Lancaster University Management School Working Paper 2006/030. On the bias of Croston's forecasting method. Ruud Teunter and Babangida Sani

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1 Lancaser Universiy Managemen School Working Paper 2006/030 On he bias of Croson's forecasing mehod Ruud Teuner and Babangida Sani The Deparmen of Managemen Science Lancaser Universiy Managemen School Lancaser LA1 4YX UK Ruud Teuner and Babangida Sani All righs reserved. Shor secions of ex, no o exceed wo paragraphs, may be quoed wihou explici permission, provided ha full acknowledgemen is given. The LUMS Working Papers series can be accessed a hp:// LUMS home page: hp://

2 On he bias of Croson s forecasing mehod Ruud Teuner a, Babangida Sani b a Deparmen of Managemen Science, Lancaser Universiy, LA1 4YX, UK b Deparmen of Mahemaical Sciences, Kano Universiy of Science and Technology, Wudil, Nigeria Absrac Croson s forecasing mehod has been shown o be appropriae in dealing wih inermien demand iems. The mehod, however, suffers from a posiive bias as shown by Syneos and Boylan (2001, 2005) who proposed a modificaion. Unforunaely, he modificaion ignores he damping effec on he bias of he probabiliy ha a demand occurs. This leads o overcompensaion and a negaive bias, which can in fac be larger han he posiive bias of he original mehod. Levén and Segersed (2004) also proposed a modified Croson mehod, bu ha suffers from an even more severe bias. Building on he resuls of Syneos and Boylan (2001, 2005), we propose a new modificaion ha akes he damping effec ino accoun. A numerical sudy confirms ha i considerably ouperforms he exising mehods. Moreover, he performance is beer over he enire range of relevan parameers, which avoids he need o use differen mehods depending on he demand caegorisaion as suggesed by Syneos e al. (2005). Keywords: Forecasing; Inermien demand; Croson s mehod 1. Inroducion I is no easy o forecas inermien demand due o is erraic and someimes lumpy naure. Many organisaions in he manufacuring and especially service indusries simply use single exponenial smoohing. However, as was firs shown by Croson (1972), his generally leads o inappropriae sock levels. Croson proposed an alernaive mehod ha akes accoun of boh demand size and iner-arrival ime beween demands. The mehod is now widely used in indusry and i is incorporaed in various bes selling forecasing sofware packages (see Syneos e al, 2005). Croson s mehod has been assessed by several auhors since The lieraure is reviewed in deail in Secion 2. Mos auhors come o he conclusion ha Croson s mehod is more suiable for inermien demand han radiional mehods such as moving average and single exponenial 1

3 smoohing. In fac, as shown by Teuner and Duncan (2006), conradicing resuls can be explained by he use of inappropriae performance measures. A disadvanage of he original Croson mehod is ha i is posiively biased. Syneos and Boylan (2001) noed his and proposed a modificaion. However, as we will show in Secion 3, ha modificaion over-compensaes, leading o a negaive bias insead. Levén and Segersed (2004) also proposed a modificaion, bu heir mehod is even more biased as we also show in Secion 3. Building on he resuls of Syneos and Boylan (2001, 2005), we propose a new modificaion of he Croson mehod in Secion 4. A numerical sudy in Secion 5 confirms ha his new mehod considerably ouperforms he exising mehods. This paper ends wih conclusions in Secion Lieraure on Croson s mehod We will review he conribuions chronologically. Rao (1973) made correcions o several expressions in Croson s paper wihou affecing he final conclusions or he forecasing procedure. Schulz (1987) presened a forecasing procedure which is basically he Croson s mehod and suggesed a base-sock invenory policy wih replenishmen delays. Willemain e al. (1994) compared Croson s mehod wih exponenial smoohing and concluded ha Croson s mehod is robusly superior o exponenial smoohing, alhough resuls wih real daa in some cases show a more modes benefi. Johnson and Boylan (1996b) obained similar resuls, bu furher showed ha Croson s mehod is always beer han exponenial smoohing when he average iner-arrival ime beween demands is greaer han 1.25 review inervals. Sani and Kingsman (1997) compared various forecasing and invenory conrol mehods on some long series of low demand real daa from a ypical spare pars depo in he UK. They concluded based on cos and service level, ha he bes forecasing mehod is moving average followed by Croson s mehod. An imporan conribuion is ha by Syneos and Boylan (2001). They show ha Croson s mehod lead o a biased esimae of demand per uni ime. They also propose a modified mehod and demonsrae he improvemen in a simulaion experimen. Snyder (2002) criically assessed Croson s mehod wih a view o overcome cerain implemenaion difficulies on he daa ses used. Snyder made correcions o he underlying heory and proposed modificaions. Ghobbar and Friend (2003) compared various forecasing mehods using real daa of aircraf mainenance repair pars from an airlines operaor. The daa is sporadic in naure and 2

