The Dynamic Newsvendor Model with Correlated Demand

Decision Sciences Volume 47 Number 1 February 2016

© 2015 Decision Sciences Institute

The Dynamic Newsvendor Model with Correlated Demand∗ Layth C. Alwan Sheldon B. Lubar School of Business, University of Wisconsin-Milwaukee, Milwaukee, WI, e-mail: [email protected]

Minghui Xu† School of Economics and Management, Wuhan University, Wuhan 430072, China, e-mail: [email protected]

Dong-Qing Yao College of Business and Economics, Towson University, Baltimore, MD, e-mail: [email protected]

Xiaohang Yue Sheldon B. Lubar School of Business, University of Wisconsin-Milwaukee, Milwaukee, WI, e-mail: [email protected]

ABSTRACT The classic newsvendor model was developed under the assumption that period-toperiod demand is independent over time. In real-life applications, the notion of independent demand is often challenged. In this article, we examine the newsvendor model in the presence of correlated demands. Specifically under a stationary AR(1) demand, we study the performance of the traditional newsvendor implementation versus a dynamic forecast-based implementation. We demonstrate theoretically that implementing a minimum mean square error (MSE) forecast model will always have improved performance relative to the traditional implementation in terms of cost savings. In light of the widespread usage of all-purpose models like the moving-average method and exponential smoothing method, we compare the performance of these popular alternative forecasting methods against both the MSE-optimal implementation and the traditional newsvendor implementation. If only alternative forecasting methods are being considered, we find that under certain conditions it is best to ignore the correlation and opt out of forecasting and to simply implement the traditional newsvendor model. [Submitted: July 12, 2013. Revised: October 11, 2013. Accepted: October 29, 2013.]

Subject Areas: Autocorrelated Newsvendor Model.

Demand,

Demand

Forecasting,

and

∗ The authors gratefully acknowledge the two anonymous referees and the associate editor for their construc-

tive comments and valuable suggestions on improving this article. Minghui Xu was supported by grants from National Science Foundation of China (Nos. 70901059 and 71371146) and the Fundamental Research Funds for the Central Universities. † Corresponding

author.

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The Dynamic Newsvendor Model with Correlated Demand

INTRODUCTION Due to fierce global competition in today’s markets, products’ life cycles are shorter and shorter. Not only fashion apparel or fresh food, but many high-tech electronics are also experiencing newsvendor phenomena. As such, newsvendor has been becoming a main building block in the research of inventory control, supply chain coordination, yield management, etc. So far, most research on newsvendor model assumes the demand is independent and identically distributed (iid). For example, Chen and Parlar (2007) studied the application of put option into the newsvendor model to improve the risk–return profile, where demand is a random variable. Besides standard newsvendor model, research extensions of newsvendor model include the price-sensitive demand newsvendor model (Kocabiyikoglu & Popescu, 2011; Wang, Lau, & Lau, 2012), multiple-class demand newsvendor model (Sen & Zhang, 1999), and newsvendor network (Van Mieghem & Rudi, 2002). More literature review can also be found in Qin, Wang, Vakharia, Chen, and Seref (2011). Among all these research studies, some papers are dedicated to studying multiperiod newsvendor model. For example, Bensoussan, Cakanyildirim, and Sethi (2007) studied the newsvendor model where only partially Markovian demand can be observed because only the demand less than inventory can be revealed. In practice, decision makers are not always perfect optimizers, which could be caused by human beings errors and biases. For example, Su (2008) adopted the quantal choice model to identify systematic biases and offered insight into when overordering and underordering could happen. In this article, we will investigate the decision bias and errors from forecast techniques perspective for the multiperiod newsvendor problem. To the best of our knowledge, only little research has been done on the demand forecast for the newsvendor model (Ozer, Uncu, & Wei, 2007; Wang, Atasu, & Kurtulus, 2012). In this article, we assume demand over time is autocorrelated. In reality, for perishables any leftover inventory at the end of each period is salvaged; while any unsatisfied demand is lost and related underage cost is incurred. However, demand over each time period is not necessarily iid, for example, the demand for fresh organic vegetables in urban China could meet this kind of demand pattern. Because of busy life schedule, high income white-collar people in China prefer to order daily fresh food online. This practice is similar to Peapod in the U.S.A. The only difference is that this kind of e-marketplace in China is government supported financially so that the government can monitor food safety and other food quality issued. In addition, the segment of high income customers is not sensitive to price, but only cares about food safety, so the demand is price-inelastic. More importantly, all the orders are placed online, so lost demand information can also be easily collected. Therefore, unlike the multiperiod newsvendor problem studied in Bensoussan, Feng, and Sethi (2011), full demand information can be observed in this e-business environment. Another example, as shown in this article, for this type of demand could be blood demand for regional hospitals in an urban city. Because this type of problem is not single-period, but repeated across certain times/seasons, different forecasting techniques can be used to help making better supply decision in each period. However, it should be recognized that due to the

