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On the variance of intermittent demand estimates
Int. J. Production Economics 128 (2010) 546–555
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
On the variance of intermittent demand estimates Aris A. Syntetos a,c,n, John E. Boylan b a b c
Centre for Operational Research & Applied Statistics, Salford Business School, University of Salford, Maxwell Building, Manchester M5 4WT, UK Buckinghamshire New University, UK BEM – Bordeaux Management School, France
a r t i c l e in f o
a b s t r a c t
Article history: Received 16 March 2010 Accepted 5 July 2010 Available online 8 July 2010
Intermittent demand occurs at random with many time periods showing no demand at all. Forecasting such demand patterns constitutes a challenging exercise because of the associated dual source of variation (demand intervals and demand sizes). Research in this area has developed rapidly in recent years with new results implemented into supply chain software solutions because of its practical implications. In an inventory context, both the accuracy of the forecasts and their variability (sampling error of the mean) have equal importance in terms of service level achievement and/or inventory cost minimisation. Although the former issue has been studied extensively (mainly building upon Croston’s model, 1972) the latter has been largely ignored. The purpose of this paper is to analyse the most wellcited intermittent demand estimation procedures in terms of the variance of their estimates. Detailed derivations are offered along with a discussion of the underlying assumptions. As such, we hope that our contribution may constitute a point of reference for further analytical work in this area as well as facilitate a better understanding of issues related to modelling intermittent demands. & 2010 Elsevier B.V. All rights reserved.
Keywords: Intermittent demand Forecasting Supply chain management Inventory management
1. Introduction Intermittent demand for products appears sporadically, with some time periods showing no demand at all. When demand occurs, the demand size may be constant or variable, perhaps highly so. Intermittent demand items may be any Stock Keeping Unit (SKU) within the range of products offered by an organisation at any level of the supply chain. Such items may collectively account for up to 60% of the total stock value (Johnston et al., 2003) and are particularly prevalent in the aerospace, automotive, military and IT sectors. They are often the items at greatest risk of obsolescence. Inventory control decisions for intermittent items are needed to determine inventory replenishment rules. These decisions can be made more intelligently if supported by more accurate and less variable demand forecasts. Improvements in forecasting and stock control may be translated to significant reductions in wastage or scrap, and very substantial cost savings with further implications for the effective management of the relevant supply chains. Replenishment requirements are calculated according to an anticipated probability distribution of demand over lead-time. Many organisations that deal with intermittent demand items face significant difficulties in organising their stock in such a way as to minimise inventory-holdings whilst achieving satisfactory service. Intermittent demand patterns are built from constituent
n
Corresponding author. Tel.: +44 161 295 5804. E-mail address: [email protected] (A.A. Syntetos).
0925-5273/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2010.07.005
elements, i.e. a demand arrival process (that results in a sequence of inter-demand intervals) and the demand sizes, when demand occurs. The dual source of variation under concern constitutes the main difficulty in forecasting the requirements for such items. In addition, the inappropriateness of standard distributions (say Normal) for representing these patterns introduces further complications into their management. The standard method used in industry to forecast intermittent demand requirements is Croston’s method (Croston, 1972). The method is incorporated in statistical supply chain forecasting software packages (e.g. Forecast Pro), and demand planning modules of component based enterprise and manufacturing solutions (e.g. Industrial and Financial Systems – IFS AB). It is also included in integrated real-time sales and operations planning processes (e.g. SAP Advanced Planning and Optimisation – APO 4.0). Croston (1972) proved the biased nature of simple exponential smoothing (SES) when applied in an intermittent demand context and he proposed a method that relies explicitly upon estimates of the inter-demand intervals and demand sizes. The method was claimed to be unbiased, but Syntetos and Boylan (2001) showed it to be positively biased (i.e. over-forecasting mean demand) and they subsequently proposed an approximately unbiased estimator: SBA (Syntetos–Boylan Approximation, Syntetos and Boylan, 2005). This estimation procedure applies a deflating factor to the Croston estimates in order to take away the bias. A little bias though still remains, on the opposite side (i.e. slightly underestimating mean demand). Syntetos (2001) also proposed an exactly unbiased modification of Croston’s method (SY method). This procedure was further evaluated by Teunter and Sani (2009)
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
and it was found to perform very well. For a comprehensive account of all recent developments in intermittent demand supply chain forecasting please refer to Fildes et al. (2008) and Syntetos et al. (2009).1 The focus of the above studies has been the bias (or the lack of it) of mean demand estimates. However, the sampling error of the mean (i.e. variability of the estimates) which is of equal importance for both forecasting and stock control purposes has not been explicitly discussed. The achievement of the service target is an important objective, and requires an unbiased forecast of mean demand, assuming that the distribution of demand is known (e.g. Poisson, Negative Binomial) and an estimate of the variance of demand. Minimisation of inventory holdings is achieved by appropriate stock rules (not discussed in this paper) and by using a minimum variance estimator of the mean demand. Consequently, it would be highly desirable to obtain minimum variance—unbiased estimator(s) of mean demand. In this paper the issue of the variance of intermittent demand estimates is explicitly addressed for SES, Croston’s method, SY and SBA. We hope that this paper may serve as a data repository for further analytical work in this area and for facilitating a better understanding of pertinent issues. Our paper concludes with a discussion of the implications of our work for supply chain forecasting practices (referring also to the bullwhip effect) and an agenda for further research in this area. The remainder of our paper is structured as follows: Section 2 the underlying model assumed for the purposes of this research is presented along with the corresponding variance of SES estimates. The variance of the estimates produced by the Croston’s method, SY and SBA are subsequently discussed in Sections 3–5, respectively. The theoretical results presented in this paper are summarised in Section 6 where the implications of this work for real world supply chain practices are also discussed. Finally, in Section 7, we present the conclusions of this research along with some important lines of further enquiry in this area.
2. An intermittent demand model Croston advocated separating the demand into two components, the inter-demand time and the size of demand, and analysing each component separately. He assumed a stationary mean model for representing the underlying demand pattern, normal distribution for the size of demand and a Bernoulli demand generation process, resulting in geometrically distributed inter-demand intervals. Three more assumptions implicitly made by Croston in developing his model are the following: independence between demand sizes and inter-demand intervals, independence of successive demand sizes and independence of successive interdemand intervals. As far as the last assumption is concerned it is important to note that the geometric distribution is characterised by a ‘memory less’ process: the probability of a demand occurring is independent of the time since the last demand occurrence, so that this distributional assumption is consistent with independent inter-demand intervals. The normality assumption is the most restrictive one for the analysis conducted by Croston, since the demand sizes may be, theoretically, represented by any probability distribution without 1 At this point it is important to note that one more modified Croston procedure has appeared in the literature (Leve´n and Segerstedt, 2004). However, this method was found to be even more biased than the original Croston’s method (Boylan and Syntetos, 2007); Teunter and Sani, 2009) and it is not further discussed in this paper.
547
affecting the results. The remaining assumptions are retained for the analysis to be conducted in this paper. If we let pt ¼the inter-demand interval that follows the geometric distribution with: E(pt)p zt ¼the demand size, when demand occurs, with a mean m and variance s2 1/p¼the underlying Bernoulli probability of demand occurrence then the demand per unit time period (Yt) can be represented by 8 1 > > > < 0, 1 p , probability of demand occurence ð1Þ Yt ¼ 1 > > , probability of demand occurence > zt , : p with EðYt Þ ¼ 0
p1 1 m þ Eðzt Þ ¼ p p p
ð2Þ
Consequently the variance of demand per unit time period is calculated as follows (Croston, 1972): VarðYt Þ ¼
p1 2 s2 m þ p p2
ð3Þ
2.1. The variance of SES estimates Croston (1972) proved the inappropriateness of exponential smoothing as a forecasting method when dealing with intermittent demands and he expressed in a quantitative form the bias associated with the use of this method when demand occurs according to the model described above. Exponential smoothing is unbiased if we consider the estimates made at the end of every forecast review period (all points in time). It is biased though if we isolate the estimates made after a demand occurrence (issue points only). The former scenario corresponds to a re-order interval/periodic review stock control system whereas the latter to a re-order level/continuous review model. The bias properties of SES are summarised in Table 1. The variance of the exponentially smoothed estimates, updated every period, is a a p1 2 s2 VarðYut Þ ¼ ð4Þ VarðYt Þ ¼ m þ 2a 2a p2 p (where a is the smoothing constant employed for updating purposes) assuming a stationary mean model and homogeneous variance of demand per unit time period. If we isolate the estimates that are made just after an issue (which are those that will be used for replenishment purposes by a continuous review stock control system) Croston showed that these estimates have the following variance (as corrected by Rao, 1973): ab2 p1 2 s2 VarðYut Þ ¼ a2 s2 þ ð5Þ m þ 2a p2 p where b ¼1 a.