4 hey showed ha moving average, Hol s and Croson s forecasing mehods are superior o oher mehods such as he exponenial smoohing. Willemain, Smar and Schwarz (2004) compared various forecasing mehods using large indusrial daa ses. They showed ha he boosrapping mehod produces more accurae forecass han boh exponenial smoohing and Croson s mehod. In an aemp o develop a forecasing procedure ha can handle boh fas moving and slow moving iems, Levén and Segersed (2004) proposed a modificaion of Croson s mehod which was hough o avoid he bias indicaed by Syneos and Boylan, The modificaion was shown o ouperform exponenial smoohing based on a simulaion experimen. Eaves and Kingsman (2004) compared various forecasing mehods using real daa from he UK s Royal Air Force. They showed ha he modified Croson s mehod by Syneos and Boylan (2001) is he bes forecasing mehod for spare pars invenory conrol. In an aemp o furher confirm he good performance of heir modified Croson s mehod, Syneos and Boylan (2005) carried ou a comparison of forecasing mehods including heirs and he original Croson s mehod. A simulaion exercise was carried ou on 3,000 producs from he auomoive indusry wih fas inermien demand. I was shown ha he modificaion is he mos accurae esimaor. In anoher sudy, Syneos, Boylan and Croson (2005) analyzed a wider range of inermien demand paerns and made a caegorisaion o guide he selecion of forecasing mehods. They indicaed ha here are demand caegories ha are beer used wih he original Croson s mehod and here are ohers ha go well wih he Syneos/Boylan modificaion. A recen comparison by Syneos and Boylan (2006) shows overall superior performance of he Syneos/Boylan modificaion, followed by simple moving average and he original Croson s mehod. Anoher comparaive sudy was conduced by Teuner and Duncan (2006), using a large daa se from he UK s Royal Air Force. Using a new performance measure ha compares arge o achieved service level, hey showed ha he original Croson s mehod as well as he Syneos & Boylan and he Levén & Segersed varians ouperform moving average and exponenial smoohing. 3. Theoreical Background p Croson s original mehod forecass separaely he ime beween consecuive ransacions and he magniude of he individual ransacions z. A he review period, if no demand occurs in a 3

5 review period hen he esimaes of he demand size and iner-arrival ime a he end of ime, pˆ respecively, remain unchanged. If a demand occurs so ha zˆ pˆ = αz = αp + (1 α) zˆ 1 + (1 α) pˆ 1, z and >0, hen he esimaes are updaed by where α is a smoohing consan beween zero and one. Hence, he forecas of demand per period a ime is given as ẑ C zˆ pˆ = Syneos & Boylan modificaion showed ha Syneos and Boylan (2001) poined ou ha Croson s original mehod is biased. They E zˆ pˆ µ p α p 1 2 α p ( C ) = E 1+ (1) and, in paricular, for α=1 ha E zˆ z 1 1 ( C ) = E = E = ln pˆ p µ. (2) p 1 p Based on (1) and ignoring he erm p 1 p, Syneos and Boylan proposed a new esimaor given as SB α zˆ = 1 2 pˆ. One can expec his new esimaor o perform beer as p 1 p ges closer o one, i.e., as he probabiliy 1/ p of posiive demand in a period ges smaller. This effec is illusraed in Figure 1, where he bias of he original Croson mehod and Syneos & Boylan modificaion are compared. Noe ha he nonmonoone behaviour is caused by he randomness of demand, as can also be seen from he differences for he wo demand series. INSERT FIGURE 1 ABOUT HERE 4

6 I is clear from Figure 1, and he above analysis explains why his also holds in general, ha Croson s original mehod has a smaller (posiive) bias if 1/ p is large (few demands are zero), and he Syneos/Boylan modificaion has a smaller bias if 1/ p is small (many demands are zero). This also explains and confirms he findings by Syneos, Boylan and Croson (2005) ha were discussed in Secion Levén & Segersed modificaion Levén and Segersed (2004) modified Croson s mehod in an aemp o obain a mehod ha works for boh slow and fas moving iems. Their esimaor is updaed as follows LS z α. = + ( 1 α) LS 1 p They referred o he above summarized resuls in Syneos and Boylan (2001) on he bias for Croson s original mehod, bu remarked ha heir esimaor does no suffer from such a bias. However, i does! Indeed, using (2) i follows ha E 1 1 ( LS ) = ln µ. p 1 p I can easily be shown ha here is always a posiive bias and ha his bias ges worse as he probabiliy 1/ p of a demand decreases. The bias is indeed very large; i is more han 50% when a demand occurs in 1 ou of 3 periods or less. Figure 2 shows ha if he probabiliy of a demand is 0.2 and he average demand size is µ = 5, hen he Levén & Segersed esimae flucuaes around 2 ( exac) whereas he expeced demand per period is µ/p = 1. E ( LS ) = o be INSERT FIGURE 2 ABOUT HERE 4. New mehod As he Syneos & Boylan mehod, he new mehod is based on (1), bu i does no ignore he erm p 1 p. Based on rewriing (1) as zˆ µ 2 p α E, pˆ p 2 p αp 5

