On the effect of non-optimal forecasting methods on supply chain downstream demand

Transcription

1 IMA Journal of Management Mathematics Advance Access published March 25, 2011 IMA Journal of Management Mathematics Page 1 of 18 doi: /imaman/dpr005 On the effect of non-optimal forecasting methods on supply chain downstream demand MOHAMMAD M. ALI AND JOHN E. BOYLAN Department of Business and Management, Buckinghamshire New University, Queen Alexandra Road, High Wycombe, Buckinghamshire, HP11 2JZ, UK Corresponding author: m.ali@bucks.ac.uk [Received on 7 April 2010; accepted on 9 February 2011] Demand information sharing is used by many organizations to counter the bullwhip effect. A stream of recent papers claims that the upstream member can mathematically infer the demand at the downstream link (downstream demand inference [DDI]) without any formal information sharing mechanism. In this paper, we investigate DDI when non-optimal forecasting methods are employed by supply chains. We show that in the case of a simple moving average forecast, the demand at the downstream link can be inferred. In the case of single exponential smoothing (SES), downstream demand cannot be inferred and thus needs to be shared. Finally, we quantify the value of sharing demand information when SES is employed. Keywords: supply chain management; bullwhip effect; downstream demand inference; forecast information sharing; single exponential smoothing; simple moving average. 1. Introduction Collaborative planning forecasting and replenishment (CPFR) is a supply chain collaboration concept that has proved to be very successful in reducing inventory costs while improving service levels, thus enhancing profit margins (Danese, 2007; Skjoett-Larsen et al., 2003). Demand information sharing is one of the enablers of CPFR. Sharing the consumer demand process requires upstream supply chain members to extract demand information from their downstream partners. Various papers show that sharing demand information helps in reducing the so-called bullwhip effect (e.g. Chen et al., 2000a; Lee et al., 2000). The bullwhip effect is a well-known phenomenon in supply chain management. The bullwhip ratio measures the demand variability amplification as one move up the supply chain. Empirical evidence (e.g. Lee et al., 1997; Mccullen & Towill, 2001; Wong et al., 2007) and mathematical models (e.g. Graves, 1999; Chen et al., 2000b; Kim et al., 2006; Luong & Phien, 2007) show that the orders placed by a retailer on its supplier tend to be much more variable than the customer demand on the retailer. This amplification of variability propagates up the supply chain as does the amplification of forecast error. Various papers (e.g. Chen et al., 2000a,b; Zhang, 2004) analyse the value of demand information sharing by showing that the bullwhip effect is reduced when the upstream member utilizes the downstream demand in its lead time forecasts. There has been extensive debate in the literature on whether formal information sharing mechanisms are required for extracting the demand at the downstream member when the supply chain members c The authors Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 2 of 18 M. M. ALI AND J. E. BOYLAN employ auto-regressive integrated moving average (ARIMA) methods with minimum mean squared error (MMSE) estimation and an order-up-to (OUT) inventory policy. Some papers (e.g. Lee et al., 1997, 2000; Chen et al., 2000a; Yu et al., 2002; Raghunathan, 2003; Cheng & Wu, 2005) have argued that the downstream member needs to share its demand information with the upstream member to reduce the bullwhip effect. On the other hand, other authors (Graves, 1999; Raghunathan, 2001; Zhang, 2004; Gaur et al., 2005; Gilbert, 2005) have argued that the orders from the downstream member to the upstream member already contain information about the market demand process. By using its order history, the upstream member can infer the demand at the downstream member. We refer to this approach as downstream demand inference (DDI). According to the DDI approach, the savings in inventory costs from demand information sharing could be obtained without any formal information sharing with the downstream member. If this were true, then investments in information systems to allow demand information sharing would be unnecessary. By relaxing some assumptions, Ali & Boylan (2011) have shown that the upstream member cannot exactly deduce the demand at the downstream member. They assume that the demand process at the retailer is an ARIMA (p, d, q) (Box et al., 1994), and the retailer uses an MMSE estimation method to forecast its lead time demand and employs an OUT inventory policy. They show that the manufacturer cannot infer the exact demand at the retailer. Thus, they conclude that the downstream member will have to share its demand with the upstream member via some formal information sharing mechanism in order to reduce the bullwhip effect. However, this conclusion is restricted to ARIMA forecasting models with MMSE estimation. Forecasting is seen as an increasingly critical organizational capability (Sanders & Manrodt, 2003) but relatively few studies have assessed the usage, familiarity and satisfaction of forecasting methods among practitioners. Table 1 provides a summary of six such surveys highlighting the three highly ranked methods, according to these criteria. It shows that practitioners are more familiar, satisfied and more likely to use simpler forecasting methods compared to sophisticated quantitative approaches such as the ARIMA methodology. Thus, these surveys reveal that, in the real world, a forecasting method is not always chosen on the basis of its optimality or accuracy but rather its simplicity, ease of use and familiarity. Specifically, all six surveys have found simple moving average (SMA) and single exponential smoothing (SES) (highlighted in Table 1) to be popular among forecasting practitioners based on the usage, familiarity and satisfaction. The choice of using non-optimal forecasting methods is quite rational. They are more intuitive, especially for those with a limited mathematical background (Boylan & Johnston, 2003). Difficult and sophisticated, but optimal, methods are seen as not worth the added effort (Sanders & Manrodt, 1994). Johnston et al. (1999) compared the forecasting accuracy of combinations of SMA, a non-optimal forecasting method, with SES, the optimal forecasting method for an ARIMA (0, 1, 1) demand process. They showed that the variance of the forecast error for the non-optimal method was typically <3% higher than the optimal method. This discussion opens up an interesting research area, namely to study DDI in supply chains employing simpler but non-optimal forecasting methods. In this paper, we look at the effects of nonoptimal forecasting methods on supply chain DDI to ascertain whether demand information sharing is required. We use the same supply chain model as in Lee et al. (2000) but relax the assumption of ARIMA forecasting. Keeping all other parameters fixed, we look at the effect of different non-optimal forecasting methods on DDI. We assume that the supply chain links use two non-optimal forecasting methods, SMA and SES to forecast lead time demand. Various studies have modelled the upstream propagation of demand based on these forecasting methods (Chen et al., 2000a,b; Alwan et al., 2003; Zhang, 2004) but