3. The variance of Croston’s estimates Croston (1972) suggested estimating the average interval between successive demand occurrences and the average size of demand when it occurs and to combine those statistics to give an unbiased estimate of the underlying mean demand. If we let Varðpt Þ ¼ ðp1Þ2 ðas assumed by CrostonÞ
ð6Þ
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Table 1 Intermittent demand estimators: bias and sampling error of the mean. Estimator
E (Y0 t)
SES—all points
¼ m/p
SES—issue points
¼ m aþ
Var (Y0 t) ¼
SBA (Syntetos and Boylan, 2005)
E m/p
m
p0 t is the exponentially smoothed inter-demand interval, updated only after demand occurs (with smoothing constant a), and z0 t is the exponentially smoothed size of demand, updated only after demand occurs (using the same smoothing constant a)we then have Eðzut Þ ¼ Eðzt Þ ¼ m
ð7Þ
Eðput Þ ¼ Eðpt Þ ¼ p
ð8Þ
Varðzut Þ ¼
a 2a
a
Varðzt Þ ¼
a 2a
s2
ð9Þ
a
Varðpt Þ ¼ ðp1Þ2 ð10Þ 2a 2a The estimation equation for Croston’s method is given by
Varðput Þ ¼
Yut ¼
zut put
¼ 2
ab p1 2 s m þ ¼ a2 s2 þ 2a p2 p a p1 2 a 2 s2 m þ s þ 2 2a p3 2a p a2 2 a p s þ pðp1Þm2 þ pðp1Þs2 að2aÞ 2 2a a4 4 p 2 a2 ¼ 1 VarðCrostonÞ 2
p
SY (Syntetos, 2001)
a
VarðYt Þ 2a
2
b
a ðp1Þ þ m 2 p 2a p ¼ m/p
Croston (Croston, 1972)
Var (Yt)
ð11Þ
and the expected estimate produced by this method has been shown to be (Syntetos and Boylan, 2005) zut m a ðp1Þ E m 2 ð12Þ þ put p 2a p The variance of the ratio of two independent random variables x1, x2 is given in Stuart and Ord (1994) as follows: # " x1 Eðx1 Þ 2 Varðx1 Þ Varðx2 Þ þ ¼ Var ð13Þ Eðx2 Þ x2 ðEðx1 ÞÞ2 ðEðx2 ÞÞ2 For x1 ¼z0 t and x2 ¼p0 t, considering Eqs. (7)–(10), the variance of the estimates produced by using Croston’s method is calculated by " # zut a ðp1Þ2 2 s2 VarðYut Þ ¼ Var m þ 2 ¼ ð14Þ 2a put p4 p assuming that the same smoothing constant value is used for updating demand sizes and inter-demand intervals and that both demand size and inter-demand interval series are not autocorrelated and have homogeneous variances. Rao (1973) pointed out that the right hand side of Eq. (14) is only an approximation to the variance. This follows since (13) is, in fact, an approximation. 3.1. The variance of inter-demand intervals The number of independent Bernoulli trials (with a specified probability of success) before the first success is represented by the geometric distribution. An alternative form of the geometric distribution involves the number of trials up to and including the first success (demand occurring period). Considering the notation used in this paper, the variability of the geometrically distributed inter-demand intervals is pðp1Þ, irrespectively of which form of the geometric distribution is utilised. Consequently (6) should be
p1 2 s2 m þ p p2
replaced by Varðpt Þ ¼ pðp1Þ
ð15Þ
3.2. The corrected variance of Croston’s method estimates By taking (15) into consideration, the variance of the demand per period estimates, using Croston’s method, would become zut a p1 2 s2 m þ 2 ð16Þ Var 3 2a p put p indicating that the approximated variance of the estimates produced by Croston’s method is in fact greater than that calculated by Croston himself, Eq. (14).2 Nevertheless, approximation (16) is still not correct. In fact there is a fundamental problem in directly applying Stuart and Ord’s result, given by (13), for the purpose of deriving the variance of the forecasts produced by a biased estimator. This is proven as follows: We apply Taylor’s theorem to a function of two variables, g(x), where, x is the vector: x ¼(x1, x2) and g(x)¼g(x1, x2)¼x1/x2 with E(x1)¼ y1 and E(x2)¼ y2. The vector y is defined as: y ¼(y1, y2), with g(y)¼(y1, y2) ¼ y1/y2 @g @g gðxÞ ¼ gðyÞ þ ðx1 y1 Þ þ ðx2 y2 Þ þ ð17Þ @y1 @y2 where g(y) ¼ y1/y2 is just the first term in the Taylor series and not necessarily the population expected value. For E½gðxÞ ¼ gðyÞ þ e
ð18Þ
where e is an error term, which according to Croston, can be neglected, we then have Var ½ gðxÞ ¼ EfgðxÞE½gðxÞg2 ¼ E½gðxÞgðyÞ2 # " 2 @g @g Eðx1 Þ 2 Varðx1 Þ Varðx2 Þ E ðx1 y1 Þ þ ðx2 y2 Þ ¼ þ @y1 @y2 Eðx2 Þ Eðx1 Þ2 Eðx2 Þ2 ð19Þ If we set, x1 ¼z0 t, the estimate of demand size, with E(z0 t) ¼ m and x2 ¼p0 t, the estimate of the inter-demand interval, with E(p0 t)¼p, so that g(x)¼Y0 t, it has been proven (Syntetos and Boylan, 2001, 2005) that EðYut Þ a
2
m p
or E½ gðxÞ agðyÞ
Eq. (10) in the original paper.
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
Based on that, we argue that the error term in Eq. (18) cannot be neglected and therefore approximation (19) cannot be used to represent the problem in hand. Our argument is discussed in greater detail in Appendix A and B, where we also derive a correct approximation (to the second order term) of the variance of Croston’s estimates. That variance expression is given by zut a ðp1Þ 2 a 2 s2 Var þ s m þ put 2a 2a p3 p2 a4 m2 1 1 2 9 1 p þ1 þ 1 ð20Þ 4 p4 p p 1ð1aÞ Syntetos (2001) showed, by means of simulation on a wide range of theoretically generated data (matching the demand patterns encountered in many supply chain organisations), that approximation (20) does not increase the accuracy of the calculated variance more than by only considering the first term of this approximation. In fact, for certain cases, approximation (20) was shown to perform worse. Taking that into account the variance of Croston’s estimates may be ‘safely’ approximated by zut a ðp1Þ 2 a 2 s2 Var s þ 2 m þ ð21Þ 3 put 2a 2a p p 4. The variance of the SY method estimates The estimation equation for the SY method discussed in Section 1 is given by Yut ¼
ð1ða=2ÞÞzut ð1ða=2ÞÞzut ¼ ð1ða=2put ÞÞput put ða=2Þ
and the expected estimate produced by this method can be shown to be as follows (Syntetos, 2001): ð1ða=2ÞÞzut m EðYut Þ ¼ E ð23Þ put ða=2Þ p In Appendix C we perform a series of calculations in order to find the variance of the estimates of mean demand produced by the SY method. The variance is approximated by h i 2 2 2 a 2 ð1ða=2ÞÞzut að2aÞ ðpða=2ÞÞ s þ pðp1Þm þ 2a pðp1Þs Var 4 4 put ða=2Þ pða=2Þ þ
a4 1ð1aÞ
4
with a zut m a m EðYut Þ ¼ E 1 2 put p 2 p2
2 1ða=2Þ m2 2 1 1 2 9 1 p þ1 6 p 1 p p pða=2Þ
ð24Þ Similarly to the previous section, Syntetos (2001) showed by means of simulation that consideration of both terms of approximation (24) does not provide, overall, a more reliable estimate of the calculated variance than when only the first term of this approximation is considered. The calculation of the variance of the SY method may be simplified, without sacrificing accuracy, by considering approximation (25) h i 2 2 2 2 ð1ða=2ÞÞzut að2aÞ ðpða=2ÞÞ s þ pðp1Þm þða=ð2aÞÞpðp1Þs Var 4 put ða=2Þ ðpða=2ÞÞ4
ð25Þ 5. The variance of the SBA method estimates The estimation procedure for the SBA method (Syntetos and Boylan, 2005) is a zut ð26Þ Yut ¼ 1 2 put
ð27Þ
The variance of the estimates produced by the SBA is calculated as a zut a2 zut VarðYut Þ ¼ Var 1 Var ¼ 1 ð28Þ 2 put 2 put Considering approximation (21) we finally have a zut að2aÞ ðp1Þ 2 a 2 s2 s þ 2 m þ ð29Þ Var 1 3 4 2 put 2a p p Extensive analysis conducted by Syntetos (2001) justified the choice of (21) instead of (20) for the purpose of approximating the variance of the SBA method. In the next section the theoretical results presented thus far in our paper are summarised followed by a discussion of their implications for supply chain forecasting practices.
6. Summary of theoretical results and discussion The statistical properties of the methods discussed thus far in this paper are summarised in Table 1. In particular, we indicate the expected estimate produced by each of these methods along with the variance of their estimates (sampling error of the mean). The variance of demand itself is presented as well. Such results enable the calculation of the theoretical one-step-ahead mean squared error (MSE) associated with SES, Croston, SY and SBA. MSE ¼ VarðdemandÞ þVarðEstimatesÞ þ Bias2
ð22Þ
549
ð30Þ
The MSE over a fixed lead time of duration L, may also be calculated as follows (see Strijbosch et al., 2000; Syntetos et al., 2005): MSEL ¼ LfLVarðEstimatesÞ þLBias2 þVarðActual DemandÞg
ð31Þ
(In both (30) and (31) the bias term is obviously zero for unbiased estimators.) Such results may be of particular value for computerised supply chain/inventory management systems relying upon analytical variance expressions, as opposed to estimation through smoothed MSE or mean absolute deviation (MAD) procedures. In addition, analytical MSE expressions enable the direct comparison of the performance of these methods through the consideration of both bias and sampling error of the mean (see also Syntetos et al., 2005). Teunter and Sani (2009) commented on the unbiased nature of the SY method and recommended that the estimator under concern receives more attention as a competitive alternative to SBA. Although SY is indeed unbiased, a detailed analysis conducted by Syntetos (2001) showed that this method compares unfavourably to SBA with regards to their MSE performance due to the increased variance of the relevant estimates. This is a very important point both from a theoretical and a practitioner’s perspective since, typically, greater emphasis is being placed on the bias (or the lack of it) associated with an estimation procedure rather than the variance of the estimates. Ideally though, both should be equally considered since they collectively determine the supply chain implications of using the method under concern for stock control purposes. 6.1. Implications for supply chain management The Bullwhip Effect is a term promoted by Lee et al. (1997) but was observed and modelled decades earlier by Forrester (1958). It occurs whenever demand becomes more variable as it proceeds through the supply chain, away from the consumer and towards
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A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
the supplier. Lee et al. (2000) discussed four causes of the Bullwhip Effect, namely: demand signal processing, rationing/ shortage gaming, order batching and price fluctuations. Demand signal processing in particular refers to the magnification in variance that occurs through the forecast error and the interaction between forecasting procedures and inventory rules at each stage of the supply chain (see also Chen et al., 2000a, b; for a review of studies in this area please refer to Syntetos et al., 2009). Such interactions account for significant discrepancies between the actual performance of an inventory system and what is theoretically expected; they are particularly prevalent in an intermittent demand context where very large forecast errors are all too common due to the difficulties associated with the forecasting task. For example, major deviations have been reported in the literature between the target and achieved customer service levels in such a context of application (Syntetos and Boylan, 2008). Forecast variance is a key contributory factor to the Bullwhip Effect, and as such any research oriented to variance related issues should be of direct interest to supply chain practitioners. Having mentioned that, no research has been conducted to-date on the behaviour of the Bullwhip Effect for supply chains dealing with intermittent demand products, for example service parts. Such products have become ubiquitous in modern societies and studying the relevant supply chains should be very important both from an academic and practitioner perspective.
