Managing Perishable Inventories in Retailing: Replenishment, Clearance Sales, and Segregation

Managing Perishable Inventories in Retailing: Replenishment, Clearance Sales, and Segregation Qing Li Peiwen Yu Xiaoli Wu March 22, 2016 Abstract We study joint replenishment and clearance sales of perishable goods under a general finite lifetime and a last-in-first-out (LIFO) issuing rule, a problem common in retailing. We show that the optimal policies can be characterized by two thresholds for each age group of inventory: a lower one and a higher one. For an age group of inventory with a remaining lifetime of two periods or longer, if its inventory level is below its lower threshold, then there is no clearance sales; if it is above its higher threshold, then it will be cleared down to the higher threshold. The optimal policy for the age group of inventory with a one-period remaining lifetime is different. Clearance sales may occur if its inventory level is above its higher threshold or below its lower threshold. The phenomenon that a clearance sale happens when the inventory is low is driven by the need to segregate the newest inventory from the oldest inventory and is unique to the LIFO issuing rule. The optimal policy requires a full inventory record of every age group and its computation is challenging. We consider two myopic heuristics that require only partial information. The first requires only the information about the total inventory and the second requires the information about the total inventory as well as the information about the inventory with a one-period remaining lifetime. Our numerical studies show that the second outperforms the first significantly and its performance is consistently very close to that of the optimal policy. 1 Introduction Many retail products, such as food items, pharmaceuticals, cut flowers, etc., have a short lifetime. Not only do these products generate a substantial amount of revenue themselves, School of Business and Management, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR. E-mail: imqli@ust.hk. School of Management, Fudan University, Shanghai, PR China. E-mail: ypw@fudan.edu.cn School of Business Administration, South China University of Technology, Guangzhou, PR China. E-mail: pearvoo@hotmail.com 1

they also drive store traffic. Indeed, empirical studies have consistently shown that customers choice of stores is influenced heavily by what perishable goods are on their shelves (Tsiros and Heilman 2005 and the literature therein). Despite its importance, the management of perishable goods is challenging. It requires proper handling and storage and the use of the right technology throughout the entire supply chain. The biggest challenge, however, perhaps stems from matching perishable supply with uncertain demand. Retailers sometimes run out of stock, which leads to revenue loss, and they sometimes have to throw away items approaching their expiration dates, which is a waste and causes environmental concerns. The amount of perishable goods thrown away by retailers is alarmingly high and has come under continuous public scrutiny in recently years (Bloom 2010, Stuart 2009). According to a recent study by Friends of the Earth, the four main supermarkets in Hong Kong throw away 87 tons of foods per day, most of which end up in landfills (Wei 2012). Therefore, better matching of supply and demand is critical to not only profit but also the environment. To reduce the mismatch, many retailers have used clearance sales as a strategy to sell items approaching their expiration dates at a reduced price. To effectively use clearance sales for reducing the mismatch, it is important for retailers to determine the right timing and depth of sales and to coordinate clearance sales with replenishment decisions. These, unfortunately, are nontrivial, and to our best knowledge, there are simply no rigorous guidance or decision support tools available. Most retailers rely on crude and untested rules of thumb (e.g., a two bin strategy under which anything that will expire in three days or less is moved to a clearance sales bin) and there is typically no coordination between clearance sales and replenishment. In this study, we consider a retailer who replenishes its perishable inventory periodically and needs to decide when and to what extent it should clear its inventory through clearance sales. Our objective is to build a mathematical model and to find the optimal strategy for both replenishment and clearance sales. In retailing, consumers observe expiration dates and control depletion. Therefore, fresher items are typically sold first on a last-in-first-out (LIFO) basis (e.g., Nahmias 1982, Cohen and Pekelman 1978). The problem we consider here is technically challenging because a) it has a multi-dimensional state space, and b) inventory systems under 2

LIFO are known to lack common technical properties needed for analysis. We show that clearance sales should always start with the oldest items. There is a threshold in age such that any items older than the threshold should be cleared and any items newer than the threshold should be carried over to the next period. Ordering of new items and clearance sales of items with a remaining lifetime of two periods or longer do not happen in the same period. These results have been shown under the first-in-first-out (FIFO) issuing rule. We show that they continue to hold true under the LIFO rule. Under the FIFO rule, it has been shown in the literature that for a given inventory of a certain age, the optimal policy on clearance sales has a clear-down-to structure; that is, there is a clear-down-to level such that a clearance sale will take place if and only if the inventory level is above that level and the clearance sale always reduces the inventory to that level. This is no longer true under the LIFO rule. In stark contrast, under the LIFO rule, there are two thresholds for each age group of inventory: a lower one and a higher one. For an age group with a remaining lifetime of two periods or longer, if its inventory level is below its lower threshold, then there is no clearance sales; if it is above its higher threshold, then it will be cleared down to the higher threshold. The optimal policy for the age group with a remaining lifetime of one period is very different, however. Clearance sales may take place if its inventory level is above its higher threshold or below its lower threshold. The lower the initial inventory level, the more new supply is needed in order to meet the demand in the current period. However, the more new supply there is, the less likely the oldest items will be used to meet demand because customers always retrieve the newest items first. The retailer is therefore better off clearing the small number of oldest items to recoup some revenue and to avoid outdating. We call this phenomenon segregation and it is unique to the inventory systems under LIFO. The optimal policy requires that we keep full record of inventory in each age group. We consider two myopic heuristics that require only partial information. The first heuristic requires only the information about the total inventory and it may generate as much as 7% less expected profit than the optimal policy. The second heuristic requires the information about the total inventory as well as the inventory with a one-period remaining life. It outperforms the first 3

heuristic significantly and its profit is consistently very close to the optimal profit. Our analysis demonstrates that the inventory with a one-period remaining lifetime (i.e., the oldest inventory) plays a qualitatively different role than inventories of other age groups. First, the optimal order quantity is monotone in the oldest inventory, but it is not necessarily so in other inventories. Second, the optimal policy on clearance sales with respect to the oldest inventory is to clear all, not clear, and then clear down to a certain level when the inventory increases. The optimal policies with respect to other inventories, however, are different. In particular, clearance sales won t happen when the inventories are low enough. Finally, it is critically important to keep record of the oldest inventory, and the performance of myopic heuristics that take advantage of that record can be consistently close to that of optimal policy. However, the value of keeping record of other inventories is insignificant. There is typically no age information in bar codes used in retailing. Including full age information in bar codes, which is needed for the optimal policy, is an industry-wide initiative and requires heavy investment and a lot of coordination. An easier but equally effective approach is to obtain the information about the oldest inventory at the store level and use that information in designing myopic heuristics. The remainder of this paper is organized as follows. We review the related literature in Section 2. We formulate the model and present some basic properties of the value function in Section 3. In Section 4, the optimal policies for replenishment and clearance sales are characterized. In Section 5, we propose two simple heuristics and compare their effectiveness in computational experiments. We conclude the paper in Section 6. The proofs that are not given in the paper can be found in the online appendix. 2 Literature Review Inventory management of perishable goods has a long history in the operations literature. Nahmias (1982) provided a review of early work. Recent reviews by Karaesmen et al. (2008) and Nahmias (2011) indicate a considerable renewed interest in the area. 4