3 EFFECT OF NON-OPTIMAL FORECASTING METHODS ON DOWNSTREAM DEMAND 3 of 18 TABLE 1 Summary of highly ranked forecasting methods in industry survey results Sr # Papers Familiarity (%) Satisfaction (%) Usage (%) 40 (SMA) 1 Sanders (1997) 32 (SES) 30 (Naive) 92 (SMA) 72 (SES) 92 (SES) 2 Mentzer & Kahn (1995) 90 (SES) 66 (RA) 69 (RA) 85 (SLP) 61 (Dec) 57 (TLA) 96 (SMA) 45 (RA) 34 (SMA) 3 Sanders & Manrodt (1994) 91 (SLP) 45 (SMA) 25 (RA) 84 (Naive) 43 (Sim) 20 (SES) 96 (SLP) 37 (SMA) 4 Sanders (1992) 94 (SMA) 35 (RA) 89 (SES) 22 (SES) 31 (Naïve) 5 Dalrymple (1987) 21 (SMA) 19 (% ROC) 85 (SMA) 67 (RA) 36 (RA) 6 Mentzer & Cox (1984) 82 (SLP) 60 (SES) 28 (TLA) 73 (SES) 58 (SMA & TLA) 22 (SMA) Legend: Dec, decomposition; RA, regression analysis; % ROC, % rate of change; SMA, simple moving average; SES, simple exponential smoothing; SLP, single line projection; Sim, simulation; TLA, trend line analysis. downstream inference remains unexplored. Because the methods are widely used in practice (Table 1), it is important to fill this research gap. We show in this paper that if the downstream member employs the SMA method, the upstream member can infer the demand present at the downstream link. This is based on the assumption that the upstream member is aware of the number of historical terms used in the SMA forecast. On the other hand, if the downstream member employs SES, the upstream member cannot infer the demand present at the downstream link, and an information system is required that facilitates demand information sharing. The remainder of this paper is organized as follows. In Section 2, we review some earlier literature that discusses the effect of non-optimal forecasting methods on upstream demand propagation. In Section 3, we present our supply chain model including the demand process, forecasting methods and inventory policy. In Section 4, we discuss the upstream propagation of demand for non-optimal forecasting methods. Based on this upstream propagation, in Section 5, we investigate DDI. We then move on to quantify the value of sharing demand information with the help of a simulation experiment in Section 6. Finally, in Section 7, we discuss the results of our research and present our conclusions. 2. Literature review Various papers have discussed the effect of non-optimal forecasting methods on upstream demand propagation. All the papers, except Alwan et al. (2003), restrict their analysis to the effect of forecasting methods on order variability. Alwan et al. (2003) not only look at the effect of forecasting methods on

4 4 of 18 M. M. ALI AND J. E. BOYLAN the bullwhip effect but also specifically address the propagation of the processes for demand and orders, including the effect on the parameters of the processes. Chen et al. (2000a) examined the ratio of upstream to downstream demand variance, or the bullwhip ratio, when the demand pattern at a retailer follows an autoregressive process of order 1 [(AR) (1) process]. They showed that when the retailer uses an SMA method to forecast its lead time demand, there is an increase in variability. This increase in variability is a function of three parameters: the number of historical terms (n) used in the SMA, the lead time (L) and the autocorrelation parameter (ρ). Chen et al. (2000b) performed a similar analysis on the AR (1) demand process based on SES. They concluded that the increase in variability is an increasing function of α, the smoothing parameter and an increasing function of L. Alwan et al. (2003) and Zhang (2004) compared the bullwhip effect for an AR (1) demand process for SMA, SES and an MSE-optimal forecasting method. They showed that the MMSE forecasting method results in lower variability (Alwan et al., 2003; Zhang, 2004) and lower inventory (Zhang, 2004) than the other methods. Using an AR (1) demand process, Stamatopoulos et al. (2006) argued that previous studies have only incorporated SES with a fixed smoothing constant. They compared the increase in variability when a best exponential smoothing method is chosen (i.e. a method that uses a smoothing constant that minimizes the mean squared error). They showed that this method results in lower variability than SES and SMA and thus can be used as an alternative to an MMSE forecasting method. All these studies show that the bullwhip effect is indeed present in supply chains irrespective of the forecasting method employed. Alwan et al. (2003) is perhaps the paper most closely linked with our research, as it studies demand propagation for an AR (1) demand process in the case of non-optimal forecasting methods. They show that an AR (1) demand process propagates into an autoregressive moving average (ARMA) (1, n) order process, if the downstream member employs SMA, of the n most recent demands, and into ARMA (1, ) if an SES is employed by the downstream member. The above studies limit their analysis to an AR (1) demand process at the downstream member. In addition, none of them have considered the effect of forecasting methods on DDI. One contribution of this paper is that we generalize the results of Alwan et al. (2003) for an ARIMA (p, d, q) process. Thus, we present a complete picture of upstream propagation for an ARIMA (p, d, q) demand process for supply chains using SMA and SES. The second contribution of our paper is that we present findings on DDI for an ARIMA (p, d, q) demand process when non-optimal forecasting methods are employed. 3. Supply chain model In this paper, we consider a single item two-stage supply chain (retailer manufacturer scenario) where the demand is realized at the retailer. The lead time (L) from the manufacturer to the retailer is fixed and known. We assume that the time series of demand (d t ), if stationary, can be represented by an ARMA (p, q), process given by d t = τ + ρ 1 d t 1 + ρ 2 d t ρ p d t p + ε t θ R 1 ε t 1 θ R 2 ε t 2 θ R q ε t q, (1) where d t, d t 1,..., d t p are the observed demands at time periods t, t 1,...,t p. τ is a positive constant and τ > 0, ρ 1, ρ 2,..., ρ p are the autoregressive parameters and θ1 R, θ 2 R,..., θ q R are the moving