7. Conclusions and extensions The area of intermittent demand forecasting has received a number of theoretical contributions in the recent years, most of which have been focusing on the issue of bias of the relevant estimation procedures. In this paper, we have argued for the additional consideration of the variance of intermittent demand estimates since this is equally important in terms of a method’s implications for stock control. Analytical expressions are offered with regards to the sampling error of the mean for four estimation procedures: SES, Croston, SY and SBA. The bias properties of these estimators are also presented and the results collectively permit the calculation of the relevant Mean Squared Error (MSE) expressions. MSE is the only mathematically tractable forecast accuracy metric
and its quantification allows comparisons to be performed between various estimators. In addition, such results may be particularly useful for computerised systems that rely upon analytical variance expressions for stock replenishment purposes. All the derivations presented in this paper are based upon Croston’s (1972) assumptions for modelling intermittent demands. Although, we feel that such derivations should facilitate further developments in this area, we cannot lose sight of the fact that Croston’s demand model is incompatible with his forecasting method (and its variants), because exponential smoothing methods are not designed for i.i.d. (identically and independently distributed) demand. Obviously, this does not mean that Croston’s method and its variants are not useful. Such methods do constitute the current state of the art in intermittent demand parametric forecasting. Nevertheless, an interesting line of further research would be to develop alternative models for intermittent demand forecasting, although one needs to consider also the following: a method that is optimal for one particular model may be severely sub-optimal for another model. Syntetos et al. (2006) argued that, for intermittent demand, robustness of a method across a wide range of possible underlying demand models is more important than optimality under one particular model. The scarcity of demand observations presents a significant inherent difficulty in identifying the demand model in the first place. In addition to further opportunities in modelling intermittent demands for forecasting purposes, modelling the effects of the relevant forecast errors in a supply chain context constitutes also an interesting line for further research. Demand signal processing has been shown to be a key determinant of the Bullwhip Effect but this issue has never been explored for supply chains that deal with intermittent demand items (such as service parts for example). Given the prevalence of such demand structures in modern organisational settings and given the difficulties associated with forecasting the relevant requirements, research in this area would appear to be merited. Acknowledgements The research described in this paper has been supported by the Engineering and Physical Sciences Research Council (EPSRC, UK) Grant nos. EP/D062942/1 and EP/G006075/1.
Appendix A. A correct approximation to the variance of Croston’s estimates We apply Taylor’s theorem to a function of two variables, g(x), where x is the vector: x¼(x1, x2) and g(x)¼(x1, x2)¼x1/x2 with E(x1) ¼ y1 and E(x2)¼ y2. The vector y is defined as: y ¼ (y1, y2), with g(y) ¼g(y1, y2)¼ y1/y2 " # @g @g 1 @2 g @2 g @2 g 2 2 gðxÞ ¼ gðyÞ þ ðx1 y1 Þ þ ðx2 y2 Þ þ ðx y Þ þ2 ðx y Þðx y Þ þ ðx y Þ þ ðA:1Þ 1 1 1 1 2 2 2 2 2 @y1 @y2 2 @y2 @y1 @y2 @y 1
2
where g(y)¼ y1/y2 is just the first term in the Taylor series and not the population expected value @g 1 ¼ @y1 y2
ðA:2Þ
@g y1 ¼ 2 @y2 y2
ðA:3Þ
@2 g
¼0
ðA:4Þ
@2 g 1 ¼ 2 @y1 @y2 y2
ðA:5Þ
@y1
2
@2 g @y2
2
¼ y1
2
y2 3
! ¼
2y1
y2 3
ðA:6Þ
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
551
therefore, considering (A.4), (A.1) becomes gðxÞ ¼ gðyÞ þ
# " 2 @g @g @ g 1 @2 g 2 ðx1 y1 Þ þ ðx2 y2 Þ þ ðx1 y1 Þðx2 y2 Þ þ ðx y Þ þ 2 2 @y1 @y2 @y1 @y2 2 @y2 2
ðA:7Þ
We set, x1 ¼ z0 t, the estimate of demand size, with E(z0 t)¼ m and x2 ¼p0 t, the estimate of the inter-demand interval, with E(p0 t)¼ p, so that g(x)¼Y0 t. It has been proven (Syntetos and Boylan, 2005) that EðYut Þ ¼ E½ gðxÞ gðyÞ þ
1 @2 g Eðx2 y2 Þ2 2 @y2 2
ðconsidering the first three terms in the Taylor series:Þ Therefore 92 8 @g @g @2 g 1 @2 g > > > > > ðx1 y1 Þ þ ðx2 y2 Þ þ ðx1 y1 Þðx2 y2 Þ þ ðx2 y2 Þ2 > > > 2 = < @y1 @y2 @y1 @y2 2 @y2 VarðYut Þ ¼ Var ½ gðxÞ ¼ E½gðxÞE½gðxÞ2 E 2 > > 1 @ g > > > > Eðx2 y2 Þ2 > > ; :2 2 @y2 8 !2 !2 < @g 2 @g 2 @2 g 1 @2 g ¼E ðx1 y1 Þ2 þ ðx2 y2 Þ2 þ ðx1 y1 Þ2 ðx2 y2 Þ2 þ ðx2 y2 Þ4 : @y1 @y2 @y1 @y2 4 @y2 !2 1 @2 g @g @g þ ½Eðx2 y2 Þ2 2 þ2 ðx1 y1 Þ ðx2 y2 Þ 4 @y2 @y1 @y2 þ2
@g @2 g @g @2 g @g @2 g ðx1 y1 Þ ðx1 y1 Þðx2 y2 Þ þ ðx1 y1 Þ ðx2 y2 Þ2 ðx1 y1 Þ Eðx2 y2 Þ2 2 2 @y1 @y1 @y2 @y1 @y1 @y2 @y2
þ2
@g @2 g @g @2 g @g @2 g ðx2 y2 Þ ðx1 y1 Þðx2 y2 Þ þ ðx2 y2 Þ3 ðx2 y2 ÞEðx2 y2 Þ2 2 @y2 @y1 @y2 @y2 @y2 @y2 @y2 2
@2 g @2 g @2 g @2 g 1 @2 g þ ðx1 y1 Þðx2 y2 Þ ðx2 y2 Þ2 ðx1 y1 Þðx2 y2 Þ Eðx2 y2 Þ2 2 2 @y1 @y2 @ 2 @y2 2 y @ y 1 2 @y2 @y2
!2 2
2
ðx2 y2 Þ Eðx2 y2 Þ
9 =
ðA:8Þ
;
Assuming that x1, x2 are independent
@g Var ½ gðxÞ @ y1
2
@g Eðx1 y1 Þ þ @ y2 2
2
@2 g Eðx2 y2 Þ þ @y1 @y2
!2
2
@g @2 g 1 @2 g Eððx1 y1 Þ ðx2 y2 Þ Þ þ Eðx2 y2 Þ3 þ @ y2 @ y2 2 4 @ y2 2 2
2
!2
1 @2 g Eðx2 y2 Þ 4 @ y2 2 4
!2 ½Eðx2 y2 Þ2 2
ðA:9Þ considering Eqs. (A.2), (A.3), (A.5) and (A.6) Var ½ gðxÞ ¼
Varðx1 Þ
y2
2
Varðx1 Þ ðEðx2 ÞÞ2
þ þ
2 2 2 y1 2 Varðx2 Þ 1 2y1 Eðx2 y2 Þ3 y1 Eðx2 y2 Þ4 y2 ½Eðx2 y2 Þ2 2 þ 4 Varðx1 ÞVarðx2 Þ þ 5 4 6 y2 y2 y2 y2 y2 6 2 2 3 2 4 ðEðx1 ÞÞ Varðx2 Þ Varðx1 ÞVarðx2 Þ 2ðEðx1 ÞÞ Eðx2 y2 Þ ðEðx1 ÞÞ Eðx2 y2 Þ ðEðx1 ÞÞ2 ½Varx2 2
ðEðx2 ÞÞ4
þ
ðEðx2 ÞÞ4
ðEðx2 ÞÞ5
þ
ðEðx2 ÞÞ6
ðEðx2 ÞÞ6
ðA:10Þ
In Appendix B it is proven that for n ¼2,3 E½xut EðxÞn ¼
an 1ð1aÞn
E½xt EðxÞn
ðA:11Þ
and also that E½xut EðxÞ4 ¼
a4 1ð1aÞ
4
E½xt EðxÞ4 þ
a4 ð1ð1aÞ2 Þ2
½Varðxt Þ2
ðA:12Þ
where xt represents the demand size (zt) or inter-demand interval (pt), x0 t is their exponentially smoothed estimate (z0 t, p0 t) and E(x) is the population expected value for either series. Consideration of (A.11) and (A.12) necessitates the adoption of the following assumptions:
no auto-correlation for the demand size and inter-demand interval series homogeneous moments about the mean for both series same smoothing constant value is used for both series Taking also into account that: Varðzt Þ ¼ s2 and Varðpt Þ ¼ pðp1Þ
552
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
(A.10) becomes Var
zut put
a s2 2a p2
þ
a 2a
m2
a 2 m2 pðp1Þ a 2 s2 pðp1Þ a3 2m2 a4 m2 a4 m2 2 þ Eðpt pÞ3 þ Eðpt pÞ4 þ p ðp1Þ2 p2 ðp1Þ2 2a 2a p6 p4 p4 1ð1aÞ3 p5 1ð1aÞ4 p6 ð1ð1aÞ2 Þ2 p6
ðA:13Þ The third moment about the mean in the geometric distribution, where: 1/p is the probability of success in each trial, is calculated as 1 1 1 p1 1 1 p1 1 2p1 ðp1Þð2p1Þ 1 þ1 ¼ ¼ ¼ 2 Eðpt pÞ3 ¼ 1 p p p3 p p3 p p p3 p p5
ðA:14Þ
and the fourth moment: Eðpt pÞ4 ¼
9ð1ð1=pÞÞ2 1ð1=pÞ 1 9ð1ð1=pÞÞ 1 1 1 4 1 1 2 9 1 p þ p2 ¼ p2 1 9 1 p þ1 þ ¼ 1 þ ¼ 1 4 2 4 2 p p p p p 1=p 1=p 1=p 1=p
ðA:15Þ
If we consider (A.14) and (A.15), and also the fact that
a4 2 2
ð1ð1aÞ Þ
¼
a 2 , 2a
(A.13) becomes Var
zut put
1 2 9 1 p þ1 p
a s2 a 2 pðp1Þ a 2 s2 pðp1Þ a3 2m2 ðp1Þð2p1Þ a4 m2 1 þ þ þ 1 m 2 4 4 3 10 4 p 2a p 2a 2a p p p 1ð1aÞ 1ð1aÞ p4
ðA:16Þ
Since the fourth part of approximation (A.16) becomes almost zero even for quite low average inter-demand intervals, finally the variance is approximated by (A.17) Var
zut put
a ðp1Þ 2 a 2 s2 a4 m2 1 1 2 9 1 p þ m þ 1 þ1 þ s 2a p p 2a p3 p2 1ð1aÞ4 p4
ðA:17Þ
This proves the result given by Eq. (20).