7 we sugges he following esimaor: zˆ pˆ ( 2 pˆ αpˆ ) ( 2 pˆ α ) ( 2 α ) zˆ α zˆ = 1 =. 2 pˆ α 2 α pˆ 2 In he nex secion, we will compare he bias of his new mehod o ha of he oher mehods in a numerical sudy. We noe ha he Levén & Segersed mehod canno be modified in a similar way, a leas no wihou abandoning is core idea of only updaing he demand per period. 5. Numerical comparison of new and exising mehods We compare he new modificaion o he original Croson s mehod, he Syneos & Boylan modificaion and he Levén & Segersed modificaion in a numerical sudy. We employ a full facorial design, where we vary he smoohing parameer α (0.1, 0.2, 0.3), he probabiliy of a demand 1/p (0.1, 0.3, 0.5, 0.7), and he ype and variance of he demand disribuion (normal wih mean 1 and variance 0.1, normal wih mean 1 and variance 0.3, discree uniform beween 1 and 2, discree uniform beween 1 and 10). We remark ha, based on he resuls in Secion 3, he main deerminans of he bias are expeced o be α and 1/p. For each of he = 48 experimens, a demand series of 10,000 periods is generaed randomly. All mehods are iniialized wih he correc values. The repored biases are averaged over all 10,000 periods. Table 1 gives he complee resuls. INSERT TABLE 1 ABOUT HERE The resuls confirm our expecaion ha he smoohing parameer, α, and he probabiliy of a demand, 1/p, are he main deerminans of he biases of he differen mehods, and ha ype and variance of he demand disribuion have lile effec. 6

8 As expeced based on he resuls in Secion 3, he Levén & Segersed modificaion performs very poorly wih an average bias of 71%. The original Croson mehod and he Syneos & Boylan modificaion perform considerably beer. However, he original Croson mehod sill performs poorly when 1/p is small (wih biases up o 18%), whereas he Syneos & Boylan modificaion performs poorly when 1/p is large (wih biases up o 12%). The performance of boh mehods deerioraes when α increases. Alhough here are a couple of experimens where eiher he original Croson or he Syneos & Boylan modificaion has he smalles (absolue) bias, he new mehod generally ouperforms he exising mehods for he enire range of considered parameer values. The average absolue bias is 1% for he new mehod as compared o 6% for he original Croson mehod and 5% for he Syneos & Boylan modificaion. 6. Conclusions Building on resuls in he lieraure, a new modificaion of Croson s mehod for forecasing inermien demand was proposed. In a comparaive numerical sudy, his new mehod was shown o significanly ouperform exising mehods. The average absolue bias for he new mehod was 1% as compared o 5%, 6% and 71% for he original Croson mehod, he Syneos & Boylan modificaion and he Levén & Segersed modificaion, respecively. Furhermore, conrary o exising mehods, performance is well for small as well as large demand inervals and does no deeriorae as he smoohing consan increases. The robusness implies ha he new mehod can be used in all cases and avoids he need for demand caegorizaion. References Croson, J.D., Forecasing and sock conrol for inermien demands. Operaional Research Quarerly 23, Eaves, A.H.C., Kingsman, B.G., Forecasing for he ordering and sock-holding of spare pars. Journal of he Operaional Research Sociey 55, Ghobbar, A.A., Friend, C.H., Evaluaion of forecasing mehods for inermien pars demand in he field of aviaion: a predicive model. Compuers and Operaions Research 30,