5 EFFECT OF NON-OPTIMAL FORECASTING METHODS ON DOWNSTREAM DEMAND 5 of 18 average parameters at the retailer. ε t, ε t 1,..., ε t q are the noise terms in the observed demands at time periods t, t 1, t 2,..., t q. The noise terms are independent and identically distributed, with mean zero and constant variance σ 2 ε. Using the backshift operator, B, equation (1) is written as follows: where ρ(b)(d t ) = θ R (B)ε t, (2) ρ(b) = 1 ρ 1 B ρ 2 B 2 ρ p B p, θ(b) = 1 θ 1 B θ 2 B 2 θ q B q. We assume that the time series of demand (d t ), if non-stationary, can be represented by an ARIMA (p, d, q), process given by ρ(b) d (d t ) = θ R (B)ε t, where = 1 B. (3) ARIMA models have proved to be useful for developing insights into the bullwhip effect (Syntetos et al., 2009). Readers interested in more details on the ARIMA (p, d, q) demand process are referred to Chatfield (2003). As discussed in Section 1, simple forecasting methods are widely used by practitioners due to familiarity, simplicity and ease of implementation. We assume that the supply chain links employ SMA and SES as their forecasting methods. In Section 1, we have presented our rationale for using these two forecasting methods. These methods are non-optimal for all ARIMA (p, d, q) processes, except when an SES is employed for an ARIMA (0, 1, 1) process. The SMA forecasting method is the arithmetic mean of the n most recent observations. Every forecasting period, the newest observation is included and the oldest is dropped out. Mathematically, n 1 ˆF t+1 = 1 d t i, (4) n where ˆF t+1 = forecast value for next period and n = number of terms in the SMA. In the SMA method, the past observations are weighted equally, whereas an SES assigns decreasing weights as the observations get older. There are two ways in which an SES forecast can be expressed. The first approach is to assume that an infinite data history (d t, d t 1, d t 2,...) is available. Then, the infinite representation of an SES is as follows: This can also be expressed recursively ˆF t+1 = α i=0 (1 α) j d t j. (5) j=0 ˆF t+1 = αd t + (1 α) ˆF t, (6) where ˆF t+1 is the forecast value for the next period, d t is the actual value of the observation in period t and α is the smoothing constant (0 < α < 1).

6 6 of 18 M. M. ALI AND J. E. BOYLAN The second approach is to assume a finite data history (d t, d t 1, d t 2,..., d 0 ). Then the finite representation of an SES is given by ˆF t+1 = αd t + α(1 α)d t 1 + α(1 α) 2 d t α(1 α) t 1 d 1 + (1 α) t d 0. (7) The finite representation is clearly more realistic; the infinite representation is more convenient for some mathematical derivations. In this paper, the finite representation is adopted. We consider a periodic review inventory system where supply chain links review their inventory levels every period. The links base their inventory replenishments on a simple OUT policy. Each link replenishes the demand during the last period plus the change being made in the OUT levels. The sequence of the events at period t in the model is as follows: before the end of time period t, t = 1, 2, 3,..., the retailer receives its order placed t L periods ago from the manufacturer. The demand (d t ) is then realized at the retailer, who observes its inventory level, fulfils the realized demand and then places an order (Y t ) to the manufacturer. Excess demand is backlogged. The order (Y t ) from the retailer becomes the demand for the manufacturer, which the manufacturer receives at the end of time period t. The manufacturer then ships the order (Y t ) to the retailer from its stocks. If the manufacturer does not have enough stocks to fulfil the order (Y t ), they arrange the extra stocks from an external source and bear a penalty cost (P) per item. At the end of every time period t, the manufacturer incurs an inventory holding cost, denoted by H per item. This sequence of events is identical to that assumed by Lee et al. (2000). We assume that both supply chain links use the same forecasting method. Similarly, we assume that both links in the supply chain are aware of and utilize an OUT inventory policy. The assumption of usage of the same forecasting method throughout the supply chain is common to all the papers discussed in Section 2. As we are critically analysing the DDI approach presented in the literature, such consistency with past papers will facilitate our study. 4. Upstream propagation of demand for non-optimal methods Upstream propagation shows the relationships between order processes and the original demand process. It not only shows the progression of demand processes through the supply chain but also is very useful in modelling and quantifying the value of demand information sharing. Alwan et al. (2003) is the only paper that looked specifically at the issue of demand propagation for non-optimal forecasting methods. They model an AR (1) process and assume that the retailer uses SMA, SES and MMSE methods to forecast lead time demand. Alwan et al. (2003) showed that an AR (1) demand process would propagate into ARMA (1, n) if the SMA method is used based on an average of n historical terms. Further, they showed that an AR (1) would propagate into ARMA (1, ) when an SES is employed. In this section, we first generalize the above results to an ARIMA (p, d, q) demand process. We use a finite representation of SES and show later how it relates to the result by Alwan et al. (2003). In Section 5, we discuss DDI for both the non-optimal forecasting methods. PROPOSITION 4.1 When an SMA forecasting method is used to forecast the lead time demand for an ARIMA (p, d, q R ) demand process at the downstream member, the order to the upstream member follows an ARIMA (p, d, n + q R ) process given by the following: ρ(b) d (Y t ) = θ M (B)a t, (8) where Y t is the order from the downstream member to the upstream member, θ M (B) is the moving average operator for the manufacturer, of the orderq M = n + q R, and a t = ( L n + 1 ) ε t.