Appendix B. The 2nd, 3rd and 4th moment about the mean for exponentially smoothed estimates If we define xu ¼
X1
EðxuÞ ¼
j¼0
að1aÞj xtj ði:e: the EWMA estimateÞ
X1 j¼0
ðB:1Þ
að1aÞj Eðxtj Þ ¼ assuming Eðxtj Þ ¼ EðxÞ for all j Z0 aEðxÞ
X1 j¼0
ð1aÞj ¼
a 1ð1aÞ
EðxÞ ¼ EðxÞ
ðB:2Þ
Therefore we can write
xuEðxÞ ¼
1 X
j
n
að1aÞ ½xtj EðxÞ, ½xuEðxÞ ¼
j¼0
8
j¼0
9n = að1aÞ ½xtj EðxÞ ; j
and
8 9n
ðB:3Þ
j¼0
Assuming series is not auto-correlated, for n ¼2,3 we then have E½xuEðxÞn ¼
1 X
an ð1aÞnj E½xtj EðxÞn
ðB:4Þ
j¼0
and assuming E[xt j E(x)]n ¼E[x E(x)]n for all jZ0, i.e. homogeneous moments of order n E½xuEðxÞn ¼
an 1ð1aÞn
E½xEðxÞn
For n ¼2 VarðxuÞ ¼
a2 1ð1aÞ2
VarðxÞ
ðB:5Þ
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
553
and for n¼3
a3
E½xuEðxÞ3 ¼
1ð1aÞ3
E½xEðxÞ3
For n ¼ 4, Eq. (B.3) becomes
4
E½xuEðxÞ ¼ E
8
j
að1aÞ ½xtj EðxÞ
j¼0
assuming no auto-correlation ¼
94 = ; 1 X
a4 ð1aÞ4j E½xtj EðxÞ4 þ
j¼0
1 X 1 X
a2 ð1aÞ2j E½xtj EðxÞ2 a2 ð1aÞ2i E½xti EðxÞ2
j¼0i¼0 4
assuming homogeneous moments about the mean ¼ a E½xtj EðxÞ4
1 X
ð1aÞ4j þ a4 ½VarðxÞ2
j¼0
1 X
ð1aÞ4j ¼
j¼0 1 X 1 X
1 X 1 X
ð1aÞ2j ð1aÞ2i
ðB:6Þ
j¼0i¼0
1
ðB:7Þ
1ð1aÞ4
ð1aÞ2j ð1aÞ2i ¼ 1 þð1aÞ2 þ ð1aÞ4 þð1aÞ6 þ
j ¼ 0i ¼ 0
þ ð1aÞ2 þð1aÞ4 þ ð1aÞ6 þ þ ð1aÞ4 þ ð1aÞ6 þ
ðB:8Þ
: : : : 1 X 1 X
ð1aÞ2j ð1aÞ2i ¼ 1 þ 2ð1aÞ2 þ3ð1aÞ4 þ 4ð1aÞ6 þ
ðB:9Þ
j¼0i¼0
If we multiply the first and the second part of Eq. (B.9) with (1 a)2, we then have 1 X 1 X
ð1aÞ2
ð1aÞ2j ð1aÞ2i ¼ ð1aÞ2 þ2ð1aÞ4 þ 3ð1aÞ6 þ
ðB:10Þ
j¼0i¼0
Subtracting (B.9) and (B.10) 1 X 1 h iX ð1aÞ2j ð1aÞ2i ¼ 1 þð1aÞ2 þ ð1aÞ4 þ ð1aÞ6 þ ¼ 1ð1aÞ2 j¼0i¼0
1 1ð1aÞ2
Therefore 1 X 1 X
ð1aÞ2j ð1aÞ2i ¼
j¼0i¼0
1
ðB:11Þ
ð1ð1aÞ2 Þ2
Considering Eqs. (B.7) and (B.11), Eq. (B.6) becomes E½xuEðxÞ4 ¼
a4 1x2 4
E½xEðxÞ4 þ
a4 ð1ð1aÞ2 Þ2
½VarðxÞ2
ðB:12Þ
This proves the results given by Eqs. (A.11) and (A.12).
Appendix C. The variance of the SY method’s estimates We set, for the problem under concern a x1 ¼ 1 zut 2 with expected value h a i a a Eðx1 Þ ¼ y1 ¼ E 1 zut ¼ 1 Eðzut Þ ¼ 1 m 2 2 2 and variance Varðx1 Þ ¼ Var
h
1
a i
a2 a2 a a2 a 2 zut ¼ 1 s Varðzut Þ ¼ 1 Varðzt Þ ¼ 1 2 2 2 2a 2 2a
ðC:1Þ
ðC:2Þ
554
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
and x2 ¼ put
a
2 with expected value
a a a ¼ Eðput Þ ¼ p Eðx2 Þ ¼ y2 ¼ E put 2 2 2 and variance
ðC:3Þ
a a a ¼ Varðput Þ ¼ Varðpt Þ ¼ pðp1Þ ðC:4Þ Varðx2 Þ ¼ Var put 2 2a 2a (for the variance derivations consider also Appendix B). The third and the fourth moments about the mean for the x2 variable (consider also Appendix A and B) are calculated as follows: a a3 a3 a3 ðp1Þð2p1Þ ¼ Eðput pÞ3 ¼ Eðpt pÞ3 ¼ Eðx2 y2 Þ3 ¼ E put p þ p5 2 2 1ð1aÞ3 1ð1aÞ3
ðC:5Þ
a a4 a4 a4 a4 1 1 2 a 2 2 9 1 p þ1 þ Eðx2 y2 Þ4 ¼ E put pþ ¼ Eðput pÞ4 ¼ Eðpt pÞ4 þ ½Varðpt Þ2 ¼ p2 1 p ðp1Þ2 4 2 2 4 2 2 2a p p 1ð1aÞ ð1ð1aÞ Þ 1ð1aÞ
ðC:6Þ (assuming that the same smoothing constant value is used for both x1 and x2 series and that both series are not auto-correlated and have homogeneous moments about the mean). Consequently we apply Taylor’s theorem to a function of two variables, g(x)¼x1/x2 with @g 1 ¼ @y1 y2
ðC:7Þ
@g y1 ¼ 2 @y2 y2
ðC:8Þ
@2 g
¼0
ðC:9Þ
@2 g 1 ¼ 2 @y1 @y2 y2
ðC:10Þ
@y1
2
@2 g @y2
2
¼ y1
!
2
y2 3
¼
2y1
ðC:11Þ
y2 3
If we consider only the first three terms, we have # " 2 @g @g @ g 1 @2 g 2 gðxÞ ¼ gðyÞ þ ðx1 y1 Þ þ ðx2 y2 Þ þ ðx1 y1 Þðx2 y2 Þ þ ðx y Þ þ 2 2 @y1 @y2 @y1 @y2 2 @y2 2
ðC:12Þ
with x1 ð1ða=2ÞÞzut m ¼E E½ gðxÞ ¼ E x2 put ða=2Þ p y1 ð1ða=2ÞÞm m a gðyÞ ¼ ¼ y2 pða=2Þ p
ðC:13Þ
and (consider also Appendix A) Var ½ gðxÞ ¼ E½gðxÞE½gðxÞ2 ( )2 y1 @g @g @2 g 1 @2 g x1 2 E þ ðx1 y1 Þ þ ðx2 y2 Þ þ ðx1 y1 Þðx2 y2 Þ þ ðx y Þ E 2 2 @y2 @y1 @y2 2 @y2 2 y2 @y1 x2
ðC:14Þ
In order to simplify somewhat the derivation of the variance of the SY method’s estimates we approximate E[g(x)] by: E½ gðxÞ gðyÞ þ
1 @2 g Eðx2 y2 Þ2 2 @y2 2
(based on (C.12) and considering the results presented by Syntetos and Boylan, 2005) Considering (A.10) and assuming that x1 ,x2 are independent, (C.14) becomes Var ½gðxÞ
Varðx1 Þ 2
ðEðx2 ÞÞ
þ
ðEðx1 ÞÞ2 Varðx2 Þ ðEðx2 ÞÞ
4
þ
Varðx1 ÞVarðx2 Þ ðEðx2 ÞÞ
4
2ðEðx1 ÞÞ2 Eðx2 y2 Þ3 5
ðEðx2 ÞÞ
þ
ðEðx1 ÞÞ2 Eðx2 y2 Þ4 6
ðEðx2 ÞÞ
ðEðx1 ÞÞ2 ½Var x2 2 ðEðx2 ÞÞ6
ðC:15Þ
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
555
Finally the variance of the estimates of the SY method is calculated as follows:
Var
1ða=2Þ zut a ð1ða=2ÞÞ2 s2 a ð1ða=2ÞÞ2 m2 pðp1Þ ð1ða=2ÞÞ2 ða=ð2aÞÞ2 s2 pðp1Þ ð1ða=2ÞÞ2 m2 a3 ðp1Þð2p1Þ þ þ put ða=2Þ 2a ðpða=2ÞÞ2 2a p5 ðpða=2ÞÞ5 ð1aÞ3 ðpða=2ÞÞ4 ðpða=2ÞÞ4 þ
a4
ð1ða=2ÞÞ2 m2
1ð1aÞ4 ðpða=2ÞÞ6
a 2 ð1ða=2ÞÞ2 m2 1 1 2 a 2 ð1ða=2ÞÞ2 m2 2 9 1 p þ1 þ p2 1 p ðp1Þ2 p2 ðp1Þ2 6 2a 2a p p ðpða=2ÞÞ ðpða=2ÞÞ6
ðC:16Þ Since the fourth part of approximation (C.16) becomes almost zero even for quite low average inter-demand intervals, and the last two terms cancel each other, finally the variance is approximated by
Var
h i 2 2 2 2 1ða=2Þ zut a a2 ðpða=2ÞÞ s þ pðp1Þm þða=ð2aÞÞpðp1Þs a4 ð1ða=2ÞÞ2 m2 2 1 1 2 9 1 p 1 þ p 1 þ1 p p put ða=2Þ 2a 2 ðpða=2ÞÞ4 1ð1aÞ4 ðpða=2ÞÞ6 ðC:17Þ
This proves the result given by Eq. (24).
References Boylan, J.E., Syntetos, A.A., 2007. The accuracy of a modified Croston procedure. International Journal of Production Economics 107, 511–517. Chen, F., Drezner, Z., Ryan, J.K., Simchi-Levi, D., 2000a. Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting, lead times, and information. Management Science 46, 436–443. Chen, F., Ryan, J.K., Simchi-Levi, D., 2000b. The impact of exponential smoothing forecasts on the bullwhip effect. Naval Research Logistics 47, 269–286. Croston, J.D., 1972. Forecasting and stock control for intermittent demand. Operational Research Quarterly 23, 289–303. Fildes, R., Nikolopoulos, K., Crone, S., Syntetos, A.A., 2008. Forecasting and operational research: a review. Journal of the Operational Research Society 59, 1150–1172. Forrester, J.W., 1958. Industrial dynamics—a major breakthrough for decision makers. Harvard Business Review 36, 37–66. Johnston, F.R., Boylan, J.E., Shale, E.A., 2003. An examination of the size of orders from customers, their characterisation and the implications for inventory control of slow moving items. Journal of the Operational Research Society 54, 833–837. Lee, H.L., Padmanabhan, V., Whang, S., 1997. Information distortion in a supply chain: the Bullwhip effect. Management Science 43, 546–558. Lee, H.L., So, K.C., Tang, C.S., 2000. The value of information sharing in a two-level supply chain. Management Science 46, 626–643.