Our work is most related to the literature that considers the LIFO issuing rule under stochastic demand. Despite the applicability of the models under LIFO, not very much research has been conducted, perhaps because of the technical difficulty. Cohen and Pekelman (1978) analyzed the evolution over time of the age distribution of inventory. Under two particular order policies-constant order quantity and fixed critical number, they determined the shortages and the number of outdated items in each period by the age distribution and related them to inventory decisions. Pierskalla and Roach (1972) and Deniz et al. (2010) considered issuing endogenously and the set of feasible issuing rules includes LIFO. The former showed that under most of the objectives, FIFO is the optimal issuing rule. The latter focused on finding heuristics to coordinate replenishment and issuing. Parlar et al. (2008) and Cohen and Pekelman (1979) compared FIFO issuance with LIFO. Motivated by blood banking and food management, Prastacos (1979) considered a two-echelon model where goods are produced at a central facility and subsequently shipped to regional centers to fulfill demand. When items are issued according to a LIFO policy, Prastacos (1979) suggested an allocation procedure, which he called the segregating policy, under which older items are sent to some locations and newer items to others. But none of the above-mentioned researchers has considered the optimal inventory ordering policy under LIFO; nor have they included clearance sales. Our paper is also related to the literature on inventory disposal. Fukuda (1961) and Angelus (2011) studied stock disposals in multiechelon systems. The former assumes excess inventory in each stage can be returned to the next stage upstream, whereas the latter allows inventory to be sold in secondary markets at each stage. Both papers assume an infinite product lifetime. Martin (1986) and Rosenfield (1989, 1992) found the optimal level of inventory to dispose of when items can perish. However, there is no replenishment decisions in their models. The disposal price in general may depend on the quantity available. Cachon and Kok (2006) offered a simple adjustment to the newsvendor model to take that into account. Li et al. (2009) considered joint inventory control and pricing for perishable goods with a two-period lifetime under the FIFO rule. Finally, Li and Yu (2014), Chen et al. (2014), and Xue et al. (2012) independently studied clearance sales of perishable goods under the FIFO rule. The former used 5

it as an example to show the application of multimodularity. But as we will see later, when the issuing rule is LIFO, the objective function is not even concave, let alone anti-multimodular. The inventory systems in which goods are sold in two prices are sometimes called two bin systems (see Han et al. 2012 and the literature therein). In summary, although the related literature is voluminous, our work is the first to study joint replenishment and clearance sales of perishable goods in retailing where goods have a general finite lifetime and are issued based on LIFO. 3 Model Description The goods can be sold either at a regular price, r, or a clearance sale price, s. Under a regular price, the demand in a period is random. For ease of exposition, we present some of the analysis under the assumption that the demand is continuous. However, this assumption is not essential. In cases where functions are not differentiable, derivatives mean left sided derivatives. Let D represent the demand and Φ its distribution function. Unmet demand is lost. The demand under a clearance sale is so large that the inventory on clearance sales will never go unsold. This assumption is common in the literature on inventory disposal. The goods have an n-period lifetime and without loss of generality, zero value after they expire. The items that expire incur an outdating cost θ per unit to be removed from the shelf and disposed of. The items that are carried over to the next period incur a holding cost h per unit. The retailer purchases the goods at a cost c per unit. Profits received in future periods are discounted by a discount factor α. We assume that r > c and s < αc h. Without the latter condition, the retailer would order an infinite amount in each period and clear all the remaining inventory after the regular demand is filled. In the model, there are only two prices the retailer can choose to sell its inventory in the planning horizon. More sophisticated pricing schemes used in services such as airlines and hotels are not common in retailing. One reason is that the process of frequently changing price tags is labor intensive and the cost relative to the value of the products is normally quite high. 6

The timing of events is as follows. At the beginning of each period, the retailer decides an order quantity, q, of new items. At the end of each period after the regular demand is realized and filled as much as possible, the retailer decides how much of the remaining inventory that has not expired, if any, should be carried over to the next period and how much should be sold at a clearance sale price. If the initial inventory in the current period is described by a vector x = (x 1, x 2,..., x ), where x i is the number of units on hand with i periods of life remaining, then after demand is realized but before the clearance sales, the system state becomes Y(q, x, D) = (Y 1, Y 2,..., Y ), where for 1 i n 2 and Y i (q, x, D) = (x i+1 (D q k=i+2 x k ) + ) + Y (q, x, D) = (q D) +. The outdated amount is where S(q, x, D) = (x 1 (D q x i ) + ) +. The dynamic programming formulation is as follows: V t (x) = max {re min(q + x i, D) cq θes(q, x, D) + Eπ t+1 (Y(q, x, D))}, (1) q 0 i=1 π t (y) = max 0 z y {s (y i z i ) h z i + αv t (z)}. (2) i=1 We assume that at the end of the planning horizon, any unsold inventory has no value; that is, V T +1 (x) = 0. i=1 As a result, π T +1 (y) = s i=1 y i. In (2), z = (z 1, z 2,..., z ), where z i represents the amount of inventory with a remaining lifetime of i periods that is carried over to the next period. The inventory sold in clearance sales is y z. Combining (1) and (2) yields π t (y) = s y i + i=1 max u t(z, q), (3) 0 z y,q 0 7

where u t (z, q) = (s + h) z i αcq + αe{r min(q + z i, D) θs(q, z, D) + π t+1 (Y(q, z, D))}. i=1 We will analyze the optimization problem (3) henceforth. The state variables are now represented by y, the inventory levels after demand is realized but before the clearance sales. Because there is no information updating between the clearance sale decision at the end of a period and the ordering decision in the next period, we can redefine time periods. In (3), both the clearance sale decision and order decision are made at the beginning of a period. This treatment can simplify the exposition and is common in the literature (e.g., Huggins and Olsen 2010, Li and Yu 2014). Denote i=1 (z t, q t ) = arg max 0 z y,q 0 u t(z, q), where z t = (z t,1, z t,2,..., z t, ). When there are multiple maximizers, (z t, q t ) is defined as the smallest in lexicographical order. The following lemma on the marginal value of initial inventory is critical for the optimal policy on clearance sales. Lemma 1 (i) s πt(y) y 1 πt(y) y 2... πt(y) y αc h; (ii) ut(z,q) z 1 (iii) ut(z,q) q ut(z,q) z 2 ut(z,q) z 2... ut(z,q) z ;... ut(z,q) z. The first two results in Lemma 1 have been shown for the FIFO case by Li and Yu (2014). They can be interpreted to mean that the marginal value of newer inventory is always greater than that of older inventory irrespective of the issuing rules. Similar inequalities to Lemma 1 (i) have been shown in the literature for the ordering region under the FIFO issuing rule, a cost minimizing objective, and without considering clearance sales (e.g., Fries (1975), Nahmias (1975), and Nandakumar and Morton (1993)). Although fresher items are always more desirable, the benefit gained from replacing an older item with a newer one is no more than c, the purchasing cost. This explains the first inequality in part (iii). 8