7 EFFECT OF NON-OPTIMAL FORECASTING METHODS ON DOWNSTREAM DEMAND 7 of 18 The proof is given in Appendix I. Alwan et al. (2003) showed that when an SES is employed on an AR (1) process, it propagates into an ARMA (1, ) process. This result was based on an infinite representation of SES. In realworld applications, it is not possible to have a time series with an infinite data history. Using a finite representation of SES as in equation (7), we now generalize the results for an ARIMA (p, d, q) demand process. PROPOSITION 4.2 When the SES forecasting method is used to forecast the lead time demand for an ARIMA (p, d, q R ) demand process at the downstream member, the order to the upstream member approximately follows an ARIMA (p, d, t 1) process: ρ(b) d (Y t ) θ M (B)a t, (9) where Y t is the order from the downstream member to the upstream member, θ M (B) is the moving average operator for the manufacturer, and is of the orderq M = t 1, and t is the current time period. The approximation is due to the presence of an extra term on the right-hand side. For an ARIMA (p, d, q) process, this extra term is as follows: L α(1 p+d α)t 1 (d 1 d 0 ) + αd 0 d(d 1)(d 2) 3! i=2 [(1 α) t i ( ρ i dρ i 1 + d(d 1) ρ i 2 2! )] ρ i ( 1) d ρ i d. (10) The proof of the approximate term (10) is given in Appendix II. It is obvious from the expression that this extra term will tend to 0 as t tends to. An upper bound for the absolute value of this expression, denoted by U, has been established (see Appendix II): ( ) U = Lα(1 α) t 1 d d 1 d 0 + L(d + 1) (1 α) t p d [1 (1 α) p+d 1 ]d D 0, (11) where D = d/2 if d is even, and D = (d + 1)/2 if d is odd. For finite values of t, the upper bound is deflated by a factor of (1 α) as t increases by 1, thus becoming negligible rapidly. For example, for 2 years of monthly data (t = 24) and for p + d = 4, if α = 0.2 (often used in practice), then (1 α) t p d = For 2 years of weekly data (t = 104), with other parameters unchanged, then (1 α) t p d is of the order of However, the extra term would not be negligible for very short histories, low-smoothing constants and long lead times. Alwan et al. (2003) used an infinite representation of SES, assuming that an infinite data history is available. If we let t tend to in our SES model for an AR (1) demand process, it propagates into an ARMA (1, ) process which is the result of Alwan et al. (2003), and is shown in Fig. 1. Thus, the result of Alwan et al. (2003) is compatible and a special case of our result in Proposition 4.2 above. 5. Downstream demand inference In this section, based on the upstream propagation of demand, we analyse the possibility of DDI. We have defined DDI as the ability of the upstream member to infer the demand present at the downstream link. We concentrate on the effects of non-optimal forecasting methods on DDI.

8 8 of 18 M. M. ALI AND J. E. BOYLAN FIG. 1. Demand characterization in case of non-optimal forecasting methods (Alwan et al., 2003). For DDI, the upstream member needs to first identify the demand process and then calculate the demand at the downstream link. These two aspects of demand inference will now be addressed. In some cases, process propagation is unique: a unique demand process at the downstream member would translate into a given demand process at the upstream member. On the other hand, in some cases, this propagation is not unique: various demand processes at the downstream member would translate into the same demand process at the manufacturer. Suppose the demand process is ARIMA (1, 1, 3) at the manufacturer. If the lead time between the manufacturer and the retailer is 1 and the supply chain links utilize MMSE forecasting methods, such a demand process could only propagate from an ARIMA (1, 1, 4) process at the retailer. This is because, for MMSE forecasts, an ARIMA (p, d, q) demand process propagates to an ARIMA (p, d, max{p + d, q L}) order process (Gilbert, 2005). On the other hand, suppose the demand process is ARIMA (1, 1, 2) at the manufacturer. Again, if we assume the lead time to be 1, this demand process could propagate from various processes at the retailer, namely ARIMA (1, 1, 2), ARIMA (1, 1, 1) and ARIMA (1, 1, 0). The accurate identification of the demand process at the downstream link depends on whether the propagation is unique. If only one demand process is possible at the downstream link, then accurate identification is feasible. On the other hand, identification is not feasible if a range of demand processes is possible at the downstream link. The calculation of the demand at the downstream member depends on the number of moving average terms at both links. If the upstream member has more than, or equal to, the number of moving average terms than the downstream link, then it can accurately calculate the demand at the downstream link. This is because the number of equations at the upstream link would be equal to or greater than the number of unknowns at the downstream link. On the other hand, if the upstream member has fewer moving average terms, then it is not possible to calculate the demand at the downstream link. This is due to the presence of more unknowns than equations at the upstream link. We will now look specifically at the effects of non-optimal forecasting methods on DDI. In Proposition 4.1, presented in Section 4, we showed that if the retailer employs SMA to forecast its lead time demand, an ARIMA (p, d, q R ) demand process would propagate into an ARIMA (p, d, q M ) process at the manufacturer, where q M = q R + n. The upstream propagation of demand is unique when the SMA method is used. A unique demand process at the downstream member would translate into a given demand process at the upstream member as shown in Fig. 2. Thus, the manufacturer would always be able to identify the demand process present at the retailer. In the case of SMA, the upstream member would be able to accurately calculate the demand at the downstream member. This is because the downstream member has fewer moving average terms (q R )

9 EFFECT OF NON-OPTIMAL FORECASTING METHODS ON DOWNSTREAM DEMAND 9 of 18 FIG. 2. Upstream propagation of demand using SMA and SES methods. than the moving average terms (q R + n) at the upstream member as shown in Fig. 2 above. Thus, the upstream member will have more equations than unknowns. Since both the process and the parameters may be accurately identified, DDI is feasible when SMA is employed. The equations for calculating the moving average terms for SMA and SES for an AR (1) process are provided in Alwan et al. (2003). Using the same procedure as Alwan et al. (2003) equations can be derived for an ARIMA (p, d, q) process. We show in this paper that the DDI approach is not possible when supply chain links use the SES forecasting method and thus organizations may not resort to the DDI approach. Therefore, it is not necessary to derive equations for moving average terms for the more general demand process. In Proposition 4.2, presented in Section 4, we showed that an ARIMA (p, d, q R ) demand process would propagate into an ARMA (p, d, q M ) process at the manufacturer plus another term, where q M = t 1. The above figure shows that the propagation of demand in the case of SES is not unique. Assuming t is larger than q R + 1, there could be a range of demand processes present at the downstream member. The upstream member, in this case, would not be able to calculate the demand, as it would not know how many demand parameters to calculate. Thus, in the case of SES, the upstream member cannot infer the demand at the downstream member. 6. Simulation experiment We now present some numerical results of our simulation experiment to illustrate the value of sharing demand information in supply chains using SES to forecast demand. We assume four demand processes at the retailer, including both stationary and non-stationary processes. These processes are AR (1), moving average MA (1), ARIMA (1, 1, 1) and ARIMA (1, 1, 2). The choice of ARIMA (p, d, q) processes are restricted to p, d, q 2, as in practice demand can usually be expressed by remaining within these constraints (Montgomery & Johnson, 1976; Box et al., 1994; Zhang, 2004). These models have also been termed as important in the literature owing to their frequent occurrence (Roberts, 1982, Box et al., 1994). In our simulation experiment, the demand parameters are selected to ensure that the demand processes are stationary and invertible. We use equation (3) to generate the demand at the retailer for each demand process. We assume that the retailer uses the SES method to forecast its lead time demand and uses the OUT inventory policy to place orders on the manufacturer as discussed earlier. The manufacturer s forecast will now depend on the supply chain strategy. The no information sharing (NIS) strategy is where the manufacturer utilizes the orders from the retailer for its lead time

