single Exponenial Smoohing. 3. The forecass produced from he exrapolaion of he wo lines are combined wih equal weighs. Thus, we invesigae he use of Thea model, which can replace SES mehod in eiher numeraor or denominaor of Croson s raio, or even in boh. Originally, he combinaion of Croson wih Thea was proposed by Nikolopoulos e al. (2007), where Thea model was applied jus for he exrapolaion of he demand sizes (numeraor). The inuiion for using his combinaion was ha Croson-Thea could pick up rends of non-saionary series, hus displaying more poen predicive power. Mehod BIAS ACCURACY ME MASE MdASE Croson-Thea(0.05, Num & Denom) E-04 Croson-Thea( opimal a, Num & Denom) E-04 Croson-Thea( opimal a, Denom) E-04 Croson-Thea( opimal a, Num) E-04 Croson-Thea(0.05, Num) E-04 Croson-Thea(0.05, Num, Round) E-04 Table 6. Combining Croson wih Thea The resuls of Croson-Thea combinaion are presened in Table 6. Each row of he able displays he resuls for a separae implemenaion of he approach, in erms of he seleced a smoohing value (0.05 or opimal, as discussed in Subsecion 4.1), he level a which Thea was applied (numeraor, denominaor or boh) and, las, he applicaion

13 (or no) of he rounding heurisic (as discussed in Subsecion 4.2) a he final forecass. The resuls indicae significan improvemens in erms of bias, when Thea model is used for he exrapolaion of inervals (denominaor). Specifically Croson- Thea( opimal a, Denom) implemenaion has he bes bias performance in comparison o all oher implemenaions presened in he curren sudy, wih an improvemen close o 50% from he second bes implemenaion (SBA( opimal a)). On he oher hand, he Croson-Thea combinaion does no pay back in erms of accuracy. The resuls indicae worse ou-of-sample accuracy performance han he insample accuracy under Naïve, when Thea model is applied o he denominaor. Lasly, he rounding effec seems o work once again as a simple self-improvemen heurisic, offering noable improvemens in he resuling accuracy merics, while keeping he bias level consan. The moderae performance of Croson-Thea combinaion in erms of accuracy could be inerpreed as lack of he rend componen in he examined empirical daase. Even if he rend componen is almos zero for he majoriy of inermien demand series, he presence of a deerminisic rend would considerably favor he use of Thea model over SES Discussion The use of differen smoohing parameers for he demand and inervals has been previously suggesed (Snyder, 2002). However, appropriaely choosing smoohing values independenly for he numeraor and he denominaor of each one of he series, via a widely used cos funcion (MSE), has no been previously proposed nor invesigaed. In fac, his pracice drives in considerable gains in erms of bias, while a he same ime no negaive impacs on accuracy are recorded. The mos imporan observaion, however, derives from comparing Tables 2 and 3. I is prey clear, ha as he saic value of he smoohing parameer (a) raises from 0.05 o 0.20, here is a significan deerioraion of he bias meric (measured as ME). However, selecing he mos suiable smoohing value independenly for each series, hrough minimizing insample error, leads o significanly beer resuls (up o 25%). This pracically means ha he use of a cos funcion for selecing he bes smoohing parameers per series can make a difference. In many indusrial applicaions, especially hose involving ime series of spare pars or SKUs, non-ineger poin forecass are considered as non realisic. The impacs of rounding he final poin forecass derived from inermien demand mehods are

