Retail Inventory Management When Records Are Inaccurate

Retail Inventory Management When Records Are Inaccurate Nicole DeHoratius, Adam J. Mersereau, and Linus Schrage The University of Chicago Graduate School of Business 5807 South Woodlawn Ave., Chicago, IL 60637 {nicole.dehoratius, adam.mersereau, linus.schrage}@chicagogsb.edu November 10, 2005 Inventory record inaccuracy is a significant problem for retailers using automated inventory management systems. While investments in preventative and corrective measures can be effective remedies, gains can also be achieved through inventory management tools that account for record errors. In this paper, we consider intelligent inventory management tools that account for record errors using a Bayesian inventory record. We assume that excess demands are lost and unobserved, in which case sales data reveal information about physical inventory positions. We show that a probability distribution on inventory levels is a sufficient summary of past sales and replenishment observations, and that this probability distribution can be efficiently updated as observations are accumulated. We also demonstrate the use of this distribution as the basis for practical replenishment and inventory audit policies, and illustrate how the needed parameters can be estimated using data from a large national retailer. Our replenishment policies avoid the problem of freezing, in which a physical inventory position persists at zero while the corresponding record is positive. In addition, simulation studies show that our replenishment policies recoup much of the cost of inventory record inaccuracy, and that our audit policies significantly outperform the popular zero-balance walk audit policy. 1. Introduction Inventory record inaccuracy, the discrepancy between the recorded inventory quantity and the actual inventory quantity physically available on the shelf, is a substantial problem in retailing. DeHoratius and Raman (2004) found inaccuracies in 65% of the nearly 370,000 inventory records observed across thirty-seven retail stores. They identified several reasons for the

inaccuracies including replenishment errors, employee theft, customer shoplifting, improper handling of damaged merchandise, imperfect inventory audits, and incorrect recording of sales. The direct financial impact of lost merchandise is substantial and has long garnered the attention of store managers. However, retailers with inaccurate inventory records may incur additional costs due to the uncertainty around their inventory positions. First, retailers may choose to buffer the added uncertainty with additional inventory, or else lose sales due to stockouts. Second, inventory record inaccuracy may undermine decision support tools such as automated replenishment and automated demand forecasting systems that do not account for the inventory uncertainties (Raman et. al. 2001). For example, in the presence of inventory record inaccuracy, automated replenishment systems may order when ordering is unnecessary or fail to order when they should. Kang and Gershwin (2005) coin the term freezing for a scenario in which the retailer has no items on the shelf (and hence no sales) but a positive inventory record (and hence no orders), resulting in a persistent physical stockout. At Gamma 1, a publicly held retailer with annual sales of roughly ten billion dollars that uses automated replenishment systems to manage store inventory, nearly 12% of the nearly 370,000 items audited across 37 stores had no inventory on the shelf, yet the recorded inventory quantity was positive (DeHoratius and Raman 2004). There are at least three ways a retailer may respond to inventory record inaccuracy: 1. Prevention: Reduce or eliminate the root causes of inventory record inaccuracy through the implementation and execution of process improvement. 2. Correction: Identify and correct existing inventory record discrepancies through auditing policies. 3. Integration: Use inventory planning and decision tools robust enough to account for the presence of record inaccuracy. Our focus is primarily on the third category, although we emphasize that these three approaches are not exclusive. Indeed, an effective inventory management strategy is likely a three-pronged approach that attacks the problem using multiple tools. We in fact provide a rule for doing selective audits. Our goal is a practical set of tools for managing inventory when records are inaccurate. To account for record inaccuracy, we propose the maintenance of a Bayesian inventory record, 1 Name disguised upon retailer s request. 2

a probability distribution that reflects the retailer s uncertainty around the true level of physical inventory on the store shelf. This represents an alternative to the point estimate of inventory commonly used in retail practice. While not eliminating the need for management attention to prevention and correction of inventory record errors, we show that the costs of inventory record inaccuracy can be mitigated through this approach. We consider multi-period inventory management of a single item with periodic review and unobserved lost sales. When unfulfilled demands are lost and unobserved, sales data inform the retailer about the available physical inventory. Intuitively, if visible sales (a censored observation of demand) are zero in a day, then a rational retailer will realize that the lack of sales may be due to a physical stockout and adjust her beliefs accordingly. Conversely, if visible sales are positive in a day, then the retailer knows that the previous physical inventory position could not have been zero. Our approach captures this intuition through Bayesian updating of our proposed inventory record. We show that the proposed probability distribution sufficiently summarizes all relevant information available to the retailer regarding the physical inventory position, and we show that the Bayes update can be performed efficiently. We make no assumptions regarding the replenishment policy, except that the retailer perfectly observes replenishment quantities and that replenishment decisions are based only on information available to the decision-maker. Given that our proposed inventory record can be efficiently stored and maintained, it remains a challenge to develop effective replenishment and audit policies based on it. Direct optimization of these policies requires solution of dynamic programs whose state is given by a probability distribution. Such optimization problems are examples of partially observed Markov decision problems (POMDPs), and remain intractable to solve for the problem sizes and horizons required by most practical inventory management applications. Rather than optimize the POMDPs directly, we seek practical heuristic policies that are easily implemented. We show in a numerical simulation that our policies consistently outperform commonly used policies that assume accurate records. Thus, our policies are capable of recouping much of the cost of inventory record inaccuracy, without implementation of inventory tracking technologies like Radio Frequency Identification (RFID). In the interest of making our proposals practical, a second challenge is estimation of the needed parameters. We present a statistical framework for estimation of the necessary model 3

parameters, and illustrate the framework by calibrating our model based on real audit data from Gamma Corporation. The remainder of the paper is organized as follows. After a brief review of relevant literature in Section 2, in Section 3 we introduce our Bayesian inventory record, prove its sufficiency, and show how it can be updated. We then address replenishment decisions based on the Bayesian record in Section 4, and audit triggering in Section 5. To implement these policies, we present in Section 6 methodologies for estimating the necessary model parameters from available data, and we illustrate our methodologies on real retail data. Calibrating our model using these estimates, we test the performance of our ordering and audit policies through numerical simulation in Section 7. We conclude in Section 8 with discussion of some future research opportunities. 2. Related Literature and Our Contributions Empirical research has provided clear evidence of the existence of inventory record inaccuracy in a number of contexts, including government agencies (Schrady 1970 and Rinehart 1960) and utilities (Redman 1995). In the retail context that is our focus, Gentry (2005) reports a discrepancy between recorded and actual inventory amounting to $142 million, or the equivalent of 21,000 ocean containers, at The Limited, a well-known apparel retailer. DeHoratius and Raman (2004) measure inventory record inaccuracy at Gamma. Ton and Raman (2005) examine the problem of misplaced products at Borders. Both papers identify several drivers of such execution problems and discuss their impact on retail product availability. The working paper of Lee and Ozer (2005) provides a recent summary of modeling research on inventory record inaccuracy. Modeling work on the issue of record inaccuracy dates to Iglehart and Morey (1972), who derive approximations of audit policies and buffer stocks that limit the probability of physical inventory stockouts. Morey (1985) provides a back of the envelope expression for service levels in terms of buffer stock and audit frequency. Morey and Dittman (1986) analyze the problem of selecting audit frequencies to achieve desired inventory record accuracy goals. More recently, Camdereli and Swaminathan (2005) examine supply chain coordination issues when the retailer must contend with inventory record inaccuracy. Focusing on the problem within a firm, Kang and Gershwin (2005) evaluate the potential cost of record inaccuracy using a simulation study and evaluate several simple heuristic remedies. Kok 4