10 10 of 18 M. M. ALI AND J. E. BOYLAN forecasts. In this case, although demand (d t ) at the retailer has been realized, it has not been shared with the manufacturer. The manufacturer could use alternative approaches other than using the retailer s orders. However, we assume in this paper that the manufacturer will base their forecasts on the orders received from the retailer. The manufacturer will use the orders for its SES forecast using the following equation: Ŷ L t = L[αY t 1 + α(1 αt)y t 2 + α(1 α) 2 Y t α(1 α) t 2 Y 1 + (1 α) t 1 Y 0 ]. (12) We compare this with forecast information sharing (FIS) strategy. This is where the retailer shares its demand information with the manufacturer and the manufacturer utilizes the retailer s demand in its lead time forecast. The forecast equation for SES in the case of the FIS approach is given as Ŷ L t = L[αd t 1 + α(1 α)d t 2 + α(1 α) 2 d t α(1 α) t 2 d 1 + (1 α) t 1 d 0 ]. (13) In both cases, we calculate the value of FIS by calculating the percentage reduction in inventory cost of the manufacturer by utilizing FIS, compared to a NIS strategy. Value of FIS =100 average inventory cost (NIS) average inventory cost (FIS). (14) average inventory cost (NIS) In order to calculate the average inventory cost, the inventory holdings are simulated at the end of every period using the newsvendor approach (see Lee et al., 2000 for details). The inventory cost for each period is then calculated, based on the value of the inventory holdings. The inventory cost is averaged across all periods. The cost parameters used in our experiment to calculate the manufacturer s inventory costs are given by a holding cost of 1 and a penalty cost of 25. Similar values have been adopted by Raghunathan (2001) and Lee et al. (2000). We also examine the effect of two parameters, manufacturer s lead time and SES smoothing parameter, on the value of FIS. We look at the effect of the manufacturer s lead time by examining values of 1, 6 and 12. The effect of the SES smoothing parameter (α) on the value of FIS is calculated by experimenting with values of 0.1, 0.3 and 0.8. Our simulation results show that the FIS strategy always results in lower inventory cost than the NIS strategy. We present our results by considering the effect of lead time and smoothing constant on the value of FIS. First, we look at the effect of the manufacturer s lead time on the value of FIS. We calculate the inventory costs of both strategies, NIS and FIS, at three different lead times namely 1, 6 and 12. The simulation result shows that the value of FIS is an increasing function of the manufacturer s lead time. It is evident from Fig. 3 that substantial benefits can be obtained by sharing demand information. There are more benefits of centralizing the demand information when lead times are longer. This is quite logical, as the forecasts for shorter lead times would be less variable than those for longer lead times, thus making demand information sharing less critical (see Lee et al., 2000; Chandra & Grabis, 2005 for details). We now discuss the effect of the smoothing constant in SES on the value of FIS. Chen et al. (2000b) have shown mathematically that for an AR (1) process, the magnitude of the bullwhip effect is an increasing function of the smoothing parameter, α, when an SES is used as the forecasting method. This implies that at higher values of α, there will be greater value of FIS. Our simulation results show that this is not only true for an AR (1) process but also the same principle applies to other ARIMA (p, d, q)

11 EFFECT OF NON-OPTIMAL FORECASTING METHODS ON DOWNSTREAM DEMAND 11 of 18 FIG. 3. Inventory cost reduction by using FIS instead of NIS: effect of lead time (L). FIG. 4. Inventory cost reduction by using FIS instead of NIS: effect of smoothing constant (α). demand processes. The following figure (Fig. 4) shows that the percentage reduction in inventory cost is an increasing function of the smoothing constant. The upstream propagation of demand (as discussed in Section 4) shows that the manufacturer s history contains information about the retailer s demand. Higher values of the smoothing constant result in lower weighting on the history and thus more value of FIS. 7. Conclusions In this paper, we have looked at the effect of non-optimal forecasting methods on DDI. We show that when the supply chain links use SMA for forecasting their lead time demand, the upstream member can accurately infer the demand present at the downstream member, owing to the unique propagation of the demand process. When the supply chain links employ SES, the upstream member would not be able to accurately infer the demand at the downstream member, owing to the non-unique demand propagation in this case.