14 empirically examined in his research. The mos significan resul is ha his echnique resuls in beer accuracy levels (up o 2%) while a he same ime no deerioraion in erms of bias is recorded. As a resul, we srongly recommend he use of his heurisic, which is regarded as appealing in boh empirical and pracical erms. Finally, an alernaive o he radiional Croson s mehod was examined in Subsecion 4.3. Originally, Croson proposed he use of SES as he exrapolaion procedure for boh decomposed series (demand and inervals). Given ha boh decomposed series represen sequences of non-zero values, we consider, for he firs ime, he full poenial of using if he Thea model as he exrapolaion echnique of eiher he nominaor, denominaor or boh. In fac, he choice of his model lies in is superior performance, as recorded in pas major inernaional forecasing compeiions. This simple echnique allows Croson s framework ac almos in an unbiased way (improvemen up o 90% from he original approach), when Thea model is used for he exrapolaion of he inervals beween non-zero demands. 5. The forecasing suppor sysem (FSS) In many applicaions, saisical forecass are produced via dedicaed and auonomous forecasing suppor sysems (FSS). This direcion serves muliple purposes. Firsly, managers and praciioners may no be familiar wih he necessary saisical background. Secondly, an auomaed FSS can handle, pre-process and forecas housands of ime series (sales or orders for SKUs) in jus a few seconds. Lasly, many feaures of he modern FSSs, such as saisical analysis, handling he impacs of special periods and inegraion of judgmenal inervenions, are regarded as necessary o he forecasing process of any indusry. The needs of he empirical analysis of he curren research led us o he design and developmen of a unique and dedicaed forecasing suppor sysem for handling daa of inermien naure. The purpose of he curren secion is o give he general guidelines owards he implemenaion of a FSS ha fully implemens he proposed heurisics analyzed in he curren research, so enabling he managers o have direc access o any gains derived hrough heir pracice in any manufacuring or indusrial seing. Moreover, we aim o give insighs o praciioners already uilizing cusomized sofware as o which direcions of addiional implemenaion or exernal modules should hey arge for exploiing hese heurisics.

15 The Inermien Demand Forecasing Sysem (IDFS) was designed following a hreeier physical archiecure (visualizaion, saisical/business and daa). The main advanage of his archiecure is ha i is easily serviceable and expandable. The firs layer of he archiecure is he user inerface, where boh graphical and numerical inerpreaion of he daa and he resuls are displayed. Furhermore, his layer enables users owards a deailed selecion of he parameers relaed o he forecasing process (forecasing mehods, horizons, hold-ou samples, error merics), along wih he usage of he hree heurisics presened in his paper. Figure 1 and Figure 2 show wo ypical screen displays of he proposed sysem. The second layer is he saisical/business ier. All he saisical and mehodological procedures are modeled and implemened in he middle layer, which includes he funcions relaed o he forecasing process. This layer includes all original forecasing mehodologies (for example SES, Thea model, Croson, SBA and Syneos) and also allows mehods o inerac wih each oher (for example, owards he formaion of Croson-Thea). Addiional adjusmens, such as he use of opimized over conac smoohing values or he rounding of he final forecass, are passing as exernal variables, hrough he ineracion wih he firs layer. In addiion, a middle ier is used o creae he base for furher exension wih exernal sofware, encompassing inerfaces, wrappers and web-services necessary for daa exchange. Lasly, he hird layer consiss of he daa ier of he applicaion. The daa base managemen sysem (daabase, views and relaions) is lying on he daa base server, a windows-based machine isolaed from he inerne in order o avoid hreas and provide saisfacory response imes. The daa layer provides he saisics/business layer wih he required hisorical daa and sores any forecass and accuracy resuls.

16 Figure 1. Graphical visualizaion of he inpu daa and parameer iniializaion Figure 2. Forecasing mehods/parameers and graphical inerpreaion of he empirical resuls