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nature of the perishablity of the blood product, any leftover inventory will be salvaged or disposed, any unsatisfied demand is lost with shortage cost or satisfied by emergency delivery with high extra cost. As such, we have a newsvendor-type problem but with period-to-period demand being correlated. The contribution of the article is twofold. Aforementioned, we first extend the demand of the newsvendor model to be autocorrelated and analyze the optimal ordering decision. Second, we compare the mismatch cost for different forecasting techniques, thus evaluate the decision performance caused by model choice. The remainder of the article is as follows. In section “The Model,” we present the autocorrelated newsvendor style demand process, and study the ordering decisions based on mean square error (MSE) optimal forecasts versus the traditional newsvendor implementation using a static long-term mean forecast. In section “Implementing Other Forecasting Methods,” we extend the newsvendor ordering decisions to the other commonly used business forecasting techniques, in particular, moving average and exponential smoothing methods, and compare the expected costs under these different forecasting schemes. All proofs are put in the Online Appendix.

THE MODEL Consider a newsvendor-type firm who sells a single product in an infinite planning horizon. There are a number of potential stochastic processes that can be assumed for the demand process, ranging from a simple iid process to a nonstationary process. In reality, there are few real-world (demand or otherwise) processes that exhibit pure iid behavior (Alwan & Roberts, 1995; Charnes, Marmorstein, & Zinn, 1995). Given a time-pattern to the demand process, the firm has an option to make an ordering decision each period based on some form of demand forecasting. One flexible correlated demand process that has been studied extensively in the supply chain literature is the AR(1) model (Kahn, 1987; Lee, Padmanabhan, & Whang, 1997; Chen, Drezner, Ryan, & Simchi-Levi, 2000; Chen, Ryan, & SimchiLevi, 2000; Lee, So, & Tang, 2000). In the following subsections, we describe the AR(1) model in more details, consider the newsvendor problem when the correlative structure is optimally exploited, and consider the newsvendor problem when the correlative structure is effectively ignored.

Motivating Example As we noted, our motivation is to consider the newsvendor problem in the context of a correlated demand process. We have made references to applications for which non-iid behavior has been encountered for perishable goods. In this section, we motivate our model development with empirical data demonstrating the reality of correlative data in newsvendor-type scenarios. Specifically, we consider blood demand collected from a large regional hospital in Milwaukee county of Wisconsin. Blood is composed of components such as red blood cells (RBC), white cells, platelets, and plasma, all of which (except plasma) are perishable, with varied lifetime between 6 hours and 35 days. The demand for blood is highly fluctuated. Hospitals and blood banks aim to

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Figure 1: Time-series plot of daily blood demand. 12

RBC O+

10 8 6 4 2 0 0

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accurately forecast the blood demand each day and then plan and manage their blood inventory. Currently the commonly adopted inventory management practice is order-up-to replenishment policy (De Kort, Janssen, & Kortbeck, 2011), which is optimal for nonperishable products. However, the value of this kind of practice is suboptimal for blood inventory management. Actually for perishables like blood, if the inventory level is too high, there is a risk of the product being outdated and limited or no value. As such, certain cost (e.g., $20–$40) will be incurred because all blood banks and centers charge hospitals fee for collection, lab testing, storage, and distribution (Prastacos, 1984). On the other hand, if the inventory level is too low and there is a shortage, emergency delivery from local or region blood banks/centers must be made, and high extra cost (e.g., $80–$100) is incurred to the hospital. This cost varies for different cities and delivery mode (e.g., air delivery for populous city in LA). Therefore, the newsvendor model can be adopted to analyze the optimal ordering policy for blood inventory. The RBC O+ transfusion daily demand data from June 2009 to January 2010 were made available from the hospital’s information system. The transfusion data considered in our analysis are categorized as “new-aged” blood cells, which are used for the most critical patients, such as cancer and immune-deficient patients. If these blood cells are unused by day’s end, then they are downgraded to the category of “medium-aged” blood cells. Currently, the hospital simply uses the data to estimate overall daily mean demand to establish inventory levels. Figure 1 shows the time-series plot of the demand data. The plot shows a high degree of variability with demand varying from as low as 0 units to as high as 12 units. Careful inspection of the plot shows hints of a nonrandom structure in the form of positively correlated meandering around the overall mean. Applying the nonparametric runs test to the data reveals 99 observed runs in the series versus an expected number of 119.125 runs under the null hypothesis of an underlying iid process. The reported p-value for the runs test is 0.008, which is highly significant at the 5% level of significance. Inspection of an autocorrelation function (ACF) and partial autocorrelation function (PACF) for the blood series would show a significant correlation at lag

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Figure 2: Estimated AR(1) model for blood series. 12 RBC O+

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1. Furthermore, the Ljung-Box Q statistics are all highly significant for all lags shown. This is further evidence that the blood series has an underlying systematic pattern. The fact that the PACF only reveals a significant lag one correlation is an indication that the data series might be well fitted with an AR(1) model. Using the ARIMA fitting procedure of SAS JMP version 11, we find the estimated AR(1) model to be dˆ t = 2.045 + 0.2659dt−1 . The model diagnostics for the estimated AR(1) are excellent as reflected by the randomness tests of the runs test, ACF, and Ljung-Box Q statistics all suggesting random residuals. Furthermore, both the Akaike’s Information Criterion (AIC) value and Bayesian Information Criterion (BIC) value for the estimated AR(1) are substantially smaller than the mean-only fitted model. Figure 2 shows the estimated AR(1) model (solid line) superimposed on the original blood series (dashed line). The plot of the fitted model clearly shows the meandering behavior to the blood series process. Basing inventory replenishment on a horizontal mean model can prove to be suboptimal. The subsequent sections will explore this issue for the more general case.