4 General Characterization Define k t = max{i : zt,i < y i }, and if the set on the right-hand side is empty, we let k t = 0. Here k t represents the remaining lifetime of the newest inventory that is sold in clearance sales. The following properties of the optimal policies have been shown by Li and Yu (2014) under the FIFO issuing rule. We here reproduce it under a different context. Theorem 1 (i) z t,j = 0 for j k t 1 and z t,j = y j for j k t + 1. (ii) If k t 2, then q t = 0. According to Theorem 1(i), all items with a remaining lifetime strictly less than k t will be sold in clearance sales and will not be carried over to the next period. All items with a remaining lifetime strictly greater than k t will be carried over to the next period. Theorem 1(ii) means that clearance sales of items with a two-period lifetime or more and ordering of new items cannot happen in the same period. The result has been shown by Li and Yu (2014) using a sample-path argument. The proof here follows directly from Lemma 1(iii). Let z k = (0, 0,..., 0, z k, y k+1, y k+2,..., y ). Because of Theorem 1, finding the optimal k and z k will lead us to the optimal policy; that is, the original maximization problem (3) reduces to max u t(z, q) = max{ max u t(z 1, q), max { max u t (z k, 0)}}. (4) 0 z y,q 0 0 z 1 y 1,q 0 2 k 0 z k y k The mapping between the optimal policies and the states is in general very complex. To see the complexity, let us take a look at a numerical example. In the example, the lifetime n is 3 and hence the state space is two dimensional. In Table 1, we report zt,2 for different y 1 and y 2. Unlike in the FIFO case, the optimal policies on clearance sales do not have a desired clear-down-to structure. As y 2 is increases, it is optimal to first have no clearance sale, then 9

clear some inventory, then have no clearance sale, and finally clear inventory down to 8. The optimal policies also do not in general have the monotonicity that we observe in the FIFO case. For example, in Table 2, zt 4,1 is first increasing and then decreasing in y 2 when y 1 = 1. In Table 3, qt 4 is first decreasing in y 2, then increasing, and finally decreasing. In Table 4, we increase the life time to four. We similarly observe that zt 4,2 is not monotone in y 3. In spite of the complexity, the mapping between the optimal policies and the state variables is not without structure. First, in Table 1, there seems to exist two thresholds in each row a lower one (i.e., 4) and a higher one (i.e., 8). If y 2 is less than the lower threshold, then it is optimal not to have clearance sales; if y 2 is greater than the higher threshold, then it is optimal to clear the inventory down to the higher threshold. The same happens for each column in Table 4. Second, we can see from Table 2 that two thresholds also exist for the oldest inventory y 1. However, the optimal policies are different. For example, for the column corresponding to y 2 = 0 in Table 2, the lower and higher thresholds are 1 and 4.4, respectively. If y 1 is less than 1, we clear all the inventory; if y 1 falls between 1 and 4.4, there is no clearance sale; and if y 1 is greater than 4.4, we clear the inventory down to 4.4. Finally, in Table 3, although the optimal order quantity is not monotone in y 2, it is monotone in y 1. These three observations can be analytically verified and we present them in Theorems 2, 4 and 5. Table 1: zt 4,2 as a function of y 1 and y 2 (lifetime n = 3) y 2 0 1 2 3 4 5 6 7 8 9 10 y 1 0 0 1 2 3 4 4.9 4.9 7 8 8 8 1 0 1 2 3 4 4.9 4.9 7 8 8 8..... 6 0 1 2 3 4 4.9 4.9 7 8 8 8 Note. T = 5, r = 3.0, s = 0.4, c = 1.0, h = 0.2, θ = 0.4, α = 0.95, D = Uniform[3, 5]....... Theorem 2 For 2 i n 1, there exist U t,i and L t,i, where U t,i L t,i and they are functions of y i+1,y i+2,...,y but independent of y 1, y 2,..., y i. (i) If y i < L t,i, then z t,i = y i; 10

Table 2: zt 4,1 as a function of y 1 and y 2 (lifetime n = 3) y 2 0 1 2 3 4 5 6 y 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 2 2 2 2 1.4 0 0 0 3 3 3 2.4 1.4 0 0 0 4 4 3.4 2.4 1.4 0 0 0 5 4.4 3.4 2.4 1.4 0 0 0 6 4.4 3.4 2.4 1.4 0 0 0 Note. T = 5, r = 3.0, s = 0.4, c = 1.0, h = 0.2, θ = 0.4, α = 0.95, D = Uniform[3, 5] Table 3: Optimal order quantity qt 4 as a function of y 1 and y 2 (lifetime n = 3) y 2 0 1 2 3 4 5 6 y 1 0 4.7 3.6 2.6 1.6 0.6 0 0 1 4.7 3.6 1.2 0.2 0.6 0 0 2 2.2 1.2 0.2 0 0.6 0 0 3 1.2 0.2 0 0 0.6 0 0 4 0.2 0 0 0 0.6 0 0 5 0 0 0 0 0.6 0 0 6 0 0 0 0 0.6 0 0 Note. T = 5, r = 3.0, s = 0.4, c = 1.0, h = 0.2, θ = 0.4, α = 0.95, D = Uniform[3, 5] (ii) If y i U t,i, then z t,i = U t,i. Theorem 2 provides a partial characterization on the optimal clearance sale policy. For an age group with remaining lifetime i 2, if its inventory level is below its lower threshold, then there is no clearance sales; if it is above its higher threshold, then the inventory will be cleared down to the higher threshold. When the inventory level falls between the two thresholds, the optimal policy is complicated. As we have seen from earlier numerical studies, it can be no clearance sales, but it can also be to clear down to a threshold value between L t,i and U t,i. However, we know that in the latter case, all inventories older than i will be cleared (Theorem 1(ii)), and in both cases, all the oldest inventory will be cleared (Theorem 3 below). Theorem 3 For any given i [2, n 1], the following statements hold. 11

Table 4: zt 4,2 as a function of y 2 and y 3 (lifetime n = 4) y 3 0 1 2 3 4 5 6 y 2 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 2 2 2 2 2 1 0 0 3 3 3 2.8 2 3 0 0 4 4 4 2.8 4 4 0 0 5 5 4 5 4.8 4 0 0 6 5 6 5.5 4.8 4 0 0 7 7 6.5 5.5 4.8 4 0 0 8 7.8 7.3 5.5 4.8 4 0 0 9 7.8 7.3 5.5 4.8 4 0 0 10 7.8 7.3 5.5 4.8 4 0 0 Note. T = 5, r = 3.0, s = 0.4, c = 1.0, h = 0.2, θ = 0.4, α = 0.95, D = Uniform[3, 5] (i) If y i L t,i, then z t,1 = 0. (ii) If y i U t,i, then zt,j = 0 for all j [1, i). (iii) z t,i is independent of y 1. Theorem 3 provides properties of optimal clearance sale quantity with respect to inventory in other age groups. For inventory with a remaining lifetime i 2, if its inventory level falls between the two thresholds, then all the oldest inventory will be cleared. If the inventory level is above the higher threshold, then all the older inventory will be cleared. Theorem 3(iii) implies that the amount of the oldest inventory only affects its own clearance sales quantity. Theorems 2 and 3 do not include the case when i = 1. In order to characterize the impact of the oldest inventory, we need the following assumption. Assumption 1 (a) The demand distribution is a Pólya frequency function of order 2 (P F 2 ); (b) s αc + αs h αθ. 12