Leve´n, E., Segerstedt, A., 2004. Inventory control with a modified Croston procedure and Erlang distribution. International Journal of Production Economics 90, 361–367. Rao, A.V., 1973. A comment on: forecasting and stock control for intermittent demands. Operational Research Quarterly 24, 639–640. Strijbosch, L.W.G., Heuts, R.M.J., van der Schoot, E.H.M., 2000. A combined forecastinventory control procedure for spare parts. Journal of the Operational Research Society 51, 1184–1192. Stuart, A., Ord, J.K., 1994. Kendall’s Advanced Theory of Statistics, vol. 1: Distribution Theory 6th ed Edward Arnold, London. Syntetos, A.A., 2001. Forecasting of intermittent demand. Unpublished Ph.D. Thesis, Brunel University of Buckinghamshire New University, UK. Syntetos, A.A., Boylan, J.E., 2001. On the bias of intermittent demand estimates. International Journal of Production Economics 71, 457–466. Syntetos, A.A., Boylan, J.E., 2005. The accuracy of intermittent demand estimates. International Journal of Forecasting 21, 303–314. Syntetos, A.A., Boylan, J.E., 2008. Demand forecasting adjustments for service-level achievement. IMA Journal of Management Mathematics 19, 175–192. Syntetos, A.A., Boylan, J.E., Croston, J.D., 2005. On the categorisation of demand patterns. Journal of the Operational Research Society 56, 495–503. Syntetos, A.A., Boylan, J.E., Croston, J.D., 2006. A note on the categorisation of demand patterns. Journal of the Operational Research Society 57, 1257–1258. Syntetos, A.A., Boylan, J.E., Disney, S.M., 2009. Forecasting for inventory planning: a 50-year review. Journal of the Operational Research Society 60, S149–S160. Teunter, R., Sani, B., 2009. On the bias of Croston’s forecasting method. European Journal of Operational Research 194, 177–183.
Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
On the variance of intermittent demand estimates Aris A. Syntetos a,c,n, John E. Boylan b a b c
Centre for Operational Research & Applied Statistics, Salford Business School, University of Salford, Maxwell Building, Manchester M5 4WT, UK Buckinghamshire New University, UK BEM – Bordeaux Management School, France
a r t i c l e in f o
a b s t r a c t
Article history: Received 16 March 2010 Accepted 5 July 2010 Available online 8 July 2010
Intermittent demand occurs at random with many time periods showing no demand at all. Forecasting such demand patterns constitutes a challenging exercise because of the associated dual source of variation (demand intervals and demand sizes). Research in this area has developed rapidly in recent years with new results implemented into supply chain software solutions because of its practical implications. In an inventory context, both the accuracy of the forecasts and their variability (sampling error of the mean) have equal importance in terms of service level achievement and/or inventory cost minimisation. Although the former issue has been studied extensively (mainly building upon Croston’s model, 1972) the latter has been largely ignored. The purpose of this paper is to analyse the most wellcited intermittent demand estimation procedures in terms of the variance of their estimates. Detailed derivations are offered along with a discussion of the underlying assumptions. As such, we hope that our contribution may constitute a point of reference for further analytical work in this area as well as facilitate a better understanding of issues related to modelling intermittent demands. & 2010 Elsevier B.V. All rights reserved.
Keywords: Intermittent demand Forecasting Supply chain management Inventory management
1. Introduction Intermittent demand for products appears sporadically, with some time periods showing no demand at all. When demand occurs, the demand size may be constant or variable, perhaps highly so. Intermittent demand items may be any Stock Keeping Unit (SKU) within the range of products offered by an organisation at any level of the supply chain. Such items may collectively account for up to 60% of the total stock value (Johnston et al., 2003) and are particularly prevalent in the aerospace, automotive, military and IT sectors. They are often the items at greatest risk of obsolescence. Inventory control decisions for intermittent items are needed to determine inventory replenishment rules. These decisions can be made more intelligently if supported by more accurate and less variable demand forecasts. Improvements in forecasting and stock control may be translated to significant reductions in wastage or scrap, and very substantial cost savings with further implications for the effective management of the relevant supply chains. Replenishment requirements are calculated according to an anticipated probability distribution of demand over lead-time. Many organisations that deal with intermittent demand items face significant difficulties in organising their stock in such a way as to minimise inventory-holdings whilst achieving satisfactory service. Intermittent demand patterns are built from constituent
n
Corresponding author. Tel.: +44 161 295 5804. E-mail address: [email protected] (A.A. Syntetos).
0925-5273/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2010.07.005
elements, i.e. a demand arrival process (that results in a sequence of inter-demand intervals) and the demand sizes, when demand occurs. The dual source of variation under concern constitutes the main difficulty in forecasting the requirements for such items. In addition, the inappropriateness of standard distributions (say Normal) for representing these patterns introduces further complications into their management. The standard method used in industry to forecast intermittent demand requirements is Croston’s method (Croston, 1972). The method is incorporated in statistical supply chain forecasting software packages (e.g. Forecast Pro), and demand planning modules of component based enterprise and manufacturing solutions (e.g. Industrial and Financial Systems – IFS AB). It is also included in integrated real-time sales and operations planning processes (e.g. SAP Advanced Planning and Optimisation – APO 4.0). Croston (1972) proved the biased nature of simple exponential smoothing (SES) when applied in an intermittent demand context and he proposed a method that relies explicitly upon estimates of the inter-demand intervals and demand sizes. The method was claimed to be unbiased, but Syntetos and Boylan (2001) showed it to be positively biased (i.e. over-forecasting mean demand) and they subsequently proposed an approximately unbiased estimator: SBA (Syntetos–Boylan Approximation, Syntetos and Boylan, 2005). This estimation procedure applies a deflating factor to the Croston estimates in order to take away the bias. A little bias though still remains, on the opposite side (i.e. slightly underestimating mean demand). Syntetos (2001) also proposed an exactly unbiased modification of Croston’s method (SY method). This procedure was further evaluated by Teunter and Sani (2009)
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
and it was found to perform very well. For a comprehensive account of all recent developments in intermittent demand supply chain forecasting please refer to Fildes et al. (2008) and Syntetos et al. (2009).1 The focus of the above studies has been the bias (or the lack of it) of mean demand estimates. However, the sampling error of the mean (i.e. variability of the estimates) which is of equal importance for both forecasting and stock control purposes has not been explicitly discussed. The achievement of the service target is an important objective, and requires an unbiased forecast of mean demand, assuming that the distribution of demand is known (e.g. Poisson, Negative Binomial) and an estimate of the variance of demand. Minimisation of inventory holdings is achieved by appropriate stock rules (not discussed in this paper) and by using a minimum variance estimator of the mean demand. Consequently, it would be highly desirable to obtain minimum variance—unbiased estimator(s) of mean demand. In this paper the issue of the variance of intermittent demand estimates is explicitly addressed for SES, Croston’s method, SY and SBA. We hope that this paper may serve as a data repository for further analytical work in this area and for facilitating a better understanding of pertinent issues. Our paper concludes with a discussion of the implications of our work for supply chain forecasting practices (referring also to the bullwhip effect) and an agenda for further research in this area. The remainder of our paper is structured as follows: Section 2 the underlying model assumed for the purposes of this research is presented along with the corresponding variance of SES estimates. The variance of the estimates produced by the Croston’s method, SY and SBA are subsequently discussed in Sections 3–5, respectively. The theoretical results presented in this paper are summarised in Section 6 where the implications of this work for real world supply chain practices are also discussed. Finally, in Section 7, we present the conclusions of this research along with some important lines of further enquiry in this area.
2. An intermittent demand model Croston advocated separating the demand into two components, the inter-demand time and the size of demand, and analysing each component separately. He assumed a stationary mean model for representing the underlying demand pattern, normal distribution for the size of demand and a Bernoulli demand generation process, resulting in geometrically distributed inter-demand intervals. Three more assumptions implicitly made by Croston in developing his model are the following: independence between demand sizes and inter-demand intervals, independence of successive demand sizes and independence of successive interdemand intervals. As far as the last assumption is concerned it is important to note that the geometric distribution is characterised by a ‘memory less’ process: the probability of a demand occurring is independent of the time since the last demand occurrence, so that this distributional assumption is consistent with independent inter-demand intervals. The normality assumption is the most restrictive one for the analysis conducted by Croston, since the demand sizes may be, theoretically, represented by any probability distribution without 1 At this point it is important to note that one more modified Croston procedure has appeared in the literature (Leve´n and Segerstedt, 2004). However, this method was found to be even more biased than the original Croston’s method (Boylan and Syntetos, 2007); Teunter and Sani, 2009) and it is not further discussed in this paper.
547
affecting the results. The remaining assumptions are retained for the analysis to be conducted in this paper. If we let pt ¼the inter-demand interval that follows the geometric distribution with: E(pt)p zt ¼the demand size, when demand occurs, with a mean m and variance s2 1/p¼the underlying Bernoulli probability of demand occurrence then the demand per unit time period (Yt) can be represented by 8 1 > > > < 0, 1 p , probability of demand occurence ð1Þ Yt ¼ 1 > > , probability of demand occurence > zt , : p with EðYt Þ ¼ 0
p1 1 m þ Eðzt Þ ¼ p p p
ð2Þ
Consequently the variance of demand per unit time period is calculated as follows (Croston, 1972): VarðYt Þ ¼
p1 2 s2 m þ p p2
ð3Þ
2.1. The variance of SES estimates Croston (1972) proved the inappropriateness of exponential smoothing as a forecasting method when dealing with intermittent demands and he expressed in a quantitative form the bias associated with the use of this method when demand occurs according to the model described above. Exponential smoothing is unbiased if we consider the estimates made at the end of every forecast review period (all points in time). It is biased though if we isolate the estimates made after a demand occurrence (issue points only). The former scenario corresponds to a re-order interval/periodic review stock control system whereas the latter to a re-order level/continuous review model. The bias properties of SES are summarised in Table 1. The variance of the exponentially smoothed estimates, updated every period, is a a p1 2 s2 VarðYut Þ ¼ ð4Þ VarðYt Þ ¼ m þ 2a 2a p2 p (where a is the smoothing constant employed for updating purposes) assuming a stationary mean model and homogeneous variance of demand per unit time period. If we isolate the estimates that are made just after an issue (which are those that will be used for replenishment purposes by a continuous review stock control system) Croston showed that these estimates have the following variance (as corrected by Rao, 1973): ab2 p1 2 s2 VarðYut Þ ¼ a2 s2 þ ð5Þ m þ 2a p2 p where b ¼1 a.