The Demand Process Suppose that the demand faced by the firm is an AR(1) demand process (Lee et al., 1997; Chen, Drezner, Ryan, & Simchi-Levi, 2000): dt+1 = τ + φdt + εt+1 , −1 ≤ φ ≤ 1, where dt is the observed demand for period t, τ > 0 is the base demand in a period, and the noise process {εt } is an independent and identically distributed normal (iidn) process with mean zero and variance σε2 . Let f (·) and F (·) be the probability density function and cumulative distribution function of εt , respectively, and F −1 (·) its inverse function. The parameter φ reflects the degree and direction of demand process correlation. When |φ| < 1, it is known that the demand process is stationary and the long-term (marginal) mean and variance are (Box, Jenkins, & Reinsel, 2008) μd =

τ σε2 , , σd2 = 1−φ 1 − φ2

(1)

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respectively. For any i ≥ 1, dt+i = τ + φτ + · · · + φ i−1 τ + φ i dt + φ i−1 εt+1 + φ i−2 εt+1 + · · · + εt+i =

1 − φi τ + φ i dt + φ i−1 εt+1 + φ i−2 εt+1 + · · · + εt+i . 1−φ

(2)

By varying the values of φ, one can represent a wide variety of process behaviors. When φ = 0, we have an iidn process with mean τ and variance σε2 . For −1 < φ < 0, the demand process is negatively correlated and will exhibit period-to-period oscillatory behavior. For 0 < φ < 1, the demand process will be positively correlated, which is reflected by a wandering or meandering sequence of observations. As |φ| approaches one, the process approaches nonstationary behavior; in particular, an AR(1) with φ = 1 represents a pure random walk model or equivalently, an ARIMA(0,1,0) process. As pointed out by Graves and Willems (2000), varying a stationary demand model, such as an AR(1) model, is an important exercise for gaining fundamental insights about the relationships between variables such as inventory and orders relative to a wide variety of demand characterizations. For a general ARIMA process, it can be shown that the minimum MSE forecast for period t + 1 is the conditional expectation of dt+1 given current and previous observations dt , dt−1 , . . .(Box et al., 2008). In the case of an AR(1) process, this implies the MSE-optimal forecast function is given by E[dt+1 |dt ]. Because E[εt+1 |dt ] = 0, it immediately follows that for an AR(1) process, the MSE-optimal forecast for period (t + 1) is dˆ t+1 = E[dt+1 |dt ] = τ + φdt .

(3)

It is useful to note that the variance of one-step ahead forecast error is given by σε2 .

MSE Forecast-Based Implementation of Newsvendor Problem We assume that the firm faces a newsvendor problem in each period. Any unsatisfied demand will be lost and incur an underage cost, and any leftover inventory will be disposed (or salvaged) with an overage cost. Suppose the cost of overage is co and the cost of underage is cu . Under such a newsvendor model setting with correlated demand process over time, the main purpose is to see if a forecast-based ordering decision will have lower expected costs and by how much given the parameters (time-series parameters and cost parameters). By observing the realized demand in period t, dt , an order quantity Q is placed to satisfy the demand in period t + 1, then the total overage and underage cost incurred at the end of period t + 1 will be C(Q|dt ) = co max{Q − dt+1 , 0} + ˆ = Q − τ − φdt , the total overage and underage cu max{dt+1 − Q, 0}. By letting Q cost can be written as follows: ˆ − εt+1 , 0} + cu max{εt+1 − Q, ˆ 0}. ˆ t ) = co max{Q C(Q|d

(4)

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The conditional expected cost is given by Qˆ ˆ − x)f (x)dx + cu ˆ t )] = co (Q E[C(Q|d −∞

∞ ˆ Q

ˆ (x)dx. (x − Q)f

(5)

The following proposition provides the optimal ordering decision and the corresponding cost when using MSE optimal forecast. Proposition 1: The optimal order quantity based on the MSE optimal forecast is cu cu ∗ −1 −1 Qt+1 (dt ) = (τ + φdt ) + F = (τ + φdt ) + σε ,(6) co + cu co + cu and the corresponding optimal cost is ˆ ∗ |dt )] = (co + cu )σε E[C(Q

∞

xd(x),

(7)

u ) −1 ( coc+c u

ˆ ∗ = F −1 (cu /(co + cu )) and (·) is the standard normal distribution funcwhere Q tion with its inverse −1 . It follows from (6) that the forecast-based single period order quantity will dynamically change to better adapt the changing of demands. The first term is the forecasted demand for period t + 1 and the second term reflects the projected service level at a period. It can also be seen that the optimal conditional expected cost is independent of dt , and only depends on the demand noise.