17 IDFS was developed using Microsof s Visual Basic.NET 2008 while he Dundas Char, for Visual Basic.NET, was employed for he sysem implemenaion, in regards o is advanced charing funcionaliy and superior graphic opions. Finally, he Microsof SQL Server 2008 R2 daabase is uilized by IDFS o sore and rerieve he required informaion for he daa analysis and forecasing. 6. Managerial Implicaions Heurisics linked wih he simpliciy of spreadshees are considered managerial appealing and flexible, especially in he case of slow-moving iems and when dealing wih problems of differen sizes (Hummel and Jesse, 1990). As a resul, opimizing a smoohing parameer and rounding final forecass seem also o be ineresing from a managerial poin of view. Boh heurisics can easily be implemened by managers and praciioners, while offering noable gains regarding accuracy and/or bias. To begin wih, opimizing procedures for smoohing parameers of exponenial smoohing mehods are implemened and auomaed in major forecasing packages. In ha sense, hese procedures can easily be employed in a pracical invenory seing, resuling in subsanial improvemens for he bias of SBA esimaor (up o 25%), while scoring good accuracy levels (compared o SBA(0.05) implemenaion). Moreover, he ask of rounding he final forecass derived from any inermien demand mehod can be easily done by use of ordinary spreadshees. This simple heurisic offers remarkable improvemens in measured forecasing accuracy, keeping, a he same ime, bias a lower levels. Forecass for inermien demand SKUs call for rounding, so as o make forecass inerpreable and direcly usable for real supply chain managemen applicaions, such as order placemen. 7. Conclusions & Perspecives The curren sudy examined he empirical effeciveness of hree empirical heurisics owards he modificaion of commonly used forecasing approaches for inermien demand. We proposed he use of non-consan a smoohing parameer, via selecing he bes a value for each series hrough in-sample opimizaion. Moreover, hese values may be differen for he demand sizes and he inervals (Snyder, 2002). The use of SES mehod for he exrapolaion of he decomposed series in Croson s framework is also examined. We considered as an alernaive he Thea model, a echnique ha

18 ouperformed all exponenial smoohing mehods in he M3 forecasing compeiion. Lasly, an inuiively appealing heurisic, concerning he rounding of he final forecass, was proposed. The resuls indicae ha opimal selecion of a smoohing values resuls in almos idenical accuracy levels, while in some cases here are significan improvemens on he bias. Thus, model opimizaion is feasible and does pay back. Rounding seems o work surprisingly well, offering noable improvemens in erms of accuracy and keeping bias consan. These resuls render his simple heurisic suiable in cases of daa ses consising of SKUs. Croson-Thea s performance was moderae, as i performed well as far as bias is concerned, bu seemed problemaic in erms of accuracy. A las, a specialized FSS for inermien demand daa was proposed. Furher research should involve he ineracion of he rounding heurisic from a heoreical poin of view. Moreover, all experimens of his sudy could be replicaed wih differen daa ses, so as o reach more general conclusions abou he hree proposed heurisics. Specifically, he Croson-Thea combinaion should be esed horoughly wih simulaed and field rended daa. Finally, i is recenly argued (Syneos e al., 2010) ha in an invenory forecasing seing exrapolaion mehods should no only be evaluaed wih respec o heir forecas accuracy bu also in erms of heir sock conrol implicaions, as measured hrough accuracy implicaion merics (such as invenory coss and service levels achieved). Exploring he effecs of he examined heurisics on sock conrol is an ineresing line for furher research and cerainly worhwhile pursuing from a praciioner s perspecive. References Ali M.M., Boylan J.E. and Syneos A.A. (2012), Forecas errors and invenory performance under forecas informaion sharing, Inernaional Journal of Forecasing, Vol. 28, pp Assimakopoulos V. and Nikolopoulos N. (2000), The hea model: a decomposiion approach o forecasing, Inernaional Journal of Forecasing, Vol. 16, pp Babai M.Z., Ali M.M. and Nikolopoulos K. (2012), Impac of emporal aggregaion on sock conrol performance of inermien demand esimaors: Empirical analysis, Omega, Vol. 40, pp Brown R. (1959) Saisical Forecasing for Invenory Conrol, McGraw-Hill, New York. Croson J.D. (1972), Forecasing and Sock Conrol for Inermien Demands, Operaional Research Quarerly, Vol. 23, pp