3. The variance of Croston’s estimates Croston (1972) suggested estimating the average interval between successive demand occurrences and the average size of demand when it occurs and to combine those statistics to give an unbiased estimate of the underlying mean demand. If we let Varðpt Þ ¼ ðp1Þ2 ðas assumed by CrostonÞ
ð6Þ
548
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
Table 1 Intermittent demand estimators: bias and sampling error of the mean. Estimator
E (Y0 t)
SES—all points
¼ m/p
SES—issue points
¼ m aþ
Var (Y0 t) ¼
SBA (Syntetos and Boylan, 2005)
E m/p
m
p0 t is the exponentially smoothed inter-demand interval, updated only after demand occurs (with smoothing constant a), and z0 t is the exponentially smoothed size of demand, updated only after demand occurs (using the same smoothing constant a)we then have Eðzut Þ ¼ Eðzt Þ ¼ m
ð7Þ
Eðput Þ ¼ Eðpt Þ ¼ p
ð8Þ
Varðzut Þ ¼
a 2a
a
Varðzt Þ ¼
a 2a
s2
ð9Þ
a
Varðpt Þ ¼ ðp1Þ2 ð10Þ 2a 2a The estimation equation for Croston’s method is given by
Varðput Þ ¼
Yut ¼
zut put
¼ 2
ab p1 2 s m þ ¼ a2 s2 þ 2a p2 p a p1 2 a 2 s2 m þ s þ 2 2a p3 2a p a2 2 a p s þ pðp1Þm2 þ pðp1Þs2 að2aÞ 2 2a a4 4 p 2 a2 ¼ 1 VarðCrostonÞ 2
p
SY (Syntetos, 2001)
a
VarðYt Þ 2a
2
b
a ðp1Þ þ m 2 p 2a p ¼ m/p
Croston (Croston, 1972)
Var (Yt)
ð11Þ
and the expected estimate produced by this method has been shown to be (Syntetos and Boylan, 2005) zut m a ðp1Þ E m 2 ð12Þ þ put p 2a p The variance of the ratio of two independent random variables x1, x2 is given in Stuart and Ord (1994) as follows: # " x1 Eðx1 Þ 2 Varðx1 Þ Varðx2 Þ þ ¼ Var ð13Þ Eðx2 Þ x2 ðEðx1 ÞÞ2 ðEðx2 ÞÞ2 For x1 ¼z0 t and x2 ¼p0 t, considering Eqs. (7)–(10), the variance of the estimates produced by using Croston’s method is calculated by " # zut a ðp1Þ2 2 s2 VarðYut Þ ¼ Var m þ 2 ¼ ð14Þ 2a put p4 p assuming that the same smoothing constant value is used for updating demand sizes and inter-demand intervals and that both demand size and inter-demand interval series are not autocorrelated and have homogeneous variances. Rao (1973) pointed out that the right hand side of Eq. (14) is only an approximation to the variance. This follows since (13) is, in fact, an approximation. 3.1. The variance of inter-demand intervals The number of independent Bernoulli trials (with a specified probability of success) before the first success is represented by the geometric distribution. An alternative form of the geometric distribution involves the number of trials up to and including the first success (demand occurring period). Considering the notation used in this paper, the variability of the geometrically distributed inter-demand intervals is pðp1Þ, irrespectively of which form of the geometric distribution is utilised. Consequently (6) should be
p1 2 s2 m þ p p2
replaced by Varðpt Þ ¼ pðp1Þ
ð15Þ
3.2. The corrected variance of Croston’s method estimates By taking (15) into consideration, the variance of the demand per period estimates, using Croston’s method, would become zut a p1 2 s2 m þ 2 ð16Þ Var 3 2a p put p indicating that the approximated variance of the estimates produced by Croston’s method is in fact greater than that calculated by Croston himself, Eq. (14).2 Nevertheless, approximation (16) is still not correct. In fact there is a fundamental problem in directly applying Stuart and Ord’s result, given by (13), for the purpose of deriving the variance of the forecasts produced by a biased estimator. This is proven as follows: We apply Taylor’s theorem to a function of two variables, g(x), where, x is the vector: x ¼(x1, x2) and g(x)¼g(x1, x2)¼x1/x2 with E(x1)¼ y1 and E(x2)¼ y2. The vector y is defined as: y ¼(y1, y2), with g(y)¼(y1, y2) ¼ y1/y2 @g @g gðxÞ ¼ gðyÞ þ ðx1 y1 Þ þ ðx2 y2 Þ þ ð17Þ @y1 @y2 where g(y) ¼ y1/y2 is just the first term in the Taylor series and not necessarily the population expected value. For E½gðxÞ ¼ gðyÞ þ e
ð18Þ
where e is an error term, which according to Croston, can be neglected, we then have Var ½ gðxÞ ¼ EfgðxÞE½gðxÞg2 ¼ E½gðxÞgðyÞ2 # " 2 @g @g Eðx1 Þ 2 Varðx1 Þ Varðx2 Þ E ðx1 y1 Þ þ ðx2 y2 Þ ¼ þ @y1 @y2 Eðx2 Þ Eðx1 Þ2 Eðx2 Þ2 ð19Þ If we set, x1 ¼z0 t, the estimate of demand size, with E(z0 t) ¼ m and x2 ¼p0 t, the estimate of the inter-demand interval, with E(p0 t)¼p, so that g(x)¼Y0 t, it has been proven (Syntetos and Boylan, 2001, 2005) that EðYut Þ a
2
m p
or E½ gðxÞ agðyÞ
Eq. (10) in the original paper.
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
Based on that, we argue that the error term in Eq. (18) cannot be neglected and therefore approximation (19) cannot be used to represent the problem in hand. Our argument is discussed in greater detail in Appendix A and B, where we also derive a correct approximation (to the second order term) of the variance of Croston’s estimates. That variance expression is given by zut a ðp1Þ 2 a 2 s2 Var þ s m þ put 2a 2a p3 p2 a4 m2 1 1 2 9 1 p þ1 þ 1 ð20Þ 4 p4 p p 1ð1aÞ Syntetos (2001) showed, by means of simulation on a wide range of theoretically generated data (matching the demand patterns encountered in many supply chain organisations), that approximation (20) does not increase the accuracy of the calculated variance more than by only considering the first term of this approximation. In fact, for certain cases, approximation (20) was shown to perform worse. Taking that into account the variance of Croston’s estimates may be ‘safely’ approximated by zut a ðp1Þ 2 a 2 s2 Var s þ 2 m þ ð21Þ 3 put 2a 2a p p 4. The variance of the SY method estimates The estimation equation for the SY method discussed in Section 1 is given by Yut ¼
ð1ða=2ÞÞzut ð1ða=2ÞÞzut ¼ ð1ða=2put ÞÞput put ða=2Þ
and the expected estimate produced by this method can be shown to be as follows (Syntetos, 2001): ð1ða=2ÞÞzut m EðYut Þ ¼ E ð23Þ put ða=2Þ p In Appendix C we perform a series of calculations in order to find the variance of the estimates of mean demand produced by the SY method. The variance is approximated by h i 2 2 2 a 2 ð1ða=2ÞÞzut að2aÞ ðpða=2ÞÞ s þ pðp1Þm þ 2a pðp1Þs Var 4 4 put ða=2Þ pða=2Þ þ
a4 1ð1aÞ
4
with a zut m a m EðYut Þ ¼ E 1 2 put p 2 p2
2 1ða=2Þ m2 2 1 1 2 9 1 p þ1 6 p 1 p p pða=2Þ
ð24Þ Similarly to the previous section, Syntetos (2001) showed by means of simulation that consideration of both terms of approximation (24) does not provide, overall, a more reliable estimate of the calculated variance than when only the first term of this approximation is considered. The calculation of the variance of the SY method may be simplified, without sacrificing accuracy, by considering approximation (25) h i 2 2 2 2 ð1ða=2ÞÞzut að2aÞ ðpða=2ÞÞ s þ pðp1Þm þða=ð2aÞÞpðp1Þs Var 4 put ða=2Þ ðpða=2ÞÞ4
ð25Þ 5. The variance of the SBA method estimates The estimation procedure for the SBA method (Syntetos and Boylan, 2005) is a zut ð26Þ Yut ¼ 1 2 put
ð27Þ
The variance of the estimates produced by the SBA is calculated as a zut a2 zut VarðYut Þ ¼ Var 1 Var ¼ 1 ð28Þ 2 put 2 put Considering approximation (21) we finally have a zut að2aÞ ðp1Þ 2 a 2 s2 s þ 2 m þ ð29Þ Var 1 3 4 2 put 2a p p Extensive analysis conducted by Syntetos (2001) justified the choice of (21) instead of (20) for the purpose of approximating the variance of the SBA method. In the next section the theoretical results presented thus far in our paper are summarised followed by a discussion of their implications for supply chain forecasting practices.
6. Summary of theoretical results and discussion The statistical properties of the methods discussed thus far in this paper are summarised in Table 1. In particular, we indicate the expected estimate produced by each of these methods along with the variance of their estimates (sampling error of the mean). The variance of demand itself is presented as well. Such results enable the calculation of the theoretical one-step-ahead mean squared error (MSE) associated with SES, Croston, SY and SBA. MSE ¼ VarðdemandÞ þVarðEstimatesÞ þ Bias2
ð22Þ
549
ð30Þ
The MSE over a fixed lead time of duration L, may also be calculated as follows (see Strijbosch et al., 2000; Syntetos et al., 2005): MSEL ¼ LfLVarðEstimatesÞ þLBias2 þVarðActual DemandÞg
ð31Þ
(In both (30) and (31) the bias term is obviously zero for unbiased estimators.) Such results may be of particular value for computerised supply chain/inventory management systems relying upon analytical variance expressions, as opposed to estimation through smoothed MSE or mean absolute deviation (MAD) procedures. In addition, analytical MSE expressions enable the direct comparison of the performance of these methods through the consideration of both bias and sampling error of the mean (see also Syntetos et al., 2005). Teunter and Sani (2009) commented on the unbiased nature of the SY method and recommended that the estimator under concern receives more attention as a competitive alternative to SBA. Although SY is indeed unbiased, a detailed analysis conducted by Syntetos (2001) showed that this method compares unfavourably to SBA with regards to their MSE performance due to the increased variance of the relevant estimates. This is a very important point both from a theoretical and a practitioner’s perspective since, typically, greater emphasis is being placed on the bias (or the lack of it) associated with an estimation procedure rather than the variance of the estimates. Ideally though, both should be equally considered since they collectively determine the supply chain implications of using the method under concern for stock control purposes. 6.1. Implications for supply chain management The Bullwhip Effect is a term promoted by Lee et al. (1997) but was observed and modelled decades earlier by Forrester (1958). It occurs whenever demand becomes more variable as it proceeds through the supply chain, away from the consumer and towards
550
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the supplier. Lee et al. (2000) discussed four causes of the Bullwhip Effect, namely: demand signal processing, rationing/ shortage gaming, order batching and price fluctuations. Demand signal processing in particular refers to the magnification in variance that occurs through the forecast error and the interaction between forecasting procedures and inventory rules at each stage of the supply chain (see also Chen et al., 2000a, b; for a review of studies in this area please refer to Syntetos et al., 2009). Such interactions account for significant discrepancies between the actual performance of an inventory system and what is theoretically expected; they are particularly prevalent in an intermittent demand context where very large forecast errors are all too common due to the difficulties associated with the forecasting task. For example, major deviations have been reported in the literature between the target and achieved customer service levels in such a context of application (Syntetos and Boylan, 2008). Forecast variance is a key contributory factor to the Bullwhip Effect, and as such any research oriented to variance related issues should be of direct interest to supply chain practitioners. Having mentioned that, no research has been conducted to-date on the behaviour of the Bullwhip Effect for supply chains dealing with intermittent demand products, for example service parts. Such products have become ubiquitous in modern societies and studying the relevant supply chains should be very important both from an academic and practitioner perspective.