Traditional Implementation of Newsvendor Problem In the traditional presentation of the newsvendor problem, demand is implicitly or explicitly assumed to be an iid process. With an iid process, the long-term mean is used as the sole single-valued forecast in the development of the optimal order quantity. In practice, it is not unexpected that managers will implement this single mean-based method ignoring the correlative structure at hand. The presence of autocorrelation can be particularly elusive in the case of stationary behavior because the process is not drifting away from the long-term mean. Nonetheless, stationarity or nonstationarity, an implementation ignoring the correlative structure is a nonoptimal forecast method. The recognition of inaccurate forecasting in practice and the study of inaccurate impact is rich in business and economics literature. In the operations management area, inaccurate forecasting—including ignoring correlation—has been studied in the context of supply chains and bullwhip effect (Lee et al., 1997; Lee et al., 2000; Chen, Drezner, Ryan, & Simchi-Levi, 2000; Chen, Ryan, & Simchi-Levi, 2000; Alwan, Liu, & Yao, 2003). Lee and Adam (1986) studied the impact of forecast errors on the performance of material requirements planning (MRP) systems. Our focus is on the impact of such forecast inaccuracies on the newsvendor problem. With this in mind, suppose the firm uses the long-term mean and variance as the basis of its optimal order quantity, that is, each period the quantity ordered would be based on a cumulative distribution function with mean and variance given by Equation (1). Such an approach means that the firm would place an order of ¯ ∗ to minimize its expected cost by assuming a prediction of μd for demand size Q

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each period with a variance of σd2 . Let H (·) be a normal demand distribution with mean μd and variance σd , which is given in (1). ¯ can be expressed as The long-term average expected cost function LC(Q) Q¯ ∞ ¯ ¯ ¯ (Q − x)dH (x) + cu (x − Q)dH (x). LC(Q) = co −∞

¯ Q

The following proposition provides the optimal ordering decision and the corresponding cost when using long-term forecast method. Proposition 2: The optimal order quantity based on the long-term forecast is c c σε τ u u ¯ ∗ = μd + σd −1 + −1 Q = , (8) co + cu 1−φ co + cu 1 − φ2 and the corresponding optimal cost is o + cu )σε ¯ ∗ ) = (c LC(Q 1 − φ2

∞

xd(x).

(9)

u ) −1 ( coc+c u

Consider the blood series example mentioned earlier to illustrate the differences in implementation between the MSE-optimal approach versus the traditional approach. We found that the series was well fitted by an AR(1) model with τ estimated to be 2.045 and φ estimated to be 0.2659. The overall estimated mean and standard deviation of the process can be found to be 2.792 and 2.61. Let us further assume cu = $80 and co = $20 which gives −1 (0.8) = 0.842. Accordingly, the ¯ ∗ = 2.792 + 0.842(2.61) = 4.99 or essentraditional implementation calls for Q tially 5 units. Referring back to Figure 1, it is worthy to note that to the untrained eye the correlative structure in the process is not obvious leaving the practitioner with a false sense of security that a static period-to-period order quantity of 5 units is appropriate. What the traditional ordering policy fails to do is to recognize the possibility of an autocorrelative structure to the demand data. As such, there is no consideration to period-by-period forecast updating. Suppose, for example, the current observed demand is 11. The forecast (conditional mean estimate) is dˆ t+1 = 2.045 + 0.2659(11) = 4.97. Because we do not know the true error standard deviation, we can use the estimated standard deviation given by the root MSE from the model fit which is reported to equal 2.521. Using this conditional mean estimate along with the estimated error standard deviation, we have an order quantity for next period to be Q∗t+1 (11) = 4.97 + 0.842(2.521) = 7.09 as opposed to the static order quantity of 5 units. Figure 3 shows the dynamic order quantities for the blood series along with the traditional implementation of a static order quantity represented by the horizontal line at 5. Clearly, the implementations of the dynamic and traditional newsvendor solutions are different. It is natural to then ask what is the expected cost differences between the two approaches? It follows from (1) that σd ≥ σε , then we can see that the optimal long-term expected cost is greater than the optimal forecast-based expected cost. We can summarize the relative cost difference by looking at the ratio of σε to σd . Or, equivalently, we can consider the long-term percentage

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Figure 3: Dynamic order quantities versus traditional implementation for blood series. 7.5 7 Dynamic Q

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Figure 4: Cost reduction (%) of implementing MSE-optimal versus traditional approach. 100

Cost Reduction (%)

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0 -1.0

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cost reduction by implementing the forecast-based method versus the traditional method (“static” order quantity). In particular, we have LC(Q ¯ ∗ ) − E[C(Q ˆ ∗ |dt )] P (φ|MSE, Static) = 100 % = 100(1 − 1 − φ 2 )%. ∗ ¯ LC(Q ) As can be seen, the cost difference is determined by the absolute value of parameter φ. Figure 4 draws the curve of P (φ|MSE, Static) against φ. When φ = 0 (i.e., the independent demand process), we know that σd = σε , and hence the forecastbased implementation and the traditional implementation would have the same expected cost implying there is no cost reduction as seen in Figure 4. However,

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as |φ| approaches 1, the percentage cost reduction increases to 100% in the limit. Intuitively, this result makes sense in that as |φ| approaches 1, the demand process is approaching nonstationarity with a variance arbitrarily large. In such a situation, the long-term mean does not exist. However, the forecast-based approach can still be implemented because the conditional mean exists.