12 12 of 18 M. M. ALI AND J. E. BOYLAN We examined the magnitude of savings in terms of reduced inventory costs, when the upstream member shares demand information with the downstream links and uses SES to forecast. The simulation results show that sharing demand information always results in lower inventory cost compared to the strategy of not sharing demand. Indeed, sharing information can result in substantial inventory cost savings, as shown by our simulations. When an upstream member uses the FIS strategy, it utilizes less variable demand in its forecasting process. This results in the link having to keep less safety stock and hence reduces inventory cost. FIS is operationally a better method too. First, this encourages companies to pay more attention to consumer demand, which may contribute to a more consumer-centric approach. Second, in this approach, the forecasting process only takes place once, either at the retailer or at the manufacturer or by the collaboration of both links. The business forecasting approach most organizations take is based on forecasting various demand forecasting units that may be stock keeping units (SKUs) or an aggregate of various SKUs (Holmstrom, 1998). The historical data for each of these units are analysed to determine the average, trend and seasonal demand components (e.g. SAP AG, 2004) and then the appropriate forecasting method is selected for each of the units based on the historical data. Thus, when a retailer places an order on the manufacturer for various products, the order generation process may involve different forecasting methods for different products, e.g. SMA for some products and SES or MMSE for others. In order to know the retailer s demand, the manufacturer, in the case of MMSE and SES, will have to make use of some formal information sharing mechanism. On the other hand, if the retailer has employed SMA for some products, the demand can be mathematically calculated. But if the manufacturer already has in place a formal mechanism (e.g. an integrated enterprise resource planning solution) for sharing information, there is no need for them to use another mechanism (mathematical calculation) for products forecasted with the SMA method. They can simply use the information system to find the demand of such products at the retailer. There could be various further avenues of research in this area. The survey of forecasting practitioners showed that most use non-optimal forecasting methods. Although we considered only SMA and SES in this paper, the survey revealed high usage of some other non-optimal forecasting methods. The concept of DDI can be further explored for these other non-optimal forecasting methods. Another interesting avenue of further research could be to consider a more complex supply chains with more than one entity at each stage. Finally, while discussing the benchmark NIS strategy, we assumed that the manufacturer will base their forecast on the orders received from the retailer. Further research is required to evaluate other approaches that the manufacturer may use in this case. REFERENCES ALI, M. M. & BOYLAN, J. E. (2011) Feasibility principles for downstream demand inference in supply chains. J. Oper. Res. Soc., 62, ALWAN, L. C., LIU, J. J. & YAO, D.-Q. (2003) Stochastic characterization of upstream demand processes in a supply chain. IIE Trans., 35, BOX, G. E. P., JENKINS, G. M. & REINSEL, G. C. (1994) Time Series Analysis: Forecasting and Control, 3rd edn. New Jersey: Prentice-Hall. BOYLAN, J. E. & JOHNSTON, F. R. (2003) Optimality and robustness of combinations of moving averages. J. Oper. Res. Soc., 54, CHANDRA, C. & GRABIS, J. (2005) Application of multi-step forecasting for restraining the bullwhip effect and improving inventory performance under autoregressive demand. Eur. J. Oper. Res., 166, CHATFIELD, C. (2003) The Analysis of Time Series: An Introduction, 6th edn. New York: Chapman & Hall.

13 EFFECT OF NON-OPTIMAL FORECASTING METHODS ON DOWNSTREAM DEMAND 13 of 18 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. Manage. Sci., 46, CHEN, F., RYAN, J. K. & SIMCHI-LEVI, D. (2000b) The impact of exponential smoothing forecasts on the bullwhip effect. Naval Res. Logistics, 47, CHENG, T. C. E. & WU, Y. N. (2005) The impact of information sharing in a two level supply chain with multiple retailers. J. Oper. Res. Soc., 56, DALRYMPLE, D. J. (1987) Sales forecasting practices results from a United States survey. Int. J. Forecasting, 3, DANESE, P. (2007) Designing CPFR collaborations: insights from seven case studies. Int. J. Oper. Prod. Man., 27, GAUR, V., GILONI, A. & SESHADRI, S. (2005) Information sharing in a supply chain under ARMA demand. Manage. Sci., 51, GILBERT, K. (2005) An ARIMA supply chain model. Manage. Sci., 51, GRAVES, S. C. (1999) A single-item inventory model for a nonstationary demand process. Manufact. Service Oper. Manage., 1, HAMILTON, J. D. (1994) Time Series Analysis. Princeton, NJ: Princeton University Press. HOLMSTROM, J. (1998) Handling product range complexity: a case study on re-engineering demand forecasting. Bus. Process Manage. J., 43, JOHNSTON, F. R., BOYLAN, J. E., MEADOWS, M. & SHALE, E. (1999) Some properties of a simple moving average when applied to forecasting a time series. J. Oper. Res. Soc., 50, KIM, J. G., CHATFIELD, D., HARRISON, T. P. & HAYYA, J. C. (2006) Quantifying the bullwhip effect in a supply chain with stochastic lead time. Eur. J. Oper. Res., 173, KLASSEN, R. D. & FLORES, B. E. (2001) Forecasting practices of Canadian firms: survey results and comparisons. Int. J. Prod. Econ., 70, LEE, H. L., PADMANABHAN, V. & WHANG, S. (1997) Information distortion in supply chains: the bullwhip effect. Manage. Sci., 43, LEE, H. L., SO, K. C. & TANG, C. S. (2000) The value of information sharing in a two-level supply chain. Manage. Sci., 46, LUONG, H. T. & PHIEN, N. H. (2007) Measure of bullwhip effect in supply chains: the case of high order autoregressive demand process. Eur. J. Oper. Res., 183, MADY, M. T. (2000) Sales forecasting practices of Egyptian public enterprises: survey evidence. Int. J. Forecasting, 16, MCCULLEN, P. & TOWILL, D. R. (2001) Practical ways of reducing bullwhip. Inst. Oper. Manage. Control, 26, MENTZER, J. T. & COX JR, J. E. (1984) Familiarity, application, and performance of sales forecasting techniques. J. Forecasting, 3, MENTZER, J. T. & KAHN, K. B. (1995) Forecasting technique familiarity, satisfaction, usage, and application. J. Forecasting, 14, MONTGOMERY, C. & JOHNSON, L. A. (1976) Forecasting and Time Series Analysis. New York: McGraw Hill. RAGHUNATHAN, S. (2001) Information sharing in a supply chain: a note on its value when demand is nonstationary. Manage. Sci., 47, RAGHUNATHAN, S. (2003) Impact of demand correlation on the value of and incentives for information sharing in a supply chain. Eur. J. Oper. Res., 146, ROBERTS, S. A. (1982) A general class of Holt-Winters type forecasting models. Manage. Sci., 28, 808. SANDERS, N. R. (1992) Corporate forecasting practices in the manufacturing industry. Prod. Inventory Manage. J., 33, SANDERS, N. R. (1997) The status of forecasting in manufacturing firms. Prod. Inventory Manage. J., 38,