7. Conclusions and extensions The area of intermittent demand forecasting has received a number of theoretical contributions in the recent years, most of which have been focusing on the issue of bias of the relevant estimation procedures. In this paper, we have argued for the additional consideration of the variance of intermittent demand estimates since this is equally important in terms of a method’s implications for stock control. Analytical expressions are offered with regards to the sampling error of the mean for four estimation procedures: SES, Croston, SY and SBA. The bias properties of these estimators are also presented and the results collectively permit the calculation of the relevant Mean Squared Error (MSE) expressions. MSE is the only mathematically tractable forecast accuracy metric
and its quantification allows comparisons to be performed between various estimators. In addition, such results may be particularly useful for computerised systems that rely upon analytical variance expressions for stock replenishment purposes. All the derivations presented in this paper are based upon Croston’s (1972) assumptions for modelling intermittent demands. Although, we feel that such derivations should facilitate further developments in this area, we cannot lose sight of the fact that Croston’s demand model is incompatible with his forecasting method (and its variants), because exponential smoothing methods are not designed for i.i.d. (identically and independently distributed) demand. Obviously, this does not mean that Croston’s method and its variants are not useful. Such methods do constitute the current state of the art in intermittent demand parametric forecasting. Nevertheless, an interesting line of further research would be to develop alternative models for intermittent demand forecasting, although one needs to consider also the following: a method that is optimal for one particular model may be severely sub-optimal for another model. Syntetos et al. (2006) argued that, for intermittent demand, robustness of a method across a wide range of possible underlying demand models is more important than optimality under one particular model. The scarcity of demand observations presents a significant inherent difficulty in identifying the demand model in the first place. In addition to further opportunities in modelling intermittent demands for forecasting purposes, modelling the effects of the relevant forecast errors in a supply chain context constitutes also an interesting line for further research. Demand signal processing has been shown to be a key determinant of the Bullwhip Effect but this issue has never been explored for supply chains that deal with intermittent demand items (such as service parts for example). Given the prevalence of such demand structures in modern organisational settings and given the difficulties associated with forecasting the relevant requirements, research in this area would appear to be merited. Acknowledgements The research described in this paper has been supported by the Engineering and Physical Sciences Research Council (EPSRC, UK) Grant nos. EP/D062942/1 and EP/G006075/1.
Appendix A. A correct approximation to the variance of Croston’s estimates We apply Taylor’s theorem to a function of two variables, g(x), where x is the vector: x¼(x1, x2) and g(x)¼(x1, x2)¼x1/x2 with E(x1) ¼ y1 and E(x2)¼ y2. The vector y is defined as: y ¼ (y1, y2), with g(y) ¼g(y1, y2)¼ y1/y2 " # @g @g 1 @2 g @2 g @2 g 2 2 gðxÞ ¼ gðyÞ þ ðx1 y1 Þ þ ðx2 y2 Þ þ ðx y Þ þ2 ðx y Þðx y Þ þ ðx y Þ þ ðA:1Þ 1 1 1 1 2 2 2 2 2 @y1 @y2 2 @y2 @y1 @y2 @y 1
2
where g(y)¼ y1/y2 is just the first term in the Taylor series and not the population expected value @g 1 ¼ @y1 y2
ðA:2Þ
@g y1 ¼ 2 @y2 y2
ðA:3Þ
@2 g
¼0
ðA:4Þ
@2 g 1 ¼ 2 @y1 @y2 y2
ðA:5Þ
@y1
2
@2 g @y2
2
¼ y1
2
y2 3
! ¼
2y1
y2 3
ðA:6Þ
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
551
therefore, considering (A.4), (A.1) becomes gðxÞ ¼ gðyÞ þ
# " 2 @g @g @ g 1 @2 g 2 ðx1 y1 Þ þ ðx2 y2 Þ þ ðx1 y1 Þðx2 y2 Þ þ ðx y Þ þ 2 2 @y1 @y2 @y1 @y2 2 @y2 2
ðA:7Þ
We set, x1 ¼ z0 t, the estimate of demand size, with E(z0 t)¼ m and x2 ¼p0 t, the estimate of the inter-demand interval, with E(p0 t)¼ p, so that g(x)¼Y0 t. It has been proven (Syntetos and Boylan, 2005) that EðYut Þ ¼ E½ gðxÞ gðyÞ þ
1 @2 g Eðx2 y2 Þ2 2 @y2 2
ðconsidering the first three terms in the Taylor series:Þ Therefore 92 8 @g @g @2 g 1 @2 g > > > > > ðx1 y1 Þ þ ðx2 y2 Þ þ ðx1 y1 Þðx2 y2 Þ þ ðx2 y2 Þ2 > > > 2 = < @y1 @y2 @y1 @y2 2 @y2 VarðYut Þ ¼ Var ½ gðxÞ ¼ E½gðxÞE½gðxÞ2 E 2 > > 1 @ g > > > > Eðx2 y2 Þ2 > > ; :2 2 @y2 8 !2 !2 < @g 2 @g 2 @2 g 1 @2 g ¼E ðx1 y1 Þ2 þ ðx2 y2 Þ2 þ ðx1 y1 Þ2 ðx2 y2 Þ2 þ ðx2 y2 Þ4 : @y1 @y2 @y1 @y2 4 @y2 !2 1 @2 g @g @g þ ½Eðx2 y2 Þ2 2 þ2 ðx1 y1 Þ ðx2 y2 Þ 4 @y2 @y1 @y2 þ2
@g @2 g @g @2 g @g @2 g ðx1 y1 Þ ðx1 y1 Þðx2 y2 Þ þ ðx1 y1 Þ ðx2 y2 Þ2 ðx1 y1 Þ Eðx2 y2 Þ2 2 2 @y1 @y1 @y2 @y1 @y1 @y2 @y2
þ2
@g @2 g @g @2 g @g @2 g ðx2 y2 Þ ðx1 y1 Þðx2 y2 Þ þ ðx2 y2 Þ3 ðx2 y2 ÞEðx2 y2 Þ2 2 @y2 @y1 @y2 @y2 @y2 @y2 @y2 2
@2 g @2 g @2 g @2 g 1 @2 g þ ðx1 y1 Þðx2 y2 Þ ðx2 y2 Þ2 ðx1 y1 Þðx2 y2 Þ Eðx2 y2 Þ2 2 2 @y1 @y2 @ 2 @y2 2 y @ y 1 2 @y2 @y2
!2 2
2
ðx2 y2 Þ Eðx2 y2 Þ
9 =
ðA:8Þ
;
Assuming that x1, x2 are independent
@g Var ½ gðxÞ @ y1
2
@g Eðx1 y1 Þ þ @ y2 2
2
@2 g Eðx2 y2 Þ þ @y1 @y2
!2
2
@g @2 g 1 @2 g Eððx1 y1 Þ ðx2 y2 Þ Þ þ Eðx2 y2 Þ3 þ @ y2 @ y2 2 4 @ y2 2 2
2
!2
1 @2 g Eðx2 y2 Þ 4 @ y2 2 4
!2 ½Eðx2 y2 Þ2 2
ðA:9Þ considering Eqs. (A.2), (A.3), (A.5) and (A.6) Var ½ gðxÞ ¼
Varðx1 Þ
y2
2
Varðx1 Þ ðEðx2 ÞÞ2
þ þ
2 2 2 y1 2 Varðx2 Þ 1 2y1 Eðx2 y2 Þ3 y1 Eðx2 y2 Þ4 y2 ½Eðx2 y2 Þ2 2 þ 4 Varðx1 ÞVarðx2 Þ þ 5 4 6 y2 y2 y2 y2 y2 6 2 2 3 2 4 ðEðx1 ÞÞ Varðx2 Þ Varðx1 ÞVarðx2 Þ 2ðEðx1 ÞÞ Eðx2 y2 Þ ðEðx1 ÞÞ Eðx2 y2 Þ ðEðx1 ÞÞ2 ½Varx2 2
ðEðx2 ÞÞ4
þ
ðEðx2 ÞÞ4
ðEðx2 ÞÞ5
þ
ðEðx2 ÞÞ6
ðEðx2 ÞÞ6
ðA:10Þ
In Appendix B it is proven that for n ¼2,3 E½xut EðxÞn ¼
an 1ð1aÞn
E½xt EðxÞn
ðA:11Þ
and also that E½xut EðxÞ4 ¼
a4 1ð1aÞ
4
E½xt EðxÞ4 þ
a4 ð1ð1aÞ2 Þ2
½Varðxt Þ2
ðA:12Þ
where xt represents the demand size (zt) or inter-demand interval (pt), x0 t is their exponentially smoothed estimate (z0 t, p0 t) and E(x) is the population expected value for either series. Consideration of (A.11) and (A.12) necessitates the adoption of the following assumptions:
no auto-correlation for the demand size and inter-demand interval series homogeneous moments about the mean for both series same smoothing constant value is used for both series Taking also into account that: Varðzt Þ ¼ s2 and Varðpt Þ ¼ pðp1Þ
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A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
(A.10) becomes Var
zut put
a s2 2a p2
þ
a 2a
m2
a 2 m2 pðp1Þ a 2 s2 pðp1Þ a3 2m2 a4 m2 a4 m2 2 þ Eðpt pÞ3 þ Eðpt pÞ4 þ p ðp1Þ2 p2 ðp1Þ2 2a 2a p6 p4 p4 1ð1aÞ3 p5 1ð1aÞ4 p6 ð1ð1aÞ2 Þ2 p6
ðA:13Þ The third moment about the mean in the geometric distribution, where: 1/p is the probability of success in each trial, is calculated as 1 1 1 p1 1 1 p1 1 2p1 ðp1Þð2p1Þ 1 þ1 ¼ ¼ ¼ 2 Eðpt pÞ3 ¼ 1 p p p3 p p3 p p p3 p p5
ðA:14Þ
and the fourth moment: Eðpt pÞ4 ¼
9ð1ð1=pÞÞ2 1ð1=pÞ 1 9ð1ð1=pÞÞ 1 1 1 4 1 1 2 9 1 p þ p2 ¼ p2 1 9 1 p þ1 þ ¼ 1 þ ¼ 1 4 2 4 2 p p p p p 1=p 1=p 1=p 1=p
ðA:15Þ
If we consider (A.14) and (A.15), and also the fact that
a4 2 2
ð1ð1aÞ Þ
¼
a 2 , 2a
(A.13) becomes Var
zut put
1 2 9 1 p þ1 p
a s2 a 2 pðp1Þ a 2 s2 pðp1Þ a3 2m2 ðp1Þð2p1Þ a4 m2 1 þ þ þ 1 m 2 4 4 3 10 4 p 2a p 2a 2a p p p 1ð1aÞ 1ð1aÞ p4
ðA:16Þ
Since the fourth part of approximation (A.16) becomes almost zero even for quite low average inter-demand intervals, finally the variance is approximated by (A.17) Var
zut put
a ðp1Þ 2 a 2 s2 a4 m2 1 1 2 9 1 p þ m þ 1 þ1 þ s 2a p p 2a p3 p2 1ð1aÞ4 p4
ðA:17Þ
This proves the result given by Eq. (20).