and Shang (2005) examine policies for triggering inventory audits in a system where excess demand is backlogged. The backlogging assumption decomposes the accumulation of inventory record discrepancy from inventory levels, and they study a dynamic programming formulation where the number of days since the last audit serves as a sufficient statistic for the distribution of record error. Lee and Ozer s (2005) survey reports on some original research into replenishment policies in which multiple sources of record inaccuracy (i.e., misplacement, shrinkage, and transaction errors) are modeled explicitly. We note that in deriving policies for the lost sales case with unobserved errors, they ultimately make a simplifying assumption that record errors grow irrespective of inventory levels, thus allowing them use a sufficient statistic for the distribution of record error similar to that used by Kok and Shang (2005). We assume a single source of inventory record inaccuracy modeled by a single, unsigned random disturbance each period. Such a model allows us to explicitly account for the interaction between inventory inaccuracy and sales observations in a lost sales environment. Our contributions are a model of a lost sales retail inventory system with record inaccuracies, the derivation of a Bayesian inventory record in such a system, and policies based on the Bayesian inventory record for both replenishment and audit triggering. Furthermore, we emphasize the practicality of our proposed methods by demonstrating the estimation of necessary parameters using data from a real retailer. Bensoussan, Cakanyildirim, and Sethi (2005) provide some technical results on Bayesian updating and replenishment in an inventory system in which the inventory manager observes only whether physical inventory levels are zero or nonzero. We note that Bayesian approaches have arisen elsewhere in the inventory management literature, most notably when the inventory manager s uncertainty is not about the inventory position but rather about some aspect of the demand distribution. We refer the reader to Lariviere and Porteus (1999), Ding et. al. (2002), Treharne and Sox (2002), and Aviv and Pazgal (2005) for recent work in this vein. 3. A Bayesian Inventory Record 3.1 Assumptions and Notation We consider management of a single SKU (Stock Keeping Unit) using periodic (assumed daily) review. We model inventory record inaccuracy by assuming an invisible demand process that 5

affects physical inventory but is not directly observable by the decision maker. Invisible demand may be positive or negative and models the combined effect of various causes of discrepancy between actual and recorded inventory. The assumed sequence of events in day t is as follows: 1. Beginning physical inventory is I t-1. 2. A replenishment of R t arrives. 3. Demand D t arrives, and sales of S t = min{ D t, I t 1 + R t } are observed. 4. Invisible demand V t arrives, yielding invisible sales U t = min{ V t, I t 1 + R t S t }. 5. Physical inventory is updated as I t = I t-1 + R t - S t - U t. 6. The decision maker places a replenishment order R t+l+1, where L is an assumed replenishment lead time in days. Both D t and V t are random variables in our model with probability mass functions d = Pr{ D t = d}, d = 0,1,, and v = Pr{ V t = v}, v =,1, 0,1,, respectively. We denote the associated cumulative distribution functions as d and v respectively. D t is assumed to be nonnegative, but V t may be positive or negative to model various sources of record inaccuracy that generate inventory record errors of either sign. For expositional and notational simplicity, we assume that D t 's and V t 's are independent and that their distributions are stationary. We revisit these assumptions in Section 3.2. The decision maker is assumed to observe only replenishments and (visible) sales, and we denote her information set at the end of day t by t = ( I 0,S 1,,S t, R 1,, R t ), reflecting inflows and outflows observed by the end of day t. The decision maker does not directly observe U t and thus does not know I t. Most inventory management systems currently maintain an inventory record that is incremented when replenishments arrive and decremented when sales are registered. We refer to this recorded inventory at the end of day t as J t, and write its dynamics as J t = J t 1 + R t S t. In contrast to the physical inventory I t, the recorded inventory J t can be maintained using the decision-maker's information state t. 6

Our goal is an implementable set of inventory management tools whose parameters can be estimated by a retailer, and thus our model necessarily represents a couple of simplifying approximations. First, we model inventory record inaccuracy through a single random process V t. In reality, there exist several causes of inaccurate inventory records, including direct losses (like theft and damage), misplacements (and recovery of lost merchandise), data entry errors, and transactional errors. Each cause of record inaccuracy may potentially affect physical and recorded inventories differently, and may require different cost accounting (Lee and Ozer 2005). We assume a single source of error for the sake of modeling simplicity but also to simplify data estimation. Upon a store audit at time t, a retailer has values I t and J t, which inform her about aggregate discrepancy but not necessarily about the various sources of error that contribute to the discrepancy. Second, we assume that invisible demand V t arrives at a single point of time in day t, after visible demand is satisfied. In reality, we might expect that visible and invisible demand arrivals would be interweaved throughout a day. Such a situation, however, is difficult to model under the lost sales assumption. Specifically, it considerably complicates the Bayesian updating of our proposed inventory record, discussed in the next section. 3.2 Maintaining a Bayesian Inventory Record The main difference between the framework of Section 3.1 and classical multiperiod periodic review inventory models is the addition of the invisible demand process. Because the information t observed by the decision maker is insufficient to update I t, the decision maker is left with uncertainty around the true inventory level I t. The information available to the decision maker nevertheless gives her information about I t. Clearly, a replenishment increments I t and a sale decrements I t. More subtly, if the decision maker observes positive sales in a period, she then knows that the previous physical inventory position could not have been zero. Conversely, no sales in a period may indicate that physical inventory is zero. We propose that the decision maker maintain as her inventory record a probability distribution to fully account for these effects. We define P t { }, ()= i Pr I t = i t the distribution of I t at time t given the information state t, and propose that P t () be used instead of J t as an inventory record for use in retail inventory management. 7

The following result implies that the distribution P t 1 () is a sufficient summary of t-1 for use in computing P t (). Proposition 1: Assume Pr{ R t = r t 1 }= Pr{ R t = r t 1, I t 1 = j} for all t, r, t-1, and j. Then Pr{ I t = i t }= Pr{ I t = i P t 1 (),S t, R t }. The proof is a special case of a result known for partially observed Markov decision problems (see Appendix A of Smallwood and Sondik 1973). In the Appendix, we provide a proof of the proposition specific to our application. The proof outlines an implementable Bayesian update procedure for computing P t () using only the previous inventory distribution P t 1 (), observed replenishment R t, and observed sales S t. In particular, P t ( 0)= v v j + R t S t j =0 P t Pr{ I t 1 = j,s t t 1, R t } Pr{ I t 1 = k,s t t 1, R t } k =0 Pr{ I t 1 = j,s t t 1, R t } ()= i j i+ Rt S t, i > 0, j =0 Pr{ I t 1 = k,s t t 1, R t } k =0 where 0 if j + R t < S t Pr{ I t 1 = j,s t t 1, R t }= P t 1 ( j) d if j + R t = S t. d S t P t 1 ( j) St if j + R t > S t The condition Pr{ R t = r t 1 }= Pr{ R t = r t 1, I t 1 = j} requires that the replenishment quantities carry no information about the physical inventory beyond what is included in t. 8