9 Johnson, F.R., Boylan, J.E., 1996a. Forecasing for iems wih inermien demand. Journal of he Operaional Research Sociey 47, Johnson, F.R., Boylan, J.E., 1996b. Forecasing inermien demand: a comparaive evaluaion of Croson s mehod. Inernaional Journal of forecasing 12, Levén, E., Segersed, A., Invenory conrol wih a modified Croson procedure and Erlang disribuion. Inernaional Journal of Producion Economics 90, Rao, A.V., A commen on: Forecasing and sock conrol for inermien demands. Operaional Research Quarerly 24, Sani, B., Kingsman, B.G., Selecing he bes periodic invenory conrol and demand forecasing mehods for low demand iems. Journal of he Operaional Research Sociey 48, Schulz, C.R., Forecasing and invenory conrol for sporadic demand under periodic review. Journal of he Operaional Research Sociey 38, Snyder, R., Forecasing sales of slow and fas moving invenories. European Journal of Operaional Research 140, Syneos, A.A., Boylan, J.E., On he bias of inermien demand esimaes. Inernaional Journal of Producion Economics 71, Syneos, A.A., Boylan, J.E., The accuracy of inermien demand esimaes. Inernaional Journal of forecasing 21, Syneos, A.A., Boylan, J.E., On he sock conrol performance of inermien demand esimaors. To appear in Inernaional Journal of Producion Economics. Syneos, A.A., Boylan, J.E., Croson, J.D., On he caegorisaion of demand paerns. Journal of he Operaional Research Sociey 56, Teuner, R.H., Duncan, L, Forecasing demand for slow moving invenory. Research Repor, Lancaser Universiy Managemen School. Willemain, T.R., Smar, C.N., Schwarz, H.F., A new approach o forecasing inermien demand for service pars invenories. Inernaional Journal of forecasing 20, Willemain, T.R., Smar, C.N., Shockor, J.H., DeSauels, P.A., Forecasing inermien demand in manufacuring: a comparaive evaluaion of Croson s mehod. Inernaional Journal of forecasing 10,

10 values Cr (series 1) Cr (series 2) S&B (series 1) S&B (series 2) /p Figure 1. Comparison of he average bias over 10,000 periods for he Croson (Cr) and Syneos & Boylan (S&B) mehods for wo randomly generaed demand series (α = 0.15, µ/p = 2, σ = 0; boh mehods iniialized using he correc values). 9

11 Comparison of forecass Value Cr S&B L&S Time Period Figure 2. Comparison of he average bias over 10,000 periods for he Croson (Cr), Syneos & Boylan (S&B) and Levén & Segersed (L&S) mehods for a randomly generaed demand series (α = 0.15, 1/p = 0.2, µ = 5, σ = 1; all mehods iniialized using he correc values). 10

12 ` Demand Value Percenage Bias S/N α 1/p Disribuion Expeced Croson Syneos & Boylan Levén & Segersed New N(1,0.1) % -4% 151% -3% N(1,0.3) % -2% 153% -1% U(1,2) % -3% 149% -3% U(1,10) % 2% 139% 3% N(1,0.1) % -3% 71% -1% N(1,0.3) % -3% 73% -1% U(1,2) % -1% 72% 1% U(1,10) % -2% 68% -1% N(1,0.1) % -3% 37% 0% N(1,0.3) % -3% 41% -1% U(1,2) % -2% 39% 1% U(1,10) % -1% 40% 2% N(1,0.1) % -3% 19% 0% N(1,0.3) % -3% 22% 0% U(1,2) % -3% 19% 1% U(1,10) % -5% 18% -1% N(1,0.1) % -3% 146% -2% N(1,0.3) % -6% 143% -5% U(1,2) % -1% 142% 0% U(1,10) % -3% 166% -1% N(1,0.1) % -4% 74% 0% N(1,0.3) % -4% 75% -1% U(1,2) % -4% 71% -1% U(1,10) % -5% 72% -1% N(1,0.1) % -5% 39% 1% N(1,0.3) % -5% 41% 1% U(1,2) % -6% 38% 0% U(1,10) % -4% 40% 2% N(1,0.1) % -7% 19% 0% N(1,0.3) % -7% 21% 0% U(1,2) % -8% 18% 0% U(1,10) % -7% 19% 1% N(1,0.1) % -2% 141% 0% N(1,0.3) % -3% 165% -1% U(1,2) % 0% 165% 2% U(1,10) % -5% 159% -3% N(1,0.1) % -5% 73% 1% N(1,0.3) % -3% 78% 2% U(1,2) % -5% 74% 1% U(1,10) % -5% 72% 1% N(1,0.1) % -10% 37% -2% N(1,0.3) % -8% 41% 1% U(1,2) % -7% 39% 1% U(1,10) % -6% 40% 3% N(1,0.1) % -12% 18% 0% N(1,0.3) % -10% 22% 1% U(1,2) % -11% 19% 1% U(1,10) % -11% 19% 0% Table 1. Comparison of biases for new and exising mehods. The smalles bias for each of he 48 examples is indicaed in bold. 11