14 14 of 18 M. M. ALI AND J. E. BOYLAN SANDERS, N. R. & MANRODT, K. B. (1994) Forecasting practices in US corporations: survey results. Interfaces, 24, SANDERS, N. R. & MANRODT, K. B. (2003) Forecasting software in practice: use, satisfaction, and performance. Interfaces, 33, SAP AG (2004) R/3 System Release 3.0F Online Documentation. Walldorf, Germany: SAP AG. SILVER, E. A., PYKE, D. F. & PETERSON, R. (1998) Inventory Management and Production Planning and Scheduling, 3rd edn. New York: John Wiley. SKJOETT-LARSEN, T., THERNOE, C. & ANDRESEN, C. (2003) Supply chain collaboration: theoretical perspectives and empirical evidence. Int. J. Phys. Distrib. Logist. Manage., 33, SPARKES, J. R. & MCHUGH, A. K. (1984) Awareness and use of forecasting techniques in British industry. J. Forecasting, 3, STAMATOPOULOS, I., TUENTER, R. H. & FILDES, R. A. (2006) The impact of forecasting on the bullwhip effect. Working Paper 2006/016. Lancaster University Management School. SYNTETOS, A. A., BOYLAN, J. E. & DISNEY, S. M. (2009) Forecasting for inventory planning: a 50-year review. J. Oper. Res. Soc., 60, S149 S160. WONG, C. Y., EL-BEHEIRY, M. M,. JOHANSEN, J. & HVOLBY, H.-H. (2007) The implications of information sharing on bullwhip effects in a toy supply chain. Int. J. Risk Assess. Manage., 7, YU, Z., YAN, H. & CHENG, T. C. E. (2002) Modelling the benefits of information sharing-based partnerships in a two-level supply chain. J. Oper. Res. Soc., 53, ZHANG, X. (2004) The impact of forecasting methods on the bullwhip effect. Int. J. Prod. Econ., 88, Appendix Suppose that the retailer uses an SMA to forecast its lead time demand. The equation for SMA for the n most recent demands is given by ˆF t+1 = n 1 The lead time demand forecast, ˆF t+1 L, is given by i=0 n 1 ˆF t+1 L = L i=0 d t i n. d t i n. (A.1) Now, the OUT level is calculated by S t = ˆF L t+1 + zσ ε, where z is the safety factor and σ ε is the standard deviation of the noise in the lead time demand (Silver et al., 1998). The order from the retailer to the manufacturer can be calculated by summing the demand at the retailer plus any change in the OUT level in the current period. Y t = S t S t 1 + d t, Y t = ˆF L t+1 + zσ ε ˆF L t zσ ε + d t,

15 EFFECT OF NON-OPTIMAL FORECASTING METHODS ON DOWNSTREAM DEMAND 15 of 18 ( ) L Y t = n + 1 d t ( ) L d t n. n Suppose d t follows an ARIMA(p, d, q) process Consider the case when d = 0. Substituting ARIMA (p, 0, q) expressions for d t, d t 1,..., d 1 in equation (A.2). ( ) L Y t = n + 1 (τ + ρ 1 d t ρ p d t p + ε t θ 1 ε t 1 θ q ε t q ) ( ) L (τ + ρ 1 d t n 1 ρ p d t n p + ε t n θ 1 ε t n 1 θ q ε t n q ) n [( ) L Y t = τ + n + 1 ρ 1 d t 1 Ln ] [( ) L ρ 1d t n n + 1 ρ p d t p Ln ] ρ pd t n p ( ) ( ) L L + n + 1 (ε t θ 1 ε t 1 θ q ε t q ) (ε t n θ 1 ε t n 1 θ q ε t n q ). n (A.2) Let a t = ( L n + 1 ) ε t ( ) L Y t = τ+ρ 1 Y t 1 + +ρ p Y t p +a t θ 1 a t 1 θ q a t q (a t n θ 1 a t n 1 θ q a t n q ). L + n Hence, Y t follows an ARIMA (p, 0, q + n) process. Suppose that if d t is ARIMA (p, d, q), then Y t is ARIMA (p, d, q +n). Now, let d t follow an ARIMA (p, d + 1, q) process, and let D t = d t d t 1. Then, D t follows an ARIMA(p, d, q) process. Y t Y t 1 = ( ) L n + 1 (d t d t 1 ) ( L n ) (d t n d t n 1 ) = ( ) L n + 1 D t L n D t n. (A.3) By inductive assumption, the process on the right-hand side is ARIMA (p, d, q + n). Hence, Y t follows an ARIMA (p, d + 1, q + n) process. The general result for d = 0, 1, 2, 3,... is proved by induction. Appendix II Suppose that the retailer uses SES to forecast its lead time demand. The SES updating equation is given by ˆF t+1 = αd t + (1 α) ˆF t. Assuming F 1 = d 0 and solving the above equation recursively we get ˆF t+1 = αd t + α(1 α)d t 1 + α(1 α) 2 d t α(1 α) t 1 d 1 + (1 α) t d 0. Calculating the lead time forecast ˆF t+1 L for SES: ˆF L t+1 = L [ αdt + α(1 α)d t 1 + α(1 α) 2 d t α(1 α) t 1 d 1 + (1 α) t d 0 ] (A.4)