Given that replenishment orders are presumably decided by the retailer herself based on t, this is not a restrictive assumption. The use of the distribution P t () as an inventory record increases the complexity of inventory record-keeping. For each SKU, our proposal requires storage of a vector of values (assuming we truncate the distributions) rather than the single value J t. Given modern data storage capacities, however, this is hardly a drawback of our proposal. Although we have assumed for the sake of notational simplicity that the random variables D t and V t are independent and that their distributions are stationary, these assumptions can be relaxed in Proposition 1. In particular, the proposition can be trivially extended to the cases where d and v are nonstationary (as long as they are known by the decision maker) and where the mass function v of V t depends on the observed sales S t in the period. 4. Replenishment Based on a Bayesian Inventory Record Classical inventory replenishment models typically assume that inventory levels are known at the time replenishment decisions are made. In this section, we present methods for generating replenishment orders based on the Bayesian inventory record we have proposed. We assume that replenishments occur daily, and orders are filled after a fixed lead time L. By assuming that store replenishments occur daily, we do not consider fixed order costs. As many large retailers use regularly scheduled shipments to replenish many SKUs, we believe it is reasonable to ignore the impact of fixed order costs on the replenishment policy for a single SKU. 4.1 Dynamic Programming for Optimal Replenishment Policies A natural framework for deriving optimal replenishment and audit policies for inventory systems is discrete-time Markov decision problems (MDPs). MDP formulations of inventory problems typically assume physical inventory I t is known and use it as the state of a discrete-time dynamic programming formulation. Since we assume I t is unknown, our problem is naturally formulated instead as a partially observed Markov decision problem (POMDP). POMDPs are described in Smallwood and Sondik (1972) and Lovejoy (1991). The usual approach to POMDPs is to reformulate them as MDPs with state space given 9

by all possible distributions of the unknown variable. In our case, Proposition 1 implies that the probability distribution P t () is a sufficient representation of the system state at day t. Unfortunately, the resulting continuous state-space MDP is subject to the so-called curse of dimensionality. Although general-purpose algorithms exist (see Lovejoy 1991), the problem remains intractable to solve for the size and time horizons we require for practical replenishment policies. We thus look to alternate means of deriving practical policies. 4.2 Myopic Replenishment Policies Assuming accurate inventory records, no fixed ordering costs, and zero lead time, the optimal replenishment policy for a single SKU is given by a newsvendor model. It is known (see Zipkin, 2000, Section 9.4.6) that a myopic (single period) solution to this problem is also optimal over an infinite horizon. When lead times are long, optimal replenishment policies for the lost sales case are expensive to compute, and the myopic policy is commonly employed as a heuristic. In this section, we examine myopic replenishment policies for the case in which inventory is tracked using a Bayesian inventory record. Assume we are in period t and we are about to choose the order quantity R t+l+1. The first period that R t+l+1 will have any effect is period t+l+1. The inventory available to satisfy visible demand in period t+l+1 is A t +R t+l+1, where A t =I t + t< s t+l (R s S s U s ), Note that for t < s t+l: R s is known, S s and U s are random variables, I t is a random variable, and R t+l+1 is the decision variable. There will be no lost sales in period t+l+1 if A t + R t+l+1 D t+l+1. Define the distribution W(R t+l+1 ) = Pr{ A t + R t+l+1 D t+l+1 } = Pr{ D t+l+1 - A t R t+l+1 } and mass function w(r t+l+1 ) = Pr{ D t+l+1 - A t = R t+l+1 }. We take a myopic view and consider only costs incurred in t+l+1. Let h be the one period cost of having a unit left over in inventory and p the penalty per unit of unsatisfied visible demand, and assume h+p 0 and h>0. If we ignore the cost impact of the invisible demand in the current period, V t+l+1, then the myopic cost function is C 1 ( R t + L +1 )= h E max{ R t + L +1 + A t D t + L +1,0} + p E max{ D t + L +1 A t R t + L +1,0}, where the expectations are with respect to the random variables A t and D t+l+1. 10

Proposition 2: The smallest integer R t+l+1 that satisfies W(R t+l+1 ) p/(h+p) minimizes C 1 ( ). R + Proof: Note that C 1 (R) = h ( R x)w( x) + p (x R)w(x), and define the forward x= x= R difference C 1 (R) = C 1 (R + 1) C 1 (R) = h W (R) p( 1 W (R)). The differences are nondecreasing, since 2 C 1 (R) = C 1 (R + 1) C 1 (R) = ( h + p) w( R+ 1) 0. Thus, the smallest integer R such that C 1 (R) 0 minimizes C 1 (R). This is equivalent to the desired condition. Such an R must exist, since lim C 1 ( R)= h > 0. R This result is equivalent to the classical critical fractile solution to the newsvendor problem, except that the demand distribution has been replaced with by distribution W. We can compute the distribution W(R t+l+1 ) by a tedious though straightforward computation of an L-fold truncated convolution, so it is relatively straightforward to compute an optimal myopic order quantity each period. As in the classic newsvendor solution, this policy can be interpreted as targeting a particular service level dictated by the critical fractile. This is precisely what we do in the simulations described later. The current day s invisible demand V t+l+1 complicates the optimal myopic replenishment quantity calculation in two ways. First, since we assume that invisible sales occur after visible sales, the invisible sales impact the quantity of goods held overnight on day t+l+1. Second, there are direct costs (or gains) associated with certain types of inventory losses (or receipts). For instance, the retailer loses the value of any stolen or damaged merchandise. We explore the impacts of these considerations on the myopic replenishment policy in Appendix B. Myopic solutions to the replenishment problem are approximations for two reasons. First, the replenishment quantity R t+l+1 may impact future replenishment decisions and thus costs beyond time t+l+1. This effect is also present in the positive lead time, lost sales model without invisible demand. The second reason is a subtle one. The choice of replenishment quantity R t+l+1 impacts the sales observation S t+l+1 and thus the shape of the distribution P t+l+1 ( ). In this way the current decision affects the information available for future decision-making. This effect is a potentially interesting direction for future work, but we do not pursue it further here. 11