The assumption that the demand distribution is P F 2 is common in the inventory literature (e.g., Li et al. 2009, Huggins and Olsen 2010, Li and Yu 2012), and the class of distributions includes many commonly used distributions. To understand Assumption 1 (b), let us consider a marginal analysis for the last period T. The left-hand side of the inequality represents the maximal marginal profit if one unit is sold at a clearance sale and one new unit is ordered to replace it. The right-hand side represents the lowest marginal profit when that existing unit is kept in inventory to fill demand. If the inequality does not hold, then clearance sales and ordering of new items can never happen in the same period. One scenario where (b) holds is when the outdating cost is high. The outdating cost accounts for all the costs related to removing and disposing of the outdated items. It can be high in countries where landfill of food waste is banned or heavily taxed. Green groups such as Friends of the Earth have regularly monitored food waste generated by major retailer outlets and publicized their findings. If we take into account the bad publicity that outdating of perishable goods may create, the outdating cost would be even higher. The characterization of the optimal policy with respect to y 1 is given in the following theorem. Theorem 4 Suppose that Assumption 1 holds. Then there exist U t,1 and L t,1, where U t,1 L t,1 and they are functions of y 2,...,y but independent of y 1. (i) If y 1 < L t,1, then z t,1 = 0; (ii) If L t,1 y 1 < U t,1, then z t,1 = y 1; (iii) If y 1 U t,1, then z t,1 = U t,1. Figure 1 shows the ideas in Theorems 2 and 4. (In this and the next figures, we have omitted the time subscripts to simplify the presentation.) Similar to the cases with respect to y i, i 2, the optimal policy with respect to y 1 is also characterized by two thresholds L t,1 and U t,1. However, the structure of the optimal policy is strikingly different. For the cases when i 2, when the inventory level is below the lower threshold, there is no clearance sale. 13

In contrast, for i = 1, when the inventory level is below the lower threshold, all the inventory should be cleared. In addition, for any i 2, the optimal policy for the region between the lower and higher thresholds is complex and describing it requires additional threshold values. The characterization in Theorem 4 for the case when i = 1 is much sharper because we know that if y 1 falls between the two thresholds, then it is optimal not to clear y 1 but instead to carry all of it over to the next period. Assumption 1 ensures a precise characterization for the area between the two thresholds. The result that clearance sales take place when inventory is either too low or too high is not sensitive to the assumption. When y 1 is very low, the retailer does not hold the oldest and newest items at the same time. This segregation of items of different ages is a phenomenon unique to the inventory systems under LIFO and can never happen under FIFO. When y 1 is very low, a large amount of additional supply (of new items) is needed to meet the current demand. However, because customers retrieve items on a LIFO basis, the probability for the y 1 units of old items to be used to fill demand is low. Clearing all of them can avoid complete losses and outdating. Such separation does not happen between the newest inventory (i.e., q) and inventory with a remaining lifetime of at least two periods (i.e., y i for i 2) because it is suboptimal to clear some or all of the inventory with a remaining lifetime of two periods or longer and then place an order in the same period (Theorem 1). The marginal value of y 1 is s, the clearance sale price, when it is sold in a clearance sale, and greater than s if it is carried over to the next period. Therefore, the structure of the optimal policies leads to nonconcavity of the optimal profit function. As a function of y 1, the optimal policy on clearance sales is to clear all, then clear none, and finally clear down to a certain level. Similar policy structure has been observed in other contexts. For example, provided with all-unit discounts, a buyer s optimal ordering strategies may exhibit an order-not order-order phenomenon (Altintas et al. 2008). Under the segregating policy proposed by Prastacos (1979), a central facility sends older items to some locations and newer to others. Our results share the same spirit with the segregating policy: There is incentive not to allow items of different ages to go to the same location at the same time. 14

Figure 1: Optimal clearance sale policies for inventory with different remaining lifetimes Theorem 5 discusses properties of the optimal order quantity with respect to inventories with different remaining lifetimes. The analysis requires the concept of single crossing property. If a function f possesses the single crossing property in (y; t), then arg min y S f(y, t) is decreasing in t (Milgrom and Shannon 1994). Recall that z 1 = (z 1, y 2..., y n 2, y ). In the following we show that max 0 z1 y 1 u t (z 1, q) is single crossing in (q; y 1 ), which leads to the monotonicity of qt in y 1. Theorem 5 (i) max 0 z1 y 1 u t (z 1, q) is single crossing in (q; y 1 ), and consequently q t (y) is decreasing in y 1 ; (ii) If y 1 U t,1, then q t (y) = 0; (iii) If y i L t,i for an i [2, n 1], then q t (y) = 0. The monotonicity of ordering quantity with respect to y 1 cannot be extended to y i for i 2, as we can see in Table 3. For example, when y 1 = 1, if y 2 increases from 3 to 4, the optimal order quantity increases from 0.2 to 0.6. When y 2 increases from 3 to 4, z T 4,2 also increases from 3 to 4 (Table 1), but z T 4,1 decreases from 1 to 0 (Table 2). This means that the total 15

inventory after clearance sales but before ordering remains the same, which is four units. When y 2 = 3, the total inventory consists of three units of y 2 and one unit of y 1. When y 2 = 4, the total inventory consists of four units of y 2 and zero unit of y 1. The more we order, the less likely the inventory with a one-period remaining lifetime will be sold and the more likely an outdating cost will be incurred. As a result, the incentive to order is lower when y 2 = 3 than when y 2 = 4. In summary, y 1 is an economic substitute of q and any other state variables. However, the same cannot be said about y i for i 2. Theorem 5(ii) and (iii) together define an ordering region; that is, ordering can happen only when inventory levels of items with a remaining lifetime of at least two periods are all below their corresponding lower thresholds and the inventory level of the oldest inventory is below its higher threshold. To understand the optimal policies better, let s consider a special case with n = 3. In this case, the state variables are (y 1, y 2 ). Figure 2 shows the optimal policies under different values of y 1 and y 2. There are five regions in total. The only region that has complex structure for clearance sales is region IV. In this region, z 1 = 0 by Theorem 3, but z 2 can be equal to either y 2, or a threshold value that is between L 2 and U 2. For example, under the parameters used in Table 1, L 2 = 4 and U 2 = 8. When y 2 is between 4 and 8, z2 is either 7, which equals y 2, or 4.9, which is a threshold value different from 4 and 8. In terms of optimal ordering policy, ordering takes place only in region I and II, and the optimal ordering quantity is decreasing in y 1 by Theorem 5(i). The general model includes the special case when the holding cost is zero and the discount factor is 1. This special case is practically as well as theoretically important. For the specific perishable goods and retail environment that we consider, the review and decision period is typically quite short (e.g., a day or even shorter). In addition, we have explicitly considered a finite lifetime and separately computed the cost of outdating, which is sometimes considered part of holding. Therefore, in our setting, the discount factor and holding cost are unlikely to be significant (Li and Yu 2012, Zhou et al. 2011 and the literature therein). In this case, there 16

Figure 2: Optimal policies for clearance sales and replenishment when lifetime n = 3 is no reason to clear any inventory sooner than one period before its expiration date; that is, max u t(z, q) = max u t(z 1, q). 0 z y,q 0 0 z 1 y 1,q 0 In this case, both the lower and higher thresholds in Theorem 2 approach infinity and therefore z t,i = y i for all i = 2, 3,...n 1. The rest of the optimal policies, z t,1 and q t, satisfy Theorems 4 and 5. 5 Heuristics and Computational Results The analysis in Sections 4 provides us with interesting insights. Some of the properties are also useful for simplifying the computation of the optimal policy. However, the computation of the optimal policy is still challenging. For example, to compute the optimal order quantity, one must solve an ()-dimensional dynamic program, which is possible only for a small n. Good 17