Appendix B. The 2nd, 3rd and 4th moment about the mean for exponentially smoothed estimates If we define xu ¼
X1
EðxuÞ ¼
j¼0
að1aÞj xtj ði:e: the EWMA estimateÞ
X1 j¼0
ðB:1Þ
að1aÞj Eðxtj Þ ¼ assuming Eðxtj Þ ¼ EðxÞ for all j Z0 aEðxÞ
X1 j¼0
ð1aÞj ¼
a 1ð1aÞ
EðxÞ ¼ EðxÞ
ðB:2Þ
Therefore we can write
xuEðxÞ ¼
1 X
j
n
að1aÞ ½xtj EðxÞ, ½xuEðxÞ ¼
j¼0
8
j¼0
9n = að1aÞ ½xtj EðxÞ ; j
and
8 9n
ðB:3Þ
j¼0
Assuming series is not auto-correlated, for n ¼2,3 we then have E½xuEðxÞn ¼
1 X
an ð1aÞnj E½xtj EðxÞn
ðB:4Þ
j¼0
and assuming E[xt j E(x)]n ¼E[x E(x)]n for all jZ0, i.e. homogeneous moments of order n E½xuEðxÞn ¼
an 1ð1aÞn
E½xEðxÞn
For n ¼2 VarðxuÞ ¼
a2 1ð1aÞ2
VarðxÞ
ðB:5Þ
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
553
and for n¼3
a3
E½xuEðxÞ3 ¼
1ð1aÞ3
E½xEðxÞ3
For n ¼ 4, Eq. (B.3) becomes
4
E½xuEðxÞ ¼ E
8
j
að1aÞ ½xtj EðxÞ
j¼0
assuming no auto-correlation ¼
94 = ; 1 X
a4 ð1aÞ4j E½xtj EðxÞ4 þ
j¼0
1 X 1 X
a2 ð1aÞ2j E½xtj EðxÞ2 a2 ð1aÞ2i E½xti EðxÞ2
j¼0i¼0 4
assuming homogeneous moments about the mean ¼ a E½xtj EðxÞ4
1 X
ð1aÞ4j þ a4 ½VarðxÞ2
j¼0
1 X
ð1aÞ4j ¼
j¼0 1 X 1 X
1 X 1 X
ð1aÞ2j ð1aÞ2i
ðB:6Þ
j¼0i¼0
1
ðB:7Þ
1ð1aÞ4
ð1aÞ2j ð1aÞ2i ¼ 1 þð1aÞ2 þ ð1aÞ4 þð1aÞ6 þ
j ¼ 0i ¼ 0
þ ð1aÞ2 þð1aÞ4 þ ð1aÞ6 þ þ ð1aÞ4 þ ð1aÞ6 þ
ðB:8Þ
: : : : 1 X 1 X
ð1aÞ2j ð1aÞ2i ¼ 1 þ 2ð1aÞ2 þ3ð1aÞ4 þ 4ð1aÞ6 þ
ðB:9Þ
j¼0i¼0
If we multiply the first and the second part of Eq. (B.9) with (1 a)2, we then have 1 X 1 X
ð1aÞ2
ð1aÞ2j ð1aÞ2i ¼ ð1aÞ2 þ2ð1aÞ4 þ 3ð1aÞ6 þ
ðB:10Þ
j¼0i¼0
Subtracting (B.9) and (B.10) 1 X 1 h iX ð1aÞ2j ð1aÞ2i ¼ 1 þð1aÞ2 þ ð1aÞ4 þ ð1aÞ6 þ ¼ 1ð1aÞ2 j¼0i¼0
1 1ð1aÞ2
Therefore 1 X 1 X
ð1aÞ2j ð1aÞ2i ¼
j¼0i¼0
1
ðB:11Þ
ð1ð1aÞ2 Þ2
Considering Eqs. (B.7) and (B.11), Eq. (B.6) becomes E½xuEðxÞ4 ¼
a4 1x2 4
E½xEðxÞ4 þ
a4 ð1ð1aÞ2 Þ2
½VarðxÞ2
ðB:12Þ
This proves the results given by Eqs. (A.11) and (A.12).
Appendix C. The variance of the SY method’s estimates We set, for the problem under concern a x1 ¼ 1 zut 2 with expected value h a i a a Eðx1 Þ ¼ y1 ¼ E 1 zut ¼ 1 Eðzut Þ ¼ 1 m 2 2 2 and variance Varðx1 Þ ¼ Var
h
1
a i
a2 a2 a a2 a 2 zut ¼ 1 s Varðzut Þ ¼ 1 Varðzt Þ ¼ 1 2 2 2 2a 2 2a
ðC:1Þ
ðC:2Þ
554
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
and x2 ¼ put
a
2 with expected value
a a a ¼ Eðput Þ ¼ p Eðx2 Þ ¼ y2 ¼ E put 2 2 2 and variance
ðC:3Þ
a a a ¼ Varðput Þ ¼ Varðpt Þ ¼ pðp1Þ ðC:4Þ Varðx2 Þ ¼ Var put 2 2a 2a (for the variance derivations consider also Appendix B). The third and the fourth moments about the mean for the x2 variable (consider also Appendix A and B) are calculated as follows: a a3 a3 a3 ðp1Þð2p1Þ ¼ Eðput pÞ3 ¼ Eðpt pÞ3 ¼ Eðx2 y2 Þ3 ¼ E put p þ p5 2 2 1ð1aÞ3 1ð1aÞ3
ðC:5Þ
a a4 a4 a4 a4 1 1 2 a 2 2 9 1 p þ1 þ Eðx2 y2 Þ4 ¼ E put pþ ¼ Eðput pÞ4 ¼ Eðpt pÞ4 þ ½Varðpt Þ2 ¼ p2 1 p ðp1Þ2 4 2 2 4 2 2 2a p p 1ð1aÞ ð1ð1aÞ Þ 1ð1aÞ
ðC:6Þ (assuming that the same smoothing constant value is used for both x1 and x2 series and that both series are not auto-correlated and have homogeneous moments about the mean). Consequently we apply Taylor’s theorem to a function of two variables, g(x)¼x1/x2 with @g 1 ¼ @y1 y2
ðC:7Þ
@g y1 ¼ 2 @y2 y2
ðC:8Þ
@2 g
¼0
ðC:9Þ
@2 g 1 ¼ 2 @y1 @y2 y2
ðC:10Þ
@y1
2
@2 g @y2
2
¼ y1
!
2
y2 3
¼
2y1
ðC:11Þ
y2 3
If we consider only the first three terms, we have # " 2 @g @g @ g 1 @2 g 2 gðxÞ ¼ gðyÞ þ ðx1 y1 Þ þ ðx2 y2 Þ þ ðx1 y1 Þðx2 y2 Þ þ ðx y Þ þ 2 2 @y1 @y2 @y1 @y2 2 @y2 2
ðC:12Þ
with x1 ð1ða=2ÞÞzut m ¼E E½ gðxÞ ¼ E x2 put ða=2Þ p y1 ð1ða=2ÞÞm m a gðyÞ ¼ ¼ y2 pða=2Þ p
ðC:13Þ
and (consider also Appendix A) Var ½ gðxÞ ¼ E½gðxÞE½gðxÞ2 ( )2 y1 @g @g @2 g 1 @2 g x1 2 E þ ðx1 y1 Þ þ ðx2 y2 Þ þ ðx1 y1 Þðx2 y2 Þ þ ðx y Þ E 2 2 @y2 @y1 @y2 2 @y2 2 y2 @y1 x2
ðC:14Þ
In order to simplify somewhat the derivation of the variance of the SY method’s estimates we approximate E[g(x)] by: E½ gðxÞ gðyÞ þ
1 @2 g Eðx2 y2 Þ2 2 @y2 2
(based on (C.12) and considering the results presented by Syntetos and Boylan, 2005) Considering (A.10) and assuming that x1 ,x2 are independent, (C.14) becomes Var ½gðxÞ
Varðx1 Þ 2
ðEðx2 ÞÞ
þ
ðEðx1 ÞÞ2 Varðx2 Þ ðEðx2 ÞÞ
4
þ
Varðx1 ÞVarðx2 Þ ðEðx2 ÞÞ
4
2ðEðx1 ÞÞ2 Eðx2 y2 Þ3 5
ðEðx2 ÞÞ
þ
ðEðx1 ÞÞ2 Eðx2 y2 Þ4 6
ðEðx2 ÞÞ
ðEðx1 ÞÞ2 ½Var x2 2 ðEðx2 ÞÞ6
ðC:15Þ
A.A. Syntetos, J.E. Boylan / Int. J. Production Economics 128 (2010) 546–555
555
Finally the variance of the estimates of the SY method is calculated as follows:
Var
1ða=2Þ zut a ð1ða=2ÞÞ2 s2 a ð1ða=2ÞÞ2 m2 pðp1Þ ð1ða=2ÞÞ2 ða=ð2aÞÞ2 s2 pðp1Þ ð1ða=2ÞÞ2 m2 a3 ðp1Þð2p1Þ þ þ put ða=2Þ 2a ðpða=2ÞÞ2 2a p5 ðpða=2ÞÞ5 ð1aÞ3 ðpða=2ÞÞ4 ðpða=2ÞÞ4 þ
a4
ð1ða=2ÞÞ2 m2
1ð1aÞ4 ðpða=2ÞÞ6
a 2 ð1ða=2ÞÞ2 m2 1 1 2 a 2 ð1ða=2ÞÞ2 m2 2 9 1 p þ1 þ p2 1 p ðp1Þ2 p2 ðp1Þ2 6 2a 2a p p ðpða=2ÞÞ ðpða=2ÞÞ6
ðC:16Þ Since the fourth part of approximation (C.16) becomes almost zero even for quite low average inter-demand intervals, and the last two terms cancel each other, finally the variance is approximated by
Var
h i 2 2 2 2 1ða=2Þ zut a a2 ðpða=2ÞÞ s þ pðp1Þm þða=ð2aÞÞpðp1Þs a4 ð1ða=2ÞÞ2 m2 2 1 1 2 9 1 p 1 þ p 1 þ1 p p put ða=2Þ 2a 2 ðpða=2ÞÞ4 1ð1aÞ4 ðpða=2ÞÞ6 ðC:17Þ
This proves the result given by Eq. (24).
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