5. Audit Triggering Based on a Bayesian Inventory Record In addition to replenishment decisions, another decision available to the retailer is the scheduling of physical audits, in which the retailer reconciles her inventory record for a particular SKU with a manual count of the physical inventory on hand. In this section, we investigate policies for triggering audits based on the distribution P t (). We assume there is a fixed cost K to audit an SKU, and we assume that audits are perfect. That is, we assume that an audit sets recorded inventory J t equal to the physical inventory I t. An audit also resets the inventory distribution P t () equal to a unit pulse at the true inventory level I t. We may schedule an audit at the end of a day (after all sales and invisible sales have occurred, but before a replenishment decision is made), and the audit is assumed to occur immediately (before we choose R t+l+1 ). In addition, we assume without loss of generality that the most recent audit occurred at time 0, so that t represents the number of days since the last audit. Just as the replenishment problem can be formulated as a partially observed Markov decision problem (POMDP), so can the audit triggering problem. Nevertheless, solving the resulting continuous state space dynamic program remains intractable. We instead look to alternate means of deriving good practical policies. The audit policy we propose is a heuristic that trades off the fixed cost of an audit with the cost of uncertainty, which we model using the notion of Expected Value of Perfect Information (EVPI), e.g., see Hillier and Lieberman (2005). Denote by r(p t ) a replenishment order based on the inventory distribution P t (), so that R t+l+1 = r(p t ). Now suppose a clerk were to observe the true inventory I t and compute a replenishment order based on the policy r( ). Denote this more informed order by r I (P t ). If the clerk does not reveal I t, then from the retailer s perspective, r I (P t ) is a random variable whose probabilities are taken from P t. Recalling our notation from Section 4.2, the one period EVPI at time t is EVPI(t) = he[max[0, r (P t ) + A t - D t+l+1 ] + pe[max[0, D t+l+1 A t r(p t )] - he[max[0, r I (P t ) + A t - D t+l+1 ] - pe[max[0, D t+l+1 A t r I (P t )]. Intuitively, the EVPI gives the single period cost benefit if our uncertainty around the true inventory level were to be resolved. Notice that just as with our derivation of the myopic policy 12

in Section 4.2, we are measuring holding and penalty costs just after visible demand occurred, thus disregarding the cost effects of invisible demand in the current period. We seek an audit triggering policy that balances the tradeoff between the cost of uncertainty and the fixed cost of audits. Suppose that EVPI is known to grow according to a function v(), where is the time since the last audit. (We distinguish between EVPI(t), the EVPI observed at time t, and v(), the assumed trajectory of EVPI growth.) Then the average daily cost f(t) of a policy that audits every T days is given by f( T)= K T + 1 T T 0 v( )d To motivate a choice for v(), Figure 1 plots an EVPI trajectory, averaged over 500 simulation paths in which the simulated retailer maintains the distribution P t ( ) and bases replenishment orders off of it to target a 96% service level. Other parameters are the same as those used in our numerical results in Section 7.4. We observe that the plot is increasing and concave; the cost of uncertainty increases as time passes since the last audit, but at a decreasing rate. Figure 1: Average EVPI trajectory under Bayes replenishment policy. A simple function of this type is v()=, for some 0 < 1. We may calibrate such a v() using the EVPI(t) observed at time t by setting v(0) = EVPI(0) and v(t) = EVPI(t), yielding () v( )= EVPI t. t Given this v(), the average daily cost if the retailer audits every T days is, 13

f(t) is minimized for f( T)= K T + 1 T EVPI () t t () = K T + EVPI t t ( ) T 0 d T + 1. ( ). T * = Kt + 1 EVPI () t This leads to a heuristic auditing policy that can be simply specified. In particular, the retailer audits when t>t *, or equivalently, when t EVPI t ()> + 1 We note that the fixed cost K of an audit may be difficult to estimate in many environments. As an alternative to the rule above, a retailer might prefer to generate daily a prioritized list of SKUs to audit. Given the analysis above, we propose generating such a list by ordering SKUs based on their values t EVPI (t). 1/ +1 K. 6. Estimating the Demand Distributions We recognize that our models and policies are useful only if a retailer is able to estimate the necessary parameters. In this section, we address the estimation of both invisible and visible demand distributions, and illustrate our methods using actual retail data from Gamma Corporation. Section 6.1 presents a simple method for estimating invisible demand parameters using readily available data and techniques easily implemented by a retail manager. Section 6.2 provides an alternative estimation technique that leverages recent empirical work on the drivers of record inaccuracy. We include a brief discussion of visible demand parameter estimation in Section 6.3. 6.1 Estimating the Mean and Variance of Invisible Demand To generate an empirical estimate of the invisible demand, we use data collected from the physical audit of every item in one retail store. This store is one of more than a thousand owned and operated by Gamma. Like most retailers, Gamma conducts periodic physical audits while 14

the store is closed to customers. During these audits every item in the store is counted. Audit results are most often used for accounting purposes (e.g., to assess whether the aggregate dollar value of the inventory on-hand matches the expected value) however these counts also document Y i, the observed discrepancy between recorded and actual inventory for each item i. Upon completion of an audit, Gamma updates its inventory records to accurately reflect the physical counts just taken 2. Our objective is to use these audit data, data that Gamma has readily available, to estimate the distribution of invisible demand at the SKU level. To accomplish this, we take a two-step approach. First, we estimate the mean E[Y i ] and variance Var[Y i ] of discrepancy for an individual SKU. Second, we use the values E[Y i ] and Var[Y i ], which correspond to discrepancies accumulated over a long audit horizon, to estimate a distribution of daily invisible demand. We select a single SKU from the audit data of one Gamma store to illustrate our methodology. At the time of the audit, this SKU was stocked out in this store and had an inventory record of eight units. To obtain estimates of the mean and variance of discrepancy, we need more information than this single point estimate of discrepancy observed at the time of the audit. Given that we have observations for all SKUs in this store, we identify those SKUs considered to be closely related to our selected SKU. We do so using the merchandising hierarchy developed by Gamma. This merchandising hierarchy classifies SKUs into small clusters or classes of similar SKUs. For example, if our selected SKU were a two-pack of AA batteries, its class would be called batteries and all SKUs that could be categorized as a battery would be included in this class. Classes in this store have 27 SKUs, on average, with a standard deviation of 32 SKUs. The class for our selected item had a total of 58 SKUs. It is our belief that these 58 items within this class are likely to be more similar to each other in their discrepancy characteristics than are items across different classes, as documented in recent empirical work (Raman et al. 2001, DeHoratius and Raman 2004). Upon identifying this class of similar SKUs, we use this set of Y i s observed at the time of the audit to calculate a mean and variance of discrepancy, -1.61 and 605.06, respectively. Recognizing that the audit data collected at Gamma is actually the accumulation of daily errors since the previous audit, we need to transform the expected discrepancy and its variance at 2 See DeHoratius (2002) for additional details on this retailer, the audit process, and the data collection methods employed. 15