heuristics are therefore still of interest. We propose two myopic heuristics with different requirements on product age information. Because the marginal values of inventories are bounded between s and αc h (Lemma 1(i)), for both heuristics, we approximate the value-to-go function by: π t+1 (y) = s + αc h 2 y i. In the first heuristic (referred to as MH 2 ), in computing the order quantity and clearance sale quantity, all inventories on hand are treated as if they would expire in one period. Let y = i=1 y i, the heuristic policies are derived from the following one-period problem: max (s + h)z αcq + αe[r min(q + z, D) θ(z (D 0 z y,q 0 q)+ ) + + s + αc h (q D) + ]. 2 Essentially, the heuristic MH 2 approximates the original problem by treating the product as if it had a two-period lifetime (hence the subscript 2). We have observed that many retailers, if not most, do not keep track of the age of their perishable items. Hence, this simple heuristic can be readily implemented without heavy investment in information technology. Our second heuristic (referred to as MH 3 ) requires additional information. In addition to the total inventory level y, we also need to keep a record of y 1, the inventory level of items with a one-period lifetime remaining. The heuristic policies are obtained by solving the following: max (s+h)(z 1+z) αcq+αe[r min(q+z 1 +z, D) θ(z 1 (D q z) + ) + + s + αc h (q+z D) + ], z 1,z,q 2 i=1 subject to the constraints: 0 z 1 y 1, 0 z y y 1, q 0. Here items with a remaining lifetime of two periods or longer are treated as if they all had a two-period lifetime remaining. Therefore, the heuristic MH 3 is based on a three-period lifetime approximation of the original problem. In the computation, we discretize the state variables with a step size δ = 0.1. The computation of the optimal policies require simulation because of the multi-dimensional state space of the problem. The structural properties we have established in the previous sections allow us to reduce the computational effort significantly. For example, suppose y 1 > 0. Then according 18

to Theorem 5, q t (y 1 δ, y 2,..., y ) q t (y). Therefore, in searching for q t (y), we only need to search in the region [0, q t (y 1 δ, y 2,..., y )]. c.v.=0.2 c.v.=0.3 c.v.=0.4 c.v.=0.5 Table 5: The performance of the heuristic MH 2. (% Errors) r = 5 r = 10 Lifetime 3 4 5 6 3 4 5 6 θ = 0.2 0.51 1.22 1.48 1.77 0.29 0.58 1.11 1.31 θ = 0.5 1.20 1.33 1.70 2.01 0.58 0.77 1.22 1.35 θ = 0.9 1.38 1.45 1.75 2.12 0.65 1.02 1.25 1.52 θ = 0.2 0.98 1.46 1.82 2.37 0.54 0.64 1.21 1.87 θ = 0.5 1.89 2.42 3.56 3.82 1.09 1.84 2.03 2.34 θ = 0.9 2.65 3.71 4.16 4.29 1.38 2.00 2.17 2.43 θ = 0.2 1.83 2.56 3.02 3.32 1.11 1.54 1.75 2.13 θ = 0.5 2.80 3.46 3.88 4.53 1.70 2.05 2.32 2.33 θ = 0.9 3.75 4.81 5.40 5.82 1.73 2.20 2.49 2.76 θ = 0.2 2.60 3.32 3.37 4.01 1.28 1.60 3.11 3.62 θ = 0.5 3.37 4.19 4.84 5.90 1.80 1.89 3.11 3.96 θ = 0.9 4.23 5.54 6.16 6.85 2.18 2.77 3.13 4.04 Note. T = 10, s = 0.4, c = 1.0, h = 0.2, α = 0.95. To test the performance of the two heuristics, we have tried various levels of cost parameters, lifetime, and demand distributions with different coefficients of variation (c.v.). Different combinations yield similar patterns in the computational experiments and hence we only report the results with the parameters in Tables 5 and 6. More specifically, we consider a 10-period horizon, T = 10. The demands follow uniform distributions with mean 5. The values taken by system parameters are: n {3, 4, 5, 6}, c.v. {0.2, 0.3, 0.4, 0.5}, r {5, 10}, and θ {0.2, 0.5, 0.9}. The discount rate α = 0.95, and the holding cost h, the ordering cost c and the clearance sales price s are held constant at 0.2, 1.0, and 0.4 respectively. In total, 96 instances are reported. We measure the performance of each heuristic policy by the following expression: Error = π T 9(0) π i T 9 (0) π T 9 (0) 100%, i {MH 2, MH 3 }, where π MH 2 T 9 ( ) and πmh 3 T 9 ( ) represent the maximal profit function under heuristic policies MH 2 and MH 3 respectively. 19

c.v.=0.2 c.v.=0.3 c.v.=0.4 c.v.=0.5 Table 6: The performance of the heuristic MH 3. (% Errors) r = 5 r = 10 Lifetime 3 4 5 6 3 4 5 6 θ = 0.2 0.08 0.11 0.12 0.21 0.01 0.01 0.05 0.15 θ = 0.5 0.05 0.11 0.11 0.15 0.01 0.01 0.07 0.12 θ = 0.9 0.06 0.15 0.16 0.14 0.01 0.01 0.07 0.11 θ = 0.2 0.04 0.05 0.15 0.13 0.03 0.11 0.14 0.19 θ = 0.5 0.05 0.03 0.11 0.09 0.03 0.15 0.15 0.13 θ = 0.9 0.05 0.05 0.15 0.09 0.07 0.11 0.13 0.12 θ = 0.2 0.08 0.13 0.14 0.17 0.10 0.17 0.20 0.23 θ = 0.5 0.02 0.13 0.18 0.21 0.09 0.15 0.20 0.23 θ = 0.9 0.04 0.15 0.18 0.26 0.06 0.13 0.22 0.22 θ = 0.2 0.18 0.20 0.55 0.65 0.14 0.17 0.45 0.52 θ = 0.5 0.20 0.25 0.43 0.59 0.11 0.15 0.32 0.43 θ = 0.9 0.23 0.38 0.48 0.58 0.12 0.21 0.41 0.52 Note. T = 10, s = 0.4, c = 1.0, h = 0.2, α = 0.95. The heuristic MH 3 performs much better than the heuristic MH 2, which only relies on the knowledge of total stock. The heuristic MH 3 is quite robust and its profit is consistently no more than 0.7% lower than the maximal profit. When the lifetime becomes longer, the performance of both heuristics deteriorates. This is expected because in both cases the longer the lifetime, the less accurate the approximations of the remaining lifetime. We have tested the effects of demand variability, which is measured by the coefficient of variation. Both heuristics perform better under low demand-variability environments. We have also tested the effects of r, the profit margin under regular sales. The heuristic MH 2 performs better when the margin is high. For dynamic programs with a large state space, heuristics that rely only on two state variables may perform surprising well if the right state variables are chosen (see, e.g., the dual-index policy in Veeraraghavan and Scheller-Wolf 2008). The excellent performance of the heuristic MH 3 also confirms this. In this case, the right state variables are the total inventory and y 1. Once we have information about these two state variables, additional information about other state variables can only lead to minor increase in profit. Retailers typically check and 20

remove the expired items manually. Putting items on clearance sales is also done manually. The information about y 1 can be obtained during these manual processes, and we believe that the additional effort should not be significant. If this is done, then the heuristic MH 3 can be implemented without the need to include the full age information in bar codes. In the above numerical studies, the longest life time is six. In Table 7, we increase the life time to seven. In this case, computing the optimal policies is no longer possible with a regular PC. We instead compute an upper bound of the optimal profit using perfect information relaxation. That is, we assume that the retailer knows all future demands before making any decisions. The maximal profit under perfect information is an upper bound for the optimal value function and can be calculated easily with a deterministic dynamic program (Brown et al. 2010). Under MH 2, one decides optimally the clearance sale quantity. To bring it closer to the two bin strategy commonly used in practice, we drop the optimization with respect to clearance sale quantity and simply move all items that will expire in m periods to the clearance sales bin. We call the revised heuristic a two bin heuristic (TB). In our studies, we choose m to be two. Obviously, TB is no better than MH 2. We compare the performance of MH 3 and TB against the upper bound. We define the performance measures as follows: Error i = πir T 14 (0) πi T 14 (0) π IR T 14 (0) 100%, i {MH 3, T B}, where πt IR 14 ( ) and πt B T 14 ( ) represent maximal profit function under perfect information relaxation and TB, respectively. We have tested 24 cases in total and the results are reported in Table 7. As we can see the from the Table, MH 3 consistently outperforms TB. It is unclear how tight the upper bound is. 6 Conclusion From the analysis in this study, we can see that the inventory with a one-period remaining lifetime deserves special attention. Unlike inventory in other age groups, there is only one opportunity to sell it, either through a regular sale or a clearance sale. Furthermore, if unsold, 21