16 16 of 18 M. M. ALI AND J. E. BOYLAN Now, the OUT level is calculated by S t = ˆF L t+1 + zσ ε, where z is the safety factor and σ ε is the standard deviation of the noise in the lead time demand. The order from the retailer to the manufacturer can be calculated by summing the demand at retailer plus any change in the OUT level in the current period Y t = S t S t 1 + d t, Y t = ˆF L t+1 + zσ ε ˆF L t zσ + ε d t. The order from the retailer to the manufacturer is Y t = L[αd t + α(1 α)d t 1 + α(1 α) 2 d t α(1 α) t 1 d 1 + (1 α) t d 0 ] L[αd t 1 + α(1 α)d t 2 + α(1 α) 2 d t α(1 α) t 2 d 1 + (1 α) t 1 d 0 ] + d t. Re-arranging the terms, we get Y t = L[α(d t d t 1 ) + α(1 α)(d t 1 d t 2 ) + α(1 α) 2 (d t 2 d t 3 ) + + α(1 α) t 2 (d 2 d 1 ) + α(1 α) t 1 (d 1 d 0 )] + d t. (A.5) Suppose d t follows an ARIMA (p, d, q) process. Consider the case when d = 0. Substituting ARIMA (p, 0, q) expressions for d t, d t 1,..., d 1 in the above, and summarizing terms that equate to ρ 1 Y t 1, ρ 2 Y t 2,..., ρ p Y t p : Y t = τ + ρ 1 Y t 1 + ρ 2 Y t ρ p Y t p + (1 + Lα)ε t [(1 + Lα)θ 1 + Lα 2 ]ε t 1 [(1 + Lα)θ 2 Lα 2 θ 1 + Lα 2 (1 α)]ε t 2 L[α(1 α) t 2 (θ 1 + 1) + α(1 α) t 3 (θ 2 θ 1 ) + α(1 α) t 4 (θ 3 θ 2 ) + ]ε [ ] 1 p + L α(1 α) t 1 (d 1 d 0 ) + αd 0 (1 α) t i ρ i, (A.6) i=2 where d j = 0 for j < 0 and ε j = 0 for j 0. Equation (A.6) is an ARIMA (p, 0, t 1) plus another term given by: L[α(1 α) t 1 (d 1 d 0 ) + αd 0 p i=2 (1 α)t i ρ i ] Therefore, if the demand at the retailer follows an ARIMA (p, 0, q) process, and the retailer uses SES to forecast, the order generated for the manufacturer would follow ARIMA (p, 0, t 1) plus an extra term. Suppose that if d t is ARIMA (p, d, q), then Y t is approximately ARIMA (p, d, t 1). Now, let d t follow an ARIMA (p, d + 1, q) process, and let D t = d t d t 1.

17 EFFECT OF NON-OPTIMAL FORECASTING METHODS ON DOWNSTREAM DEMAND 17 of 18 Then, D t follows an ARIMA(p, d, q) process: Y t Y t 1 = L[α(d t d t 1 ) + α(1 α)(d t 1 d t 2 ) + α(1 α) 2 (d t 2 d t 3 ) + + α(1 α) t 2 (d 2 d 1 ) + α(1 α) t 1 (d 1 d 0 )] + d t L[α(d t 1 d t 2 ) + α(1 α)(d t 2 d t 3 ) + α(1 α) 2 (d t 3 d t 4 ) + + α(1 α) t 3 (d 2 d 1 ) + α(1 α) t 2 (d 1 d 0 )] d t 1 Y t Y t 1 = L[α(D t D t 1 ) + α(1 α)(d t 1 D t 2 ) + α(1 α) 2 (D t 2 D t 3 ) + + α(1 α) t 2 (D 2 D 1 ) + α(1 α) t 1 (d 1 d 0 )] + D t. By inductive assumption, the process on the right-hand side is approximately ARIMA (p, d, t 1). Hence, Y t is approximately ARIMA (p, d + 1, t 1). The general result, for d = 0, 1, 2, 3,..., is proved by induction. We will now derive the expression for the extra term in the case of an ARIMA (p, d, q) process. Substituting ARIMA (p, 1, q) expressions for d t, d t 1,..., d 1 in the equation (A.5) and summarizing terms that equate to ρ 1 (Y t 1 Y t 2 ), ρ 2 (Y t 2 Y t 3 ),..., ρ p (Y t p Y t p 1 ) we get (Y t Y t 1 ) = τ + ρ 1 (Y t 1 Y t 2 ) + ρ 2 (Y t 2 Y t 3 ) + + ρ p (Y t p Y t p 1 ) + (1 + Lα)ε t [(1 + Lα)θ 1 + Lα 2 ]ε t 1 [(1 + Lα)θ 2 Lα 2 θ 1 + Lα 2 (1 α)]ε t 2 L[α(1 α) t 2 (θ 1 + 1) + α(1 α) t 3 (θ 2 θ 1 ) + α(1 α) t 4 (θ 3 θ 2 ) + ]ε 1 + L α(1 p+1 α)t 1 (d 1 d 0 ) + αd 0 [(1 α) t i (ρ i ρ i 1 )]. (A.7) i=2 Equation (A.7) is ARIMA (p, 1, t 1) plus an extra term. The extra term as shown in equation (A.7) is given by L α(1 p+1 α)t 1 (d 1 d 0 ) + αd 0 [(1 α) t i (ρ i ρ i 1 )]. i=2 Using a similar argument for an ARIMA (p, d, q) process, the extra term is given by L α(1 p+d ( α)t 1 (d 1 d 0 ) + αd 0 [(1 α) t i d(d 1) ρ i dρ i 1 + ρ i 2 2! i=2 )] } d(d 1)(d 2) ρ i ( 1) d ρ i d. 3! (A.8) Now, we will establish an upper bound for the absolute value of this extra term for an ARIMA (p, d, q) process. This upper bound assumes that d t 0 for t 0. Let E be the value of the extra term, and let U be an upper bound of its absolute value, to be determined: p+d E < Lα(1 α) t 1 d 1 d 0 +Lαd 0 i=2 (1 α) t i [ ρ i + d ρ i 1 + ] d(d 1) ρ i ρ i d. 2!

18 18 of 18 M. M. ALI AND J. E. BOYLAN Let the maximum modulus of all the rho terms, in all summations, be P = max{ ρ i } i=p+d i=2 d. Clearly, P < 1. Now, let D = d/2 if d is even, and let D = (d + 1)/2 if d is odd. The maximum coefficient of the terms in the second part of the above expression is then: ( ) d D Hence, ( E < Lα(1 α) t 1 d p+d d 1 d 0 + Lα(d + 1) (1 α) D) t i d 0 and an upper bound, U, is given by U = Lα(1 α) t 1 d 1 d 0 + L(d + 1) i=2 ( d D) (1 α) t p d [1 (1 α) p+d 1 ]. (A.9)