the time of the audit into a measure of the daily mean and variance of invisible demand. For the purposes of parameter estimation, we implicitly ignore the fact that the discrepancy is Y i is really the accumulation of censored invisible demand realizations, and instead assume that invisible sales are equal to invisible demands. We model the daily invisible demand impacting item i as the difference of two Poisson random variables with means i and μ i,. Recalling that the variance of a Poisson random variable equals its mean and given the last audit was days ago, then the method of moments estimators of i and μ i solve: ( i μ i ) = E[Y i ], and ( i + μ i ) = Var[Y i ]. The solution is i = (E[Y i ] + Var[Y i ])/(2); μ i = i - E[Y i ]/; There is a sensible solution, i.e., i, μ i 0, if Var[Y i ] E[Y i ]. For the particular store we are using as our example, there were =523 days since the last audit. We find a solution i = 0.58 and μ i = 0.58 for our particular SKU in the data set. We use these parameters in the numerical simulations described in Section 7. 6.2. Alternative Estimation Methods An alternative estimation method to the one described above capitalizes upon recent empirical work on record inaccuracy. This work documents the likelihood of record inaccuracy (Raman et al. 2001), its magnitude (DeHoratius and Raman 2004), and several product characteristics associated with these measures. The advantage of using such models to estimate the mean and variance of Y i is that they draw upon the information provided from every observation in a given store. If we define E[Y i ] = q 1 *E[Y i Y i > 0]+q -1 *E[Y i Y i <0] where q 1 = Pr{Y i > 0} = Pr{Y i > 0 Y i 0}(1-q 0 ), q -1 = Pr{Y i < 0} = Pr{Y i < 0 Y i 0}(1-q 0 ), and 16

q 0 = Pr{Y i = 0} we can then use existing models to estimate each parameter noted above with the exception of Pr{Y i > 0} and Pr{Y i < 0}. Our overall statistical model of Y i is represented in Figure 2. Specifically, Raman et al. (2001) identify several drivers of the likelihood of record inaccuracy occurrence including item cost, annual selling quantity, and product category. Using their logistic regression model, we estimate the probability of record accuracy, q 0, for all items in our selected store including our chosen SKU (Figure 2, A). DeHoratius and Raman (2004) present several factors correlated with the magnitude of record inaccuracy including item cost and annually selling quantity. Again, we apply a similar regression model to estimate the conditional expected magnitude of discrepancy, E[Y i Y i > 0] and E[Y i Y i <0] (Figure 2, C and D), for our selected SKU. Figure 2: Data Model of Discrepancy To our knowledge, there is no empirical evidence identifying the determinants of positive, Pr{Y i > 0}, or negative discrepancies, Pr{Y i < 0}. In the data available to us, factors such as item cost and annual selling quantity were not correlated with the sign of a discrepancy. In the absence of such evidence, we estimate the likelihood of a record having a negative or a positive discrepancy using a simple Bernoulli probability model (Figure 2, B). Specifically, we 17

calculate the percentage of records with positive (negative) discrepancies in the specific store and product category of interest. Using this simple probability and the output from the series of regression models available for this approach, we derive an estimate of E[Y i ] of 0.53 for our select SKU. In addition to an estimate of E[Y i ], we also need an estimate of Var[Y i ]. To estimate the variance of discrepancy, Var[Y i ], we use the following formula derived from the theorem of conditional variance. Var[Y i ] = q 1 *Var[Y i Y i >0]+q -1 *Var[Y i Y i <0]*q 1 *E[Y i Y i >0] 2 + q -1 *E[Y i Y i < 0] 2 - {q 1 *E[Y i Y i > 0]+q -1 *E[Y i Y i < 0]} 2 Here we need to estimate two new parameters, namely Var[Y i Y i >0] and Var[Y i Y i <0]. To do so, we use a method similar to that described in Section 6.1. We isolate those items within the same product class as our selected SKU. We then categorize each item in this product class as accurate or inaccurate. For those that are inaccurate, we estimate the average squared residual with respect to the fitted value of our select item separately for items with positive and negative discrepancies. We then derive an estimate of Var[Y i ] of 799. As noted in Section 6.1, the number of days since the last audit was 523 resulting in a solution for i of 0.77 and μ i of 0.76 for our selected SKU i in the data set, which differ somewhat from the results found in Section 6.1. Although we ran numerical results under both sets of assumptions and found the results to be qualitatively similar in both cases, we chose to present in Section 7 the results using the parameters estimated in Section 6.1. We provide a sensitivity analysis in Section 7.2 demonstrating the robustness of our model to uncertainty in the estimated parameters. Note that sensitivity run B in that section uses parameters similar to those estimated here. The estimation methods we employ rely on audit data Gamma collected on a regular but infrequent basis. However, retail decision makers could potentially put processes in place to collect data to better estimate the daily invisible demand process. One such process is cycle counting. During the cycle counting process, inventory counts are taken for a select number of items and records are adjusted should any discrepancies exist. Because these adjustments can be scheduled more frequently than the annual physical audit, the data collected during this process 18

can provide retailers with a better understanding of the daily invisible demand process. Moreover, it is plausible to have store employees count inventory daily for a limited sample of items so as to better understand the true daily invisible demand processes. 6.3 Estimating the Demand Distribution Our Bayesian adjustment process assumes we know the distribution of daily demand, so we need a way of estimating this distribution. Unfortunately, as is the case with many inventory management systems, information on stock-out events is not available. In newspaper sales data used by Yang and Schrage (2005), where stock-out data were available, it was found that the simple mean of the daily sales was about 1% to 6% lower than the estimated mean demand if stock-outs were properly taken into account in the estimation process. Agrawal and Smith (1996), Nahmias and Smith (1994), and Nahmias (1994) describe how to estimate demand in the face of stockouts. Nevertheless, we treat the sales data as demand data. We illustrate demand distribution estimation with sales of two different SKUs at Gamma. We have data on quantity sold for each SKU in the Gamma store of interest. The mean units sold per day is 3.022 for SKU 1 and 2.051 for SKU 2. The obvious distribution to consider for demand data is the Poisson distribution, in which case the variance should equal the mean. The variances, however, for SKUs 1 and 2, are 9.795 and 3.912 respectively. The most noticeable defect of the Poisson distribution for these data was that the number of days with zero sales is underpredicted, which is of particular concern in our Bayesian updating procedure. The actual number of days with zero sales is 66 and 35 respectively, whereas the simple Poisson predicts only 17.6 and 22.5 days with zero sales. We compare several alternative distributions for these datasets below: Distribution Predicted Days of Zero Demand Prob{Chi-Statistic is this large} SKU 1 SKU 2 SKU 1 SKU 2 Simple Poisson 17.6 22.5 <0.001 <0.001 Spiked Poisson 155.2 59.9 <0.001 <0.001 Compound Poisson/G 87.1 42.7 0.03 0.01 Negative Binomial 74.2 40.6 0.46 0.03 Table 1: Fitting of Visible Demand Distributions 19