IMPLEMENTING OTHER FORECASTING METHODS Smoothing-based methods, such as moving average and exponential smoothing, are widely employed for forecasting purposes in many supply chain and operations management applications in industry, largely due to their simplicity and ease of implementation. Given the widespread use of these forecast methods, researchers of operations management problems have based their studies on the moving average method and/or the exponential smoothing method (Chen, Drezner, Ryan, & Simchi-Levi, 2000; Chen, Ryan, & Simchi-Levi, 2000; Alwan et al., 2003). These studies focused on an order-up-to policy within a supply-chain context with focus on the bullwhip effect. In this section, we will, however, consider both moving average method and exponential smoothing method to gain insights on their applications in the context of the newsvendor problem.

The Moving Average Method The moving average forecasting model of length k (MA(k), k ≥ 1) can be expressed as follows: dt + dt−1 + · · · + dt−k+1 dˆ t+1 = . k

(10)

To evaluate the cost function, we need to consider the forecast error for the above MA moving average model, which is et+1 = dt+1 − dˆ t+1 . It should be recognized that an AR(1) can be re-expressed in terms of errors (current and past): ∞

dt+1 =

τ + φ j εt+1−j , 1 − φ j =0

(11)

σ2

τ with mean E[dt+1 ] = 1−φ and variance Var(dt+1 ) = 1−φε 2 . MA ) is a linear combination Because the moving average forecast error (et+1 of normal variants, it also follows the normal distribution. By some calculations (please refer to Online Appendix), we know that the expected value (mean) of the forecast error for the moving-average method is 0 and the standard deviation of MA et+1 is k(1 − φ 2 ) − 2φ(1 − φ k )[1 + k(1 − φ)] 1 1 + . (12) σeMA = σε 1 − φ2 k 2 (1 − φ)2 MA is a normal random variable with mean of zero and standard Therefore, et+1 MA deviation of σe . Similar to Proposition 2, we have the following result.

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Proposition 3: Based on the moving average forecast MA(k) (k ≥ 1), in the context of the newsvendor problem, the optimal order quantity would be:

k−1 j =0 dt−j ∗ Qt+1 (dt , dt−1 , · · · , dt−k+1 ) = k 1 k(1 − φ 2 ) − 2φ(1 − φ k )[1 + k(1 − φ)] −1 1 + + σε 1 − φ2 k 2 (1 − φ)2 cu , (13) cu + co and the corresponding optimal average cost is

E[C(Q∗t+1 (dt , dt−1 , · · · , dt−k+1 ))]

= (co +

cu )σeMA

∞

xd(x). (14) u ) −1 ( cuc+c o

The Exponential Smoothing Method The simple exponential smoothing (SES) model is given by dˆ t+1 = αdt + (1 − α)dˆ t ,

(15)

where α (0 ≤ α ≤ 1) is “smoothing constant.” Even though the SES method, and to a lesser extent the moving-average method, has flexibility for adapting to a variety of correlated demand processes, it is MSE optimal for only one underlying time-series process, namely, a first-order integrated moving average, denoted by ARIMA(0,1,1) (Graves, 1999; Box et al., 2008). An ARIMA(0,1,1) is a nonstationary process that can be interpreted as a random-walk trend plus a random disturbance from the trend. With that said, its widespread usage as an all-purpose forecasting method in supply chain and operations management makes it important to study. By recursively applying (15) back t periods and by substituting in (11), we can express the SES forecast in terms of the initial starting forecast value (dˆ 1 ) and past errors: t dˆ t+1 = α (1 − α)t−i di + (1 − α)t dˆ 1 . i=1

The mean forecast can easily be shown to be E(dˆ t+1 ) = μd + (1 − α)t (dˆ 1 − μd ). As a result, the expected value (mean) of the forecast errors for the SES method can be seen to be SES E[et+1 ] = E[dt+1 − dˆ t+1 ] = E[dt+1 ] − E[dˆ t+1 ]

= μd − [μd + (1 − α)t (dˆ 1 − μd )] = (1 − α)t (μd − dˆ 1 ). Thus, for a finite t and if dˆ 1 = μd then mean forecast error is not equal to 0. We will assume that t is sufficiently large and dˆ 1 is judiciously chosen so that the mean forecast error is either equal to 0 or any difference from 0 is negligible.

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For large t, the variance of the forecast errors can be shown to approach σε2 α − 3αφ + α 2 φ SES lim Var(et+1 ) = . 1+ t→∞ 1 − φ2 (2 − α)(1 − φ + αφ) As with the moving-average method, because the forecast error for the SES method SES ) is a linear combination of normal variants, it also follows the normal distri(et+1 SES is a normal random variable with bution. In summary, for sufficiently large t, et+1 mean of 0 and standard deviation of α − 3αφ + α 2 φ 1 σeSES = σε . (16) 1+ 2 1−φ (2 − α)(1 − φ + αφ) Summarizing the above results, we get the following proposition, which can be derived similarly to Proposition 3. Proposition 4: Based on the SES forecast model given in (15), suppose that the initial starting forecast is chosen as the long-term mean μd , the order quantity for the newsvendor problem would be Q∗t+1 (dt , dˆ t , α) = [αdt + (1 − α)dˆ t ] cu α − 3αφ + α 2 φ 1 −1 + σε , (17) 1 + 1 − φ2 (2 − α)(1 − φ + αφ) cu + co and the corresponding optimal average cost is

E[C(Q∗t+1 (dt , dˆ t , α))] = (co + cu )σeSES

∞

xd(x).

(18)

u ) −1 ( cuc+c o

Study of Expected Cost Under Different Forecasting Schemes Earlier for φ = 0, we demonstrated that the long-term expected cost associated with dynamic stocking decisions based on MSE-optimal forecasting is always better than a static stocking decision based on the overall mean demand. In this section, we will study the expected cost for the alternative methods of the moving average and SES relative to the MSE-optimal method and relative to the static stocking decision. Specifically, we will investigate the relative cost for different model parameters of the moving-average method and SES method at different levels of underlying demand correlation. Cost Comparison between MSE-Optimal Forecast and the Moving Average Method To demonstrate that MSE-optimal implementation outperforms the movingaverage implementation based on long-term expected costs, it suffices to show that σeMA ≥ σε for all φ ∈ (−1, 1) and k ≥ 1. It follows from (12) that σeMA ≥ σε is equivalent to 1 k(1 − φ 2 ) − 2φ(1 − φ k )[1 + k(1 − φ)] 1+ ≥ 1, 1 − φ2 k 2 (1 − φ)2

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which is equivalent to demonstrating that T (k, φ) = k(1 − φ 2 ) + k 2 φ 2 (1 − φ)2 − 2φ(1 − φ k )[1 + k(1 − φ)] ≥ 0. By means of rearrangement, T (k, φ) can be rewritten as follows: T (k, φ) = (1 − φ)[φ 2 (1 − φ)k 2 + (2φ 2 + 1 − φ)k − 2φ]. Let W (k, φ) = φ 2 (1 − φ)k 2 + (2φ 2 + 1 − φ)k − 2φ, implying that T (k, φ) = (1 − φ)W (k, φ). For all φ ∈ (−1, 1), we see that (1 − φ) > 0, which means we need only to show that W (k, φ) ≥ 0. For a given φ, W (k, φ) is a quadratic function of k. For all φ ∈ (−1, 1), the quadratic coefficient term of φ 2 (1 − φ) is positive and, thus, W (k, φ) is convex with the minimum attained 2 +1−φ) at k = −(2φ , which is a negative value for φ ∈ (−1, 1). We now recognize 2φ 2 (1−φ) that at φ = 1, W (1, φ) = 0 and furthermore it is an inflection point with the W (1, φ) convex for φ < 1 implying W (1, φ) > 0 for all φ ∈ (−1, 1). Coupling these results, we can now say that W (k, φ) ≥ 0 for k ≥ 1 and φ ∈ (−1, 1). Hence, σeMA ≥ σε for k ≥ 1 and φ ∈ (−1, 1). To study the degree to which MSE-optimal implementation improves upon the moving-average dynamic ordering policy, we can consider the long-term percentage cost reduction: P (k, φ|MSE, MA) = 100

E[C(Q∗ (dt , dt−1 , · · · , dt−k+1 ))] − E[C(Q ˆ ∗ |dt )] t+1 % E[C(Q∗t+1 (dt , dt−1 , · · · , dt−k+1 )]

= 100 1 −

k 2 (1 − φ)2 (1 − φ 2 ) %. k 2 (1 − φ)2 + k(1 − φ 2 ) − 2φ(1 − φ k )[1 + k(1 − φ)]

Figure 5 shows the curves of P (k, φ|MSE, MA) against φ for different values of k . Figure 5 clearly shows the distinct advantage of using the MSE-optimal ordering policy over the moving-average policy in terms of cost reduction. Other than the iid case, the moving-average approach is only competitive when the demand process is highly correlated and k = 1 (i.e., the naive forecast).

Cost Comparison between Traditional Implementation and the Moving Average Method Considering the moving-average method for small k is reasonably competitive with the optimal forecast for highly correlated demand process while the overall mean forecast is more suitable for near iid conditions, we intuitively expect differing advantages between static implementation and dynamic implementation based on the moving-average method. By using (12), we can measure long-term percentage cost change by implementing the moving-average dynamic ordering policy versus static ordering policy: P (k, φ|MA, Static)

¯ ∗ ) − E[C(Q∗ (dt , dt−1 , · · · , dt−k+1 ))] LC(Q t+1 = 100 % ¯ ∗) LC(Q

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Figure 5: Cost Rreduction (%) of implementing MSE-optimal versus movingaverage approach. 90

k=1 k=5 k=15

80

Cost Reduction (%)

70 60 50 40 30 20 10 0 -1.0

-0.5

0.0 phi

0.5

1.0

Figure 6: Cost reduction (%) of implementing moving-average versus traditional approach. k=1 k=5 k=15

Cost Reduction (%)

50

0

-50

-100

-1.0

-0.5

0.0

0.5

1.0

phi

= 100 1 −

k 2 (1 − φ)2 + k(1 − φ 2 ) − 2φ(1 − φ k )[1 + k(1 − φ)] %. k 2 (1 − φ)2

Figure 6 shows the curves of P (k, φ|MA, Static) against φ for different values of k. Figure 6 shows that the dynamic order quantity based on the moving average does not uniformly dominate a static order quantity. Interestingly, we find that the

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moving average approach shows cost reductions when the demand process has a lag 1 autocorrelation of around 0.5 or greater for the different values of k. When process has weak positive correlation or negative correlation (weak or strong) then the moving average method is at a distinct disadvantage to fixed-order quantity implementation with the disadvantage worsening as k decreases.

Cost Comparison between MSE-Optimal Forecast and the Exponential Smoothing Method Similar to abovementioned, we would need to demonstrate that the ratio of SES V ar(et+1 ) to the error variance is greater than or equal to 1. Referring to (16), we have α − 3αφ + α 2 φ 1 ≥ 1, (19) 1 + 1 − φ2 (2 − α)(1 − φ + αφ) which is equivalent to φ 2 (2 − α)(1 − φ + αφ) + α − 3αφ + α 2 φ ≥ 0, or (1 − φ)[φ(1 + φ)α 2 + (1 + φ)(1 − 3φ)α + 2φ 2 ] ≥ 0. Let S(α) = φ(1 + φ)α 2 + (1 + φ)(1 − 3φ)α + 2φ 2 1 − 3φ 2 (1 − φ)(φ 2 + 4φ − 1) . = φ(1 + φ) α + + 2φ 4φ Clearly, S(α) is a quadratic function of α with minimum point (3φ − 1)/(2φ) and the minimum (1 − φ)(φ 2 + 4φ − 1)/(4φ). When 0 ≤ φ < 1/3, S(α) is increasing in α ∈ [0, 1], which in turn implies that S(α) ≥ S(0) = 2φ 2 ≥ 0 for all α ∈ [0, 1]. When 1/3 ≤ φ ≤ 1, it then follows that S(α) ≥ 0 for all α ∈ [0, 1] because φ 2 + 4φ − 1 > 0 in the range 1/3 ≤ φ ≤ 1. To study the degree to which MSE-optimal implementation improves upon the exponential smoothing dynamic ordering policy, we can consider the long-term percentage cost reduction:

ˆ ∗ |dt )] E[C(Q∗t+1 (dt , dˆ t , α))] − E[C(Q P (α, φ|MSE, SES) = 100 % E[C(Q∗t+1 (dt , dˆ t , α))] = 100 1 −

1 − φ2 1+

α−3αφ+α 2 φ (2−α)(1−φ+αφ)

%.

Figure 7 shows the curves of P (α, φ|MA, SES) against φ for different values of α. Clearly, for strongly negative demand processes, the exponential smoothing method is at a clear disadvantage as reflected by the large cost savings associated with switching to the MSE-optimal approach. For strong positive autocorrelation (φ > 0.5), the smoothing method approach is fairly competitive for large α. If the newsvendor is implemented with a small α for strong positive, the practitioner is severely penalized relative to using an optimal forecast.

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Figure 7: Cost reduction (%) of implementing MSE-optimal versus exponential smoothing approach. 90 alpha=0.1 alpha=0.5 alpha=0.9

80

Cost Reduction (%)

70 60 50 40 30 20 10 0 -1.0

-0.5

0.0

0.5

1.0

phi

Cost Comparison between Traditional Implementation and the Exponential Smoothing Method As we observed with the moving-average method, we would suspect for similar reasonable conditions for which the exponential smoothing method will be at an advantage over the traditional approach and conditions for which it is not an advantage. By using (16), we can measure long-term percentage cost change by implementing the exponential smoothing dynamic ordering policy versus static ordering policy: LC(Q ¯ ∗ ) − E[C(Q∗ (dt , dˆ t , α))] t+1 % P (α, φ|SES, Static) = 100 ¯ ∗) LC(Q

= 100 1 −

α − 3αφ + α 2 φ %. 1+ (2 − α)(1 − φ + αφ)

Figure 8 shows the curves of P (α, φ|SES, Static) against φ for different values of α. Like the moving-average implementation, Figure 8 shows that the dynamic order quantity based on the exponential smoothing does not uniformly dominate a static order quantity. Interestingly, we find like the moving average method, the forecasting method is in favor for positive autocorrelation of around 0.5 or greater. Furthermore, we find that for moderate to strong positive autocorrelation, there is little difference when the smoothing constant is chosen to be around 0.5 or larger. When process has weak positive correlation or negative correlation (weak or strong), then the exponential smoothing method is at a distinct disadvantage

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Figure 8: Cost reduction (%) of implementing exponential smoothing versus traditional approach. alpha=0.1 alpha=0.5 alpha=0.9

Cost Reduction (%)

50

0

-50

-100 -1.0

-0.5

0.0

0.5

1.0

phi

to fixed order quantity implementation with the disadvantage worsening as α increases.

CONCLUDING REMARKS In this article, motivated by a real practical example with available empirical data, we proposed the consideration of dynamic period-to-period ordering decisions as opposed to the traditional static order decision for the classic newsvendor decision. As demonstrated, a dynamic ordering decision based on an MSE-optimal forecast can provide the practitioner with substantial cost savings over the traditional implementation. In general, the newsvendor decision based on an MSE-optimal forecast will always provide a cost savings over the traditional implementation across different levels of correlation. Given the results of improved performance using a forecasting-based implementation, it may leave the practitioner with the sense that simpler forecasting methods are sufficient. Indeed, we find that many firms and practitioners rely on all-purpose forecasting models like moving-average and smoothing methods. Given this fact, we added to our study the analysis of the relative performance of these common forecasting alternatives. Our results show that except for a narrow range of conditions (e.g., little correlation or strong positive correlation with certain values for k or α), the all-purpose models are at a distinct disadvantage relative to the MSE-optimal forecast methods. Interestingly, our results show that management might be best to choose no forecasting and stick with the traditional implementation of the newsvendor model. In particular, we observed that for negative correlation up through moderate positive correlation, management is best to avoid the commonly used forecasting methods

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of moving average and exponential smoothing, that is, use no forecasting methods other than simply using the overall average as a basis for order setting. One argument that might be given against the implementation of optimal modeling is that it requires statistical skills in time-series modeling, including knowledge of model identification, model estimation, and tests for model adequacy, that are beyond the skill set of a typical operations manager. However, we believe that the industrial use of more sophisticated time-series models is steadily growing because of two reasons. First, with the rise of business analytics movement, the requirement of intense statistical training, is increasingly becoming commonplace. Second, modern computational tools are readily available to make possible automated implementation of time-series modeling including the general class of ARIMA models. There are commercial programs (e.g., Forecast Pro and Autobox) that are designed to automate model identification, model fitting, and forecasting (Goodrich, 2000; Reilly, 2000). In addition, the free open source program R provides automatic ARIMA fitting (Hyndman & Khandakar, 2008). In summary, with the results of this article, operations managers can make an informed choice: a choice that pins simplicity against operational benefits associated with implementing a most accurate forecasting system.

SUPPORTING INFORMATION Additional supporting information may be found in the online version of this article at the publisher’s website: Online Appendix

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Su, X. (2008). Bounded rationality in newsvendor models. Manufacturing & Service Operations Management, 10(4), 566–589. Van Mieghem, J. A., & Rudi, N. (2002). Newsvendor networks: Inventory management and capacity investment with discretionary activities. Manufacturing & Service Operations Management, 4(4), 313–335. Wang, T., Atasu, A., & Kurtulus¸, M. (2012). A multiordering newsvendor model with dynamic forecast evolution. Manufacturing & Service Operations Management, 14(3), 472–484. Wang, J., Lau, A., & Lau, H. (2012). Practical and effective contracts for the dominant retailer of a newsvendor product with price-sensitive demand. International Journal of Production Economics, 138(1), 46–54. Layth C. Alwan is an associate professor of Supply Chain, Operations Management and Business Statistics, Sheldon B. Lubar School of Business, University of Wisconsin-Milwaukee. He received a B.A., a B.S., an M.B.A., and Ph.D., all from the University of Chicago, and an M.S. from DePaul University. His research interests include forecast-based statistical process control and stochastic modeling in supply chain systems. He is an author of several textbooks in the areas of business statistics and quality control. His work has been published in journals such as Journal of Business and Economic Statistics, Journal of Royal Statistical Society, IIE Transactions, Production and Operations Management, and European Journal of Operational Research. Minghui Xu is a professor in the School of Economics and Management at Wuhan University. He received the B.S. degree, M.S. degree and Ph.D. degree, in 1998, 2001 and 2005, respectively, all from Wuhan University. During September 2005September 2006, he conducted postdoctoral research at the Chinese University of Hong Kong. After that, he joined Wuhan University. His research interests include stochastic modeling in logistics and supply chain systems, inventory control and pricing, marketing-operations interfaces, and operations management with risk considerations. His work has been published in journals such as Operations Research, Naval Research Logistics, IIE Transactions, European Journal of Operational Research. Dong-Qing Yao is a Professor at the College of Business and Economics of Towson University, USA. His research interests are supply chain management. His papers have been published in journals such as IIE Transactions, The Journal of Operational Research Society, Annals of Operations Research, OMEGA, European Journal of Operational Research, International Journal of Production Economics, Journal of Organizational Computing and Electronic Commerce, etc. Xiaohang Yue is an associate professor of operation management at the University of Wisconsin-Milwaukee. He has published in various journals such as Production and Operations Management, Naval Research Logistics, IIE Transactions, European Journal of Operational Research, IEEE, IJPR, IJPE, ANOR, Omega, JORS, JBR.