Table 7: The performance of the heuristics MH 3 and TB (lifetime n = 7) Case c.v θ r Error MH3 Error T B 1 0.2 0.2 5 11.55 15.17 2 0.2 0.2 10 11.29 13.94 3 0.2 0.5 5 11.60 15.70 4 0.2 0.5 10 11.52 14.55 5 0.2 0.9 5 11.25 15.68 6 0.2 0.9 10 11.30 14.16 7 0.3 0.2 5 13.21 17.52 8 0.3 0.2 10 12.72 16.07 9 0.3 0.5 5 13.75 17.34 10 0.3 0.5 10 13.21 16.71 11 0.3 0.9 5 13.91 18.69 12 0.3 0.9 10 13.82 17.09 13 0.4 0.2 5 14.48 18.21 14 0.4 0.2 10 13.20 17.18 15 0.4 0.5 5 14.72 19.34 16 0.4 0.5 10 14.23 17.99 17 0.4 0.9 5 14.06 19.07 18 0.4 0.9 10 14.01 17.62 19 0.5 0.2 5 16.72 21.16 20 0.5 0.2 10 16.12 19.02 21 0.5 0.5 5 16.84 21.03 22 0.5 0.5 10 15.96 19.83 23 0.5 0.9 5 16.70 22.65 24 0.5 0.9 10 16.14 20.09 Note. T = 15, s = 0.4, c = 1.0, h = 0.2, α = 0.95 it will lead to outdating costs. Consequently, it affects the optimal policy on clearance sales as well as on ordering differently than the inventory in other age groups. From a practical standard point, it is important to keep an accurate record of the inventory with a one-period remaining lifetime and take advantage of the information in making decisions because, as we have shown in the numerical studies, the heuristic using that information is almost as good as the optimal policy but the heuristic without using that information is much worse. Our work demonstrates that, despite their complexity, perishable goods inventory problems in a retail 22

context can be analyzed rigorously. The benefit of having a rigorous decison tool for guiding decisions is substantial. In our previous analysis, we have made the following three assumptions and they deserve further discussion. First, we assumed that the demand in clearance sales is infinite. Although this is a common assumption in the literature on inventory disposal, we do sometimes observe that cleared items go unsold, which indicates a finite demand. Because clearance sales in the future are constrained by a finite demand, the retailer may start clearance sales sooner than what our current model would predict. Second, we assumed that the regular demand and clearance sales demand are independent. If the clearance sales are through a third party, or the cleared items are removed from shelf and put at a different location at the retail store, the influence of the clearance sale decision on regular demand should be small. If the cleared items and regular items are displayed on the same shelf, the clearance sales may cannibalize the regular sales. In this case, the incentive for clearance sales will be lower, though it will not disappear. Finally, we assumed a LIFO issuing rule. Although it has been well accepted in the literature that if consumers control issuing and they can observe expiration dates, then LIFO issuing will be the result, in reality, the consumer behavior is complex. For example, some may only do a limited search, others may make mistakes in reading the expiration dates. To capture this complex consumer behavior, a very general mixed issuing rule is needed. These assumptions allow us to have a sharper characterization of the optimal policies. The main points in the paper are not driven by these assumptions and removing these assumptions can only obfuscate the central issues and add complexity to the model. Retailing, in particular retailing of perishable goods, is rich in data. We believe that this is an area where business analytics can make a big difference. More needs to be done in this area, and our work has opened up many pathways of inquiry. 23

Appendix Proof of Theorem 1: The proof of part (i) is similar to the proof of Theorem 2 of Li and Yu (2014), hence it is omitted. (ii) If k t 2, then z t,k t < y kt. The proof is by contradiction. Suppose that q t > 0. Let f(δ) = u t (z t + δe kt, q t δ). We know from Lemma 1(iii) that f(δ) is an increasing function for any 0 < δ < min{qt, y kt zt,k t }. Therefore, (z t, qt ) cannot be a maximizer and we must have qt = 0. Proof of Theorem 2: (i) Recall that z i = (0, 0,..., 0, z i, y i+1, y i+2,..., y ). Define L t,i = inf{z i 0 : u t (z i, 0)/ z i < 0}. Then u t (z i, 0) is increasing in z i for z i < L t,i. We let the lower threshold L t,i = L t,i if y k < L t,k for all k > i and L t,i = 0 otherwise. It is obvious that L t,i is a function of y i+1,...y, but independent of y 1,.., y i. By definition, we can see that y i < L t,i implies that y k < L t,k for all k i. Hence if y i < L t,i, then u t ((0,..., 0, y i,..., y ), 0) u t ((0,..., 0, 0, y i+1,..., y ), 0) u t ((0,..., 0, 0, 0, y i+2,..., y ), 0)... u t ((0,...0, y ), 0). Therefore, max k i { max 0 z k y k u t (z k, 0)} = max k i {u t((0,..., 0, y k,..., y ), 0)} = u t ((0,..., 0, y i,..., y ), 0). (5) According to (4), max u t(z, q) = max{ max u t(z 1, q), max { max u t (z k, 0)}} 0 z y,q 0 0 z 1 y 1,q 0 k 2 0 z k y k = max{ max u t(z 1, q), max { max u t (z k, 0)}, max { max u t (z k, 0)}} 0 z 1 y 1,q 0 2 k i 1 0 z k y k k i 0 z k y k = max{ max u t(z 1, q), max { max u t (z k, 0)}, u t ((0,..., 0, y i,..., y ), 0)}. 0 z 1 y 1,q 0 2 k i 1 0 z k y k 24

From the last equation, we can see that z t,i = y i. (ii) Define Ũt,i = arg max zi 0 u t (z i, 0). If y i Ũt,i, then max 0 z i y i u t (z i, 0) = u t ((0,..., 0, Ũt,i, y i+1,..., y ), 0). From Lemma 1(ii), u t (z, q) is increased if we simultaneously decrease z j and increase z i by the same amount for any j < i. Similarly, because of Lemma 1(iii), u t (z, q) is increased if q is decreased and z i is increased by the same amount. Therefore, for y i Ũt,i, i 1 u t ((z 1,..., z i 1, y i, y i+1,..., y ), q) u t ((0,..., 0, y i + q + z k, y i+1,..., y ), 0) This implies and Therefore, if y i Ũt,i, then k=1 u t ((0,..., 0, Ũt,i, y i+1,..., y ), 0) = max 0 z i y i u t (z i, 0). max u t(z 1, q) max u t (z i, 0), 0 z 1 y 1,q 0 0 z i y i max u t (z k, 0) max u t (z i, 0) for 2 k i 1. 0 z k y k 0 z i y i max u t(z, q) = max{ max u t(z 1, q), max { max u t (z k, 0)}, max u t (z i, 0), 0 z y,q 0 0 z 1 y 1,q 0 k [2,i 1] 0 z k y k 0 z i y i max { max u t (z k, 0)}} k i+1 0 z k y k We let the upper threshold U t,i = Ũt,i if = max{ max 0 z i y i u t (z i, 0), max k i+1 { max 0 z k y k u t (z k, 0)}} = max{u t ((0,..., 0, Ũt,i, y i+1,..., y ), 0), max k i+1 { max 0 z k y k u t (z k, 0)}}. u t ((0,..., 0, Ũt,i, y i+1,..., y ), 0) max k i+1 { max 0 z k y k u t (z k, 0)} and U t,i = 0 otherwise. Therefore, we have z t,i = U t,i when y i U t,i. Proof of Theorem 3: (i) According to (4), if max u t(z 1, q) = max u t((0, y 2,.,,, y ), q), 0 z 1 y 1,q 0 q 0 25

then z t,1 = 0. Consider the derivative of u t(z 1, q) with respect to z 1 : u t (z 1, q) = (αr s h) α(r + θ)φ(q + z 1 + y i ), z 1 which is a decreasing function of q, z 1, and y i for 2 i n 1. Therefore, if y i > L t,i for some i 2, we have u t (z 1, q) z 1 = u t((z 1, y 2,..., y i, y i+1,..., y ), q) z 1 u t((0,..., 0, L t,i, y i+1,..., y ), 0) z 1 u t((0,..., 0, L t,i, y i+1,..., y ), 0) z i 0. The second inequality comes from Lemma 1(ii). The last inequality holds because of the definition of L t,i. So we have z t,1 = 0 if y i > L t,i for some i 2. According to the definition of L t,i in Theorem 2 (i), if there exists some i 2 such that y i > L t,i, then either L t,i = L t,i or there exists some k > i, where y k > L t,k. Hence, we can conclude that z t,1 = 0 if y i > L t,i for some i 2. (ii) The result is obvious from the proof of Theorem 2 (ii). (iii) If for all i 2 the inventory level y i L t,i, then z t,i is independent of y 1 because z t,i = y i; otherwise, is still independent of y 1 because z t,1 = 0. Since the thresholds L t,i only depend on the level of newer inventory, we can conclude that z t,i does not depend on y 1 for any i 2. The proof of Theorem 4 requires the following lemma: Lemma 2 Let ẑ 1 t (y, q) = arg max 0 z1 y 1 u t (z 1, q) and ẑ 1 t = (ẑ 1 t, y 2, y 3,..., y ). Suppose that Assumption 1 holds. The function u t (ẑ 1 t, q) is quasiconcave in q for q [0, a i=1 y i), quasiconvex for q [a i=1 y i, a y i) and quasiconcave for q [a y i, + ), where a = Φ 1 ( r (s+h)/α r+θ ). Proof of Lemma 2: Consider the derivative of u t (z 1, q) with respect to z 1 : u t (z 1, q) = (αr s h) α(r + θ)φ(q + z 1 + y i ). z 1 26

Then ẑ 1 t (y, q) = y 1 if q a i=1 y i; a y i q if a i=1 y i < q a y i; 0 if q > a y i. Because Y(q, x, D) is independent of x 1, Y(q, ẑ 1 t, D) = Y(q, y, D). Now u t (ẑ 1 t, q) can be expressed as follows: if q a i=1 y i, u t (ẑ 1 t, q) = (s + h) i=1 y i αcq + αe[r min(q + i=1 +θ(q + y i D) + + π t+1 (Y(q, y, D))]; if a i=1 y i < q a y i, y i, D) θ(q + i=1 y i D) + u t (ẑ 1 t, q) = (s+h)(a q) αcq+αe[r min(a, D) θ(a D) + +θ(q+ y i D) + +π t+1 (Y(q, y, D))]; if q > a y i, u t (ẑ 1 t, q) = (s + h) y i αcq + αe[r min(q + y i, D) + π t+1 (Y(q, y, D))]. The expression of u t (ẑ 1 t, q) for q > a y i can be obtained by setting y 1 = 0 in the function u t (ẑ 1 t, q) for q < a i=1 y i. Hence, we only need to prove: (i) (s + h) i=1 y i αcq + αe[r min(q + i=1 y i, D) θ(y 1 (D q y i) + ) + D) + + π t+1 (Y(q, y, D))] is quasiconcave in q; (ii) (s+h)(a q) αcq+αe[r min(a, D) θ(a D) + +θ(q+ y i D) + +π t+1 (Y(q, y, D))] is quasiconvex in q. Recall that if D is a P F 2 random variable and f(x) is quasiconcave, then Ef(x D) is quasiconcave. Therefore, to show (i), it suffices to show that f(q) = cq + r min(q + is quasiconcave in q, where i=1 y i, 0) θ(q + i=1 y i ) + + θ(q + y i ) + + π t+1 ( q) q = ((y 2 ( q y i ) + ) +, (y 3 ( q y i ) + ) +,..., (y ( q) + ) +, q + ). i=3 27 i=4 (6)

The function f(q) can be expressed in a piecewise manner as, (r c)q + r i=1 y i + π t+1 (0, 0,..., 0) if q i=1 y i; (c + θ)q θ i=1 y i + π t+1 (0, 0,..., 0) if i=1 y i < q y i; cq θy 1 + π t+1 (q + y i, 0,..., 0) if y i < q i=3 y i; f(q) = cq θy 1 + π t+1 (y 2, q + i=3 y i, 0,..., 0) if i=3 y i < q i=4 y i;... cq θy 1 + π t+1 (y 2,..., y n 2, q + y, 0) if y < q 0; cq θy 1 + π t+1 (y 2,..., y n 2, y, q) if q > 0. Notice that f(q) is increasing for q i=1 y i and decreasing for q > i=1 y i since π t(y) y i αc h. For (ii), it suffices to show that is quasiconvex in q, where g(q) = ((s + h)/α c)q + θ(q + y i ) + + π t+1 ( q) q = ((y 2 ( q y i ) + ) +, (y 3 ( q y i ) + ) +,..., (y ( q) + ) +, q + ). i=3 i=4 The function g(q) can be expressed in a piecewise manner as, s q + π t+1 (0, 0,..., 0) if q y i; θ y i + (s + θ)q + π t+1 (q + y i, 0,..., 0) if y i < q i=3 y i; θ g(q) = y i + (s + θ)q + π t+1 (y 2, q + i=3 y i, 0,..., 0) if i=3 y i < q i=4 y i;... θ y i + (s + θ)q + π t+1 (y 2,..., y n 2, q + y, 0) if y < q 0; θ y i + (s + θ)q + π t+1 (y 2,..., y n 2, y, q) if q > 0, where s = (s + h)/α c < 0. Also g(q) is decreasing for q y i and increasing for q > y i since πt(y) y i s. Proof of Theorem 4: We consider two cases. In the first case, we assume there exists some k [2, n 1] such that y k > L t,k where L t,k is defined in the proof of Theorem 2. We know that z t,1 = 0 in this case. Hence we only need to define L t,1 = U t,1 = +. For the second case, we consider the situation where y k L t,k for all k [2, n 1]. Then z t,k = y k for all k [2, n 1]. Define q t (y 2, y 3,..., y ) = arg max { (s + h) y i αcq + αe[r min(q + y i, D) + π t+1 (Y(q, y, D))]}. q 0 28

Notice that q t is independent of y 1. We will prove the following statements. (i) If y i a, z t,1 = 0 and q t = q t. (ii) If q t a y i, then for y 1 < a y i, z t,1 = y 1 and q t satisfies q t a i=1 y i. For y 1 a y i, z t,1 = a y i and q t = 0. (iii) If q t > a y i > 0, and u t ((a y i, y 2, y 3,..., y ), 0) > u t ((0, y 2, y 3,..., y ), q t ), then there exists a b t (0, a y i), such that for y 1 [0, b t ), z t,1 = 0 and q t = q t. For y 1 [b t, a y i), z t,1 = y 1 and q t satisfies q t a i=1 y i. For y 1 [a y i, ), z t,1 = a y i and q t = 0. (iv) If q t > a y i > 0, and u t ((a y i, y 2, y 3,..., y ), 0) u t ((0, y 2, y 3,..., y ), q t ), then z t,1 = 0 and q t = q t. The two thresholds L t,1 and U t,1 are defined as follows. If the conditions in part (ii) hold, then L t,1 = 0 and U t,1 = a y i; if the conditions in part (iii) hold, then L t,1 = b t and U t,1 = a y i; and let L t,1 = U t,1 = + for part (i) and (iv). Because all the conditions used in defining L t,1 and U t,1 are independent of y 1, the two thresholds are functions of y 2,...,y, and are independent of y 1. Define f 1 (y 1 ) = max q a y i u t(ẑ 1 t, q) and f 2 = max q>a y i u t(ẑ 1 t, q). It is easy to see that f 2 is independent of y 1. The function f 1 (y 1 ) is increasing in y 1 because u t (ẑ 1 t, q) = max z1 y 1 u t (z 1, q). (i) If y i a, then u t (ẑ 1 t, q) = (s + h) y i αcq + αe[r min(q + y i, D) + π t+1 (Y(q, y, D))] for all q, hence q t = q t. The optimal policy of clearance sales follows from (6). (ii) If q t a y i, since u t (ẑ 1 t, q) is quasiconcave in q a i=1 y i and quasiconvex in q [a i=1 y i, a y i), then we know that u t (ẑ 1 t, q) is decreasing for all q a i=1 y i. That is, the optimal q must satisfy q t a i=1 y i. The optimal policy of clearance sales follows from (6). The case when y 1 a y i can be similarly proved. 29

(iii) If q t > a y i > 0, then when y 1 = 0, u t (ẑ 1 t, q) is increasing in q for q a y i; that is, f 1 (0) f 2. When y 1 = a y i, u t (ẑ 1 t, q) is quasiconvex in q a y i and quasiconcave in q a y i. If, in addition, u t ((a y i, y 2, y 3,..., y ), 0) > u t ((0, y 2, y 3,..., y ), q t ), then we have Therefore, there exists a b t f 1 (a y i ) = u t ((a y i, y 2, y 3,..., y ), 0) > u t ((0, y 2, y 3,..., y ), q t ) = f 2. (0, a y i) such that f 2 f 1 (y 1 ) for y 1 [0, b t ) and f 2 f 1 (y 1 ) for y 1 [b t, a y i). The optimal clearance sale policy follows from (6). For y 1 > a y i, we can similarly show that q t = 0 and z t,1 = a y i. (iv) If q t > a y i > 0 and u t ((a y i, y 2, y 3,..., y ), 0) u t ((0, y 2, y 3,..., y ), q t ), then f 2 f 1 (a y i) and hence f 2 f 1 (y 1 ) for all y 1. In this case, the optimal policies are z t,1 = 0 and q t = q t for all y 1. Proof of Theorem 5: (i) Note that in the proof of Theorem 4, we have defined ẑ 1 t such that max 0 z1 y 1 u t (z 1, q) = u t (ẑ 1 t, q). We write ẑ 1 t as ẑ 1 t (q, y 1 ) to emphasize its dependence on (q, y 1 ). Suppose y 1 1 > y2 1 and q 1 > q 2. Define δ(y 1 ) = u t (ẑ 1 t (q 1, y 1 ), q 1 ) u t (ẑ 1 t (q 2, y 1 ), q 2 ). To show that u t is single crossing in (q; y 1 ), we need to show that δ(y1 2) < 0 implies that δ(y1 1 ) < 0, and δ(y 2 1 ) 0 implies that δ(y1 1 ) 0. Suppose δ(y2 1 ) 0 and we show that δ(y1 1 ) 0. We first consider several simple cases. If q 2 a y i holds, or, q 2 < a y i < q 1 and y 2 1 +q 2 a y i hold simultaneously, then u t (ẑ 1 t (q 1, y 1 ), q 1 ) and u t (ẑ 1 t (q 2, y 1 ), q 2 ) are not related to y 1, and the result holds naturally; if q 2 < a y i < q 1 and y 1 1 + q 2 a y i hold, then u t (ẑ 1 t (q 1, y 1 ), q 1 ) is not related to y 1 and the result holds because u t (ẑ 1 t (q 2, y 1 ), q 2 ) increases in y 1 for y 2 1 + q 2 < y 1 1 + q 2 a y i; if q 2 < a y i < q 1 and y 1 1 + q 2 > a y i > y 2 1 + q 2 hold, the result holds because a y i q 2 is the maximizer of the concave function (s + h)y 1 + αre min(q 2 + i=1 y i, D) αθe(q 2 + i=1 y i D) + in y 1. Hence we have shown that the result holds under both q 2 a y i and q 2 < a y i < q 1. Next we consider the remaining three cases under q 2 < q 1 a y i. 30

Case 1. If y 2 1 +q 2 a y i, then for both y 1 1 and y2 1, δ(y 1) is independent of y 1. Hence, the result obviously holds. Case 2. If y1 2 + q 2 < a y i < y1 1 + q 1, then αc(q 1 q 2 ) + αe[θ(y1 2 (D q 2 y i) + ) + θ(y1 2 (D q 1 y i) + ) + +r min(q 1 + y1 2 + y i, D) r min(q 2 + y1 2 + y i, D) +π t+1 (Y(q 1, y, D)) π t+1 (Y(q 2, y, D))] if y1 2 + q 1 a y i; δ(y1) 2 = αc(q 1 q 2 ) (s + h)(a q 1 y1 2 y i) αe[r min(a, D) r min(q 2 + y1 2 + y i, D) θ(a D) + + θ(q 2 + y1 2 + y i D) + +θ(q 1 + y i D) + θ(q 2 + y i D) + +π t+1 (Y(q 1, y, D)) π t+1 (Y(q 2, y, D))] if y1 2 + q 1 > a y i, and δ(y1) 1 = αc(q 1 q 2 ) (s + h)(a q 1 y 1 1 y i) αe[r min(a, D) r min(q 2 + y 1 1 + y i, D) θ(a D) + + θ(q 2 + y 1 1 + y i D) + +θ(q 1 + y i D) + θ(q 2 + y i D) + +π t+1 (Y(q 1, y, D)) π t+1 (Y(q 2, y, D))] if y 1 1 + q 2 a y i; αc(q 1 q 2 ) (s + h)(q 2 q 1 ) +αe[θ(q 1 + y i D) + θ(q 2 + y i D) + +π t+1 (Y(q 1, y, D)) π