The Spiked Poisson has two parameters. With probability p 1 the demand is chosen to be zero, and with probability 1-p 1 the demand is chosen from a Poisson distribution with mean μ 1. For the Compound Poisson/G distribution, the number of orders per day is chosen from a Poisson distribution with parameter μ 2, while the number of units demanded in each order is chosen from a geometric distribution with parameter p 2. A negative binomial random variable is a sum of k 3 geometric random variables, each with mean (1-p 3 )/p 3. Agrawal and Smith (1996) point out that the negative binomial distribution can also be viewed as a Compound Poisson distribution in which batches arrive in a Poisson stream at rate k 3 *ln(p 3 ), and the batch size follows a Logarithmic distribution, see Zipkin (2000), with mean (1-p 3 )/[-p 3 *ln(p 3 )]. In all cases but the simple Poisson, the two distribution parameters were chosen so as to match the mean and variance of the actual distribution. We measured appropriateness of each distribution by computing a Chi-square goodness of fit statistic and present its associated probability in Table 1. By this measure, the negative binomial gives the best fit for both SKU s as indicated by its higher probabilities in Table 1. In addition, the fraction of days with zero demand is best predicted by the negative binomial. The negative binomial distribution corresponding to SKU 2 is the one used later in the simulations in Section 7.2. 7. Evaluating Performance In this section we discuss the performance of our proposed methods, first through a theoretical result that the Bayesian inventory record avoids freezing, and then through numerical simulations. 7.1 Freezing We define freezing as a state in which an inventory system places no orders to replenish a SKU even though there is zero physical inventory. The term was coined by Kang and Gershwin (2005), who show through numerical simulation that freezing can be a costly result of record inaccuracy. In addition, we have heard anecdotal concerns about such situations from retail managers. Freezing occurs when the inventory system registers a recorded inventory large enough to prevent replenishment, even though there is no physical inventory and thus no sales to trigger changes in the recorded inventory. For any reasonable replenishment policy, we note that a 20

frozen state can persist only when there are no negative invisible demands (that is, no invisible receipts) when the physical inventory is zero. If invisible receipts are possible, such receipts will allow sales, triggering a replenishment order. However, freezing remains a possibility when there is only nonnegative invisible demand. The following proposition indicates that any reasonable policy based on the probability distribution P t ( ) will avoid a persistent frozen state. This result holds whether or not the decision maker has accurate estimates of the invisible demand process parameters, as long as the assumed invisible demand process is nonnegative and not always zero. Proposition 3: Suppose v =0 for v<0 (no invisible receipts), 0 <1, and either v has finite support or P t ( ) has finite support for all t. Then the system may not persist in a frozen state under any replenishment policy that replenishes when E[I t ]< for any >0. Proof: Our proof supposes we are in a persistent frozen state and demonstrates a contradiction. Suppose the system is frozen from time onward, so we have S t =0 and R t =0 for t >. For t> and j>0 and based on the updating procedure developed in Section 3.2, we have Pr{ I t-1 =j, S t t-1, R t } = P t-1 (j) 0, thus For i>0 we have Pr{ I t 1 = j t 1,S t, R t }= P t 1 ( j) 0 P t 1 ( 0)+ 0 1 P t 1 0 ( ( )). P t 0 ()= i P t 1 ( 0)+ 0 1 P t 1 0 ( ( )) 0P t 1 ()+ i j i P t 1 ( j) j >i Since 0 P t-1 (0) P t-1 (0), we then have 0 + 0 P t-1 (0) 0 + P t-1 (0) which in turn implies 0 0 (1- P t-1 (0)) + P t-1 (0). Thus the first term in the expression for P t (i) is no greater than 1, yielding the inequality P t (i) P t-1 (i) 0 + t-1 (i) where t-1 (i) = j>i j-i P t-1 (j).. We use induction to show that P t (i)0 as t for all i>0. First, t-1( ) has finite support, thus for some I sufficiently large we have ()= i 0 for ii. Thus P t ( I ) P t 1 ( I ) 0 and P t t ( I ) 0 for any t>+1. Second, observe that for any i>0, if there exists a T such that for 21

t 1 t>t, t 1 ()< i c T 0 t for some constant c T, then P t () i ( 1 + c T ) 0 for all t>t. Thus an inductive argument on I, I-1, I-2,, 1 gives us that for all i > 0, P t (i)0 as t. Then E[I t ]0 for t. Our policy would eventually place an order when E[I t ]<, thus contradicting that the system persists in a frozen state. This proposition offers a retailer a reliable method of avoiding freezing, namely using the distribution P t (), updated using a nonnegative invisible demand process, as the basis of replenishment decisions. 7.2 Replenishment Policies: A Simulation Study We test the performance of an ordering policy based on our proposed Bayesian inventory record using a simulation study. Following an initial audit, we simulate demands, invisible demands, and replenishments for 90 days using three classes of replenishment policies. For the purpose of the simulation, replenishments are assumed to be placed daily and arrive after a lead time of 0 days. Full: We apply a base stock policy as if the retailer knows at all times the true inventory I t. The base stock level depends on a target service level. This class of policies are as if the retailer audits her inventory position on a daily basis, or as if the retailer installs an ideal RFID technology that gives her perfect visibility of the inventory process. The Full policies are included as benchmarks. Bayes: We manage inventory using inventory distribution P t (). We place an order at time t so as to target a particular service level, taking into account both inventory and visible demand uncertainties. This replenishment policy is discussed in Section 4.2. Naive: We include base stock replenishment policies based on the recorded inventory J t. Base stock levels are chosen corresponding to a variety of target service levels. This policy neglects the effects of inventory record inaccuracy and reflects common industry practice. 22

For the purpose of the simulation, we assume daily demands D t are independent negative binomial random variables with parameters 2.27 and 0.53 (implying a mean of 2.05), and daily invisible demand V t is modeled as the difference of two Poisson random variables with means =0.58 and μ=0.58. (We drop the subscripts on and μ for the sake of simplicity.) Parameters were fit using the methods described in Section 6 for a particular SKU in Gamma s data set. Each simulation is of 500 sample paths. We present two plots of the simulation results. For several target service levels 99%, 98%, 96%, 92%, and 86% Figure 3 plots average actual physical inventory held versus achieved fill rate for each of the three replenishment policies. The figure illustrates the classic tradeoff between inventory stock and fill rate, showing efficient frontiers of tradeoffs achievable by the various replenishment methods. The figure shows that the Bayes replenishment policies achieve a tradeoff between inventory stock and fill rate that dominates the performance achievable using the Naïve policies. That is, Bayes policies are capable of achieving higher fill rates than Naïve policies while holding less inventory. The inventory savings can be substantial. To achieve a fill rate of 93%, for example, the Bayes method achieves 20% less holding cost than the Naïve method on average over the 90 day horizon. The class of Full policies outperforms both the Bayes and Naïve classes of replenishment policies, indicating that even if a retailer accounts for inventory record inaccuracy by maintaining a distribution of inventory, the retailer still has incentive to reduce record inaccuracy through preventative and corrective measures. Figure 3: Average (actual) inventory versus observed fill rate. 23