19 De Gooijer J.G and Hyndman R.J. (2005), 25 years of ime series forecasing, Inernaional Journal of Forecasing, Vol. 22, pp Fildes R., Nikolopoulos K., Crone S.F. and Syneos A.A. (2008) Forecasing and operaional research: A review, Journal of he Operaional Research Sociey, Vol. 59, pp Huang L.T., Hsieh I.C. and Farn C.K. (2011), On ordering adjusmen policy under rolling forecas in supply chain planning, Compuers and Indusrial Engineering, Vol. 60, pp Hummel J.W. and Jesse R.R. (1990), A spreadshee heurisic approach for he socking and reenion of slow-moving, obsolescen iems, Compuers and Indusrial Engineering, Vol. 18, pp Hyndman R.J. and Koehler A.B. (2006), Anoher look a measures of forecas accuracy, Inernaional Journal of Forecasing, Vol. 22, pp Johnson F.R. and Boylan J.E. (1996), Forecasing inermien demand: A comparaive evaluaion of Croson s mehod, Inernaional Journal of Forecasing, Vol. 12, pp Johnson F.R., Boylan J.E. and Shale E.A. (2003), An examinaion of he size of orders from cusomers, heir characerizaion and he implicaions for invenory conrol of slow moving iems, Journal of he Operaional Research Sociey, Vol. 54, pp Makridakis S., Wheelwrigh S.C. and Hyndman R.J. (1998), Forecasing: Mehods and Applicaions (3rd ed.), Wiley, New York, NY. Makridakis S. and Hibon M. (2000), The M3-Compeiion: resuls, conclusions and implicaions, Inernaional Journal of Forecasing, Vol. 16, pp Nikolopoulos K., Syneos A.A. and Babai M.Z. (2007), A new inermien demand approach via combining Croson s mehd and he Thea model, paper presened a he 22nd European Conference on Operaional Research EURO XXII, July 8-11, 2007, Prague, Czech Republic. Nikolopoulos K., Syneos A.A., Boylan J.E., Peropoulos F. and Assimakopoulos V. (2011), An aggregae disaggregae inermien demand approach (ADIDA) o forecasing: an empirical proposiion and analysis, Journal of he Operaional Research Sociey, Vol. 62, pp Snyder R. (2002), Forecasing sales of slow and fas moving invenories, European Journal of Operaional Research, Vol. 140, pp Syneos A.A. and Boylan J.E. (2001), On he bias of inermien demand esimaes, Inernaional Journal of Producion Economics, Vol. 71, pp Syneos A.A. (2001), Forecasing for Inermien Demand. Brunel Universiy: Unpublished Ph.D hesis. Syneos A.A. and Boylan J.E. (2005), The accuracy of inermien demand esimaes, Inernaional Journal of Forecasing, Vol. 21, pp Syneos A.A., Boylan J.E. and Croson J.D. (2005), On he caegorizaion of demand paerns, Journal of he Operaional Research Sociey, Vol. 56, pp

20 Syneos A.A and Boylan J.E. (2010), On he variance of inermien demand esimaes, Inernaional Journal of Producion Economics, Vol. 128, pp Syneos A.A., Nikolopoulos K. and Boylan J.E. (2010), Judging he judges hrough accuracy-implicaion merics: he case of invenory forecasing, Inernaional Journal of Forecasing, Vol. 26, pp Teuner R. and Sani B. (2009), On he bias of Croson s forecasing mehod, European Journal of Operaional Research, Vol. 194, pp Teuner R.H., Syneos A.A. and Babai M.Z. (2011), Inermien demand: Linking forecasing o invenory obsolescence, European Journal of Operaional Research, Vol. 214, pp Willemain T.R., Smar C.N., Shockor J.H. and DeSauels P.A. (1994), Forecasing inermien demand in manufacuring: A comparaive evaluaion of Croson s mehod, Inernaional Journal of Forecasing, Vol. 10, pp Williams T.M. (1984), Sock conrol wih sporadic and slow-moving demand, Journal of he Operaional Research Sociey, Vol. 35, pp