For each of the three classes of policies, we select points in Figure 3 giving similar fill rates, and we plot the average dynamics of inventory growth over time in Figure 4. Since the Full policy is a base stock policy based on the true physical inventory position, it maintains a steady inventory level over time. The Bayes and Naïve policies require physical inventory growth over time to buffer the added uncertainty and maintain service quality, but the inventory growth under the Naive policy occurs at a faster rate. While the physical inventory under the Naïve policy appears to grow in a linear fashion, the Bayes policy s inventory trajectory appears concave, indicating that inventory growth under the Bayes policy diminishes over time. Intuitively, we can think of the Bayes retailer gradually approaching a state in which P t ( ) is very dispersed, and the retail manager has little useful information about the true inventory position. Naïve (observed FR=0.929) Bayes (observed FR=0.932) Full (observed FR=0.942) Figure 4: Average (actual) inventory accumulation over time. The simulation results presented in Figures 3 and 4 model a specific SKU from the Gamma data set for which the mean invisible demand is close to zero, though there is substantial variance in the invisible demand process. In addition, we perform simulations in which we assume different invisible demand parameters. We assume the same negative binomial visible demand distribution as before, and use two alternative invisible demand parameter sets: one in which invisible demands are more likely to be positive (=0.58 and μ=0.41), and one in which invisible demands are likely to be negative (=0.40, μ=0.58). In Figure 5, we show the efficient frontiers for these simulation runs. 24

Figure 5: Plots of observed fill rate versus physical inventory held for the three classes of replenishment policies. The two cases include one in which invisible demands tend to be positive (left: =0.58, μ=0.41) and one in which invisible demands tend to be negative (right: =0.40, μ=0.58). For the Full and Naïve replenishment policy classes, both plots in Figure 5 appear similar to the plot in Figure 3, illustrating that the Bayes replenishment policies are robust to changes in the invisible demand distribution. We observe that the class of Full information policies outperforms the class of Bayes policies which in turn outperforms the Naïve policies, the same ordering as in Figure 3. In addition, the tradeoff between fill rate and physical inventory achieved by the Naïve replenishment policies appears to be quite sensitive to the invisible demand process parameters. In Section 6 we discuss methods for estimating the invisible demand parameters from data. A retailer would likely have uncertainty around any such estimates. Here we examine the sensitivity of the replenishment policies to imperfect invisible demand distribution parameter estimates. We simulate the system using the same invisible demand parameters as previously, but for the purpose of generating replenishments under the Bayes policy, we use the following parameter estimates. The assumed parameters are obtained by adjusting the and μ up and down by 30%. 25

Simulation run True True μ Assumed Assumed μ Base case (Fig. 3) 0.58 0.58 0.58 0.58 Sensitivity run A 0.58 0.58 0.40 0.41 Sensitivity run B 0.58 0.58 0.75 0.75 Sensitivity run C 0.58 0.58 0.58 0.41 Sensitivity run D 0.58 0.58 0.40 0.58 Table 2: Sensitivity Analysis Parameters As previously, all results represent 500 sample paths and a simulation horizon of 90 days. The visible demand process is as described in Section 6. Analogous to the plot in Figure 3 for the Base case set of parameters, Figure 6 shows the efficient frontiers of tradeoffs between fill rate and physical inventory held for each of the sensitivity runs. Figure 6: Plots of physical inventory held versus observed fill rates for four sensitivity runs, in which the assumed invisible demand distributions do not match the true invisible demand distribution. True =0.58, μ=0.58. Top left (A): assumed =0.40, μ=0.41. Top right (B): assumed =0.75, μ=0.75. Bottom left (C): assumed =0.58, μ=0.41. Bottom right (D): assumed =0.40, μ=0.58. 26

We observe that the plots are qualitatively similar to Figure 3. Inaccurate estimates of invisible demand variability seem to have little effect on the performance of the Bayes replenishment policies, although the tradeoff achieved by the Bayes policies does seem to be affected by bias in the invisible demand parameter estimates. In particular, when the true invisible process has less invisible demand than the retailer believes (as in run C), the Bayes policies lead to higher inventories and higher fill rates, as the retailer orders more than is necessary given the true invisible demand process. Conversely, when the true invisible process has more invisible demand than the retailer believes (as in run D), then the Bayes policies lead to lower inventories and lower fill rates. The class of Bayes policies still outperforms the class of Naïve policies for all choices of parameters. 7.3 Audit Triggering Polices: A Simulation Study We also use simulation to evaluate the audit triggering policy described in Section 5. For purposes of comparison, we simulate four regimes of implementable replenishment and audit policies. In particular, we consider the Naïve replenishment policy with no auditing and with zero-balance walks, in which SKUs are audited when their post-replenishment physical inventory levels are zero. Zero-balance walk programs are in place at a number of prominent retailers as an audit mechanism that is simple to implement and intuitively appealing. In combination with the Bayes replenishment policies, we implement no auditing and the EVPIbased auditing policy discussed in Section 5 with = 0.33. For each of the four inventory management regimes, we compute the inventory management cost incurred, including penalty costs from stockouts (assumed $0.75 per item of lost sales), holding costs (assumed $0.03 per item per day), and audit costs. To illustrate dependence of the policies on fixed audit cost, we assume five choices for the audit cost K required to reconcile the inventory record for a single SKU: $0.25, $0.50, $0.75, $1.10, $1.50. Figure 7 plots the average inventory management cost per day for each regime as a function of audit cost K. In order to capture the long run effects of audits on the overall inventory management cost, all points in Figure 5 were generated over a 360-day horizon, averaged over 500 sample paths. We observe that the Bayes replenishment policy combined with the EVPIbased audit policy achieves by far the lowest inventory management cost of the four regimes, although its cost increases with the fixed audit cost K. The simulation results also show the 27

Bayes replenishment policy with no auditing outperforming the Naïve replenishment policy with zero-balance walks. For the parameters used, intelligent replenishment that recognizes the existence of errors is preferable to a common auditing heuristic with a replenishment policy that ignores errors. 0.5 Avg daily inv mgmt cost (lost sales+holding+audits) 0.45 0.4 0.35 0.3 0.25 naive, none naive, zbw bayes, none bayes, evpi 0.2 0 0.5 1 1.5 2 Cost K of a single audit ($) Figure 7: Plot of average daily inventory management costs as a function of the fixed cost K of a SKU audit for four different inventory management regimes. 8. Concluding Remarks The fundamental conclusion of our work is that a Bayesian inventory management record that accounts for potential sources of record error is a viable alternative to the traditional point estimate inventory record frequently used in inventory management models and in practice. Such a Bayesian record can be efficiently stored and maintained, and we have demonstrated good replenishment and audit policies based on this proposed record. Furthermore, we have demonstrated that the required parameters can be estimated from existing data sources. A few algorithmic and modeling questions remain. First, our work has focused on the single SKU problem, and thus implicitly ignores demand substitution effects that can complicate Bayesian record updating. Second, our work leaves open the potential for algorithmic work on the specific partially observed Markov decision problems (POMDPs) of determining optimal replenishment and audit policies based on our proposed inventory record. Third, we believe that there is room for more extensive data collection by retailers that could potentially enable improved parameter estimation and improved models of the discrepancy process. 28

Finally, we stress that our proposal does not remove the retailer s incentive to prevent and correct the root causes of inventory record inaccuracy. We therefore recommend that future work continue to examine ways to prevent record inaccuracy, whether through technological solutions such as RFID, correction mechanisms such as inventory audits, or process design improvements. 29

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Appendix A: Proof of Proposition 1 Conditioning on I t-1, P t ( I t = i t )= Pr{ I t = i t, I t 1 = j} Pr I t 1 = j t j =0 { } = Pr{ I t = i t 1, R t,s t, I t 1 = j} Pr I t 1 = j t 1,S t, R t j =0 { } { } = Pr{ U t = j i + R t S t I t 1 = j, R t,s t } Pr I = j,s t 1 t t 1, R t j =0 Pr{ S t t 1, R t } The first probability in the square bracket can be computed directly from the invisible demand distribution. In particular, if i=0, then Pr{ U t = j i + R t S t I t 1 = j, R t,s t }= v, v j i+ R t S t if i>0, then Pr{ U t = j i + R t S t I t 1 = j, R t,s t }= j i+ Rt S t. The term Pr{ I t 1 = j,s t t 1, R t } can be computed as, if j+r t < S t, then Pr{ I t 1 = j,s t t 1, R t }= 0, if j+r t = S t, then Pr{ I t 1 = j,s t t 1, R t }= Pr{ I t 1 = j t 1, R t } d = P t 1 ( j) d S d t, d S t if j+r t > S t, then Pr{ I t 1 = j,s t t 1, R t }= Pr{ I t 1 = j t 1, R t } St = P t 1 ( j) St. where we use the fact that Pr{ I t 1 = j t 1, R t }= P t 1 ( j), which follows from our assumption on the replenishment policy. Specifically, Pr{ I t 1 = j t 1, R t }= Finally, the term Pr S t t 1, R t = = Pr{ R t t 1, I t 1 = j} Pr{ I t 1 = j t 1 } Pr{ R t t 1, I t 1 = k} Pr I t 1 = k t 1 Pr{ R t t 1 } Pr{ I t 1 = j t 1 } Pr{ R t t 1 } Pr{ I t 1 = k t 1 } k k k { } { } Pr I t 1 = j t 1 Pr I t 1 = k t 1 = P t 1 ( j). { } is equal to Pr{ I t 1 = j,s t t 1, R t } j =0 { }. Thus, P t () depends 32

only on t through P t 1 (), R t, and S t. Appendix B: Myopic Replenishment Accounting for Impact of V t+l+1 As mentioned in Section 4.2, the invisible demand V t+l+1 impacts the myopic replenishment policy in two ways: by altering the holding costs following day t+l+1 (since merchandise lost need not be held and merchandise gained must be held), and through the potential of the replenishment policy to impact direct losses. Including the first effect results in a replenishment problem with a convex cost function, and the optimal replenishment policy is nearly as easy to identify as under the assumptions of Proposition 2. In particular, define C 2 (R t+l+1 ) = E[C 2 (R t+l+1 V t+l+1 )], where the expectation is with respect to the random variable V t+l+1 and C 2 (R t+l+1 V t+l+1 ) is defined as C 2 ( RV t + L +1 )= h E max{ R + A t D t + L +1 V t + L +1,0} { } + p E max D t + L +1 A t R,0, for V t + L +1 > 0 h { V t + L +1 + E max{ R + A t D t + L +1,0} }+ p E max{ D t + L +1 A t R,0}, for V t + L +1 0. Proposition B1: The smallest non-negative integer R t+l+1 that satisfies minimizes C 2 ( ). h(1-0 ) E[W(R t+l+1 -V t+l+1 ) V t+l+1 >0] + h 0 E[W(R t+l+1 )] p[1-w(r t+l+1 )] 0 Proof: Note that for V t+l+1 > 0, and for V t+l+1 0, RV t+l+1 + C 2 ( RV t + L +1 )= h ( R V t + L +1 x)w( x) + p ( x R)w( x), x= R + C 2 ( RV t + L +1 )= hv t + L +1 + h ( R x) w( x)+ p ( x R) w( x). The conditional forward differences are then C 2 ( RV t + L +1 )= hw ( R V t + L +1) p 1 W( R) for V t + L +1 > 0 hw ( R) p 1 W( R) for V 0. t + L +1 Taking expectations gives the unconditional expected forward differences x= x= R x= R 33

C 2 (R) = h(1-0 ) E[W(R-V t+l+1 ) V t+l+1 >0] + h 0 E[W(R)] p[1-w(r)], which are nondecreasing since 2 C 2 (R) = h(1-0 ) E[w(R-V t+l+1 +1) V t+l+1 >0] + h 0 E[w(R+1)] + pw(r+1) 0. Thus, the smallest nonnegative integer R such that C 2 (R) 0 minimizes C 2 (R). Note that such an R must exist, since lim C 2 (R) = h 1 0 R ( )+ h 0 = h > 0. In order to incorporate the direct costs and gains from invisible demand, assume q 0 is the direct cost from an invisible sale (and assume q is the direct gain from a unit of invisible receipt). Including the direct gains and losses from invisible demand in the myopic replenishment policy requires the minimization of the cost function C 3 (R t+l+1) = C 2 (R t+l+1) + E[ d(r t+l+1 V t+l+1) ], where { } d( RV t + L +1 )= q E min V,max{A t + L +1 t D t + L +1 + R,0}, for V t + L +1 > 0 q E V t + L +1 V t + L +1 0, for V t + L +1 0 RV = q t+l+1 1 R V w( x) q ( R x)w( x), for V t + L +1 > 0 x= x= RV t+l+1 q E V t + L +1 V t + L +1 0, for V t + L +1 0 The term E[ d(r t+l+1 V t+l+1) ] does not necessarily exhibit increasing differences, so we do not necessarily get a base-stock myopic policy of the form of Propositions 2 and B1. However, E[d(R t+l+1 V t+l+1)] is an nondecreasing function of R t+l+1, since E[d(R t+l+1 V t+l+1)] = q(1-0 ) (E[W(R t+l+1) - W(R t+l+1 - V t+l+1 ) V t+l+1 > 0]) 0. Proposition B2: Suppose R t+l+1 minimizes C 3 ( ) and Q t+l+1 minimizes C 2 ( ). Then R t+l+1 Q t+l+1 unless C 3 (R t+l+1 ) = C 2 (Q t+l+1 ). Proof: The result follows directly from the fact that E[d(R t+l+1 V t+l+1 )] is a nondecreasing function of R t+l+1. When C 2 ( ) and C 3 ( ) have unique minima, Proposition B2 says that the optimal replenishment quantity not accounting for direct invisible demand costs is no smaller than the optimal replenishment quantity if we do account for direct invisible demand costs. Proposition B2 34

formalizes the intuition that a manager may choose to stock fewer items so that there are fewer items lost to invisible demand (theft, demand, etc.). This incentive exists in our model due to the assumption that invisible demand in a day is capped by the number of units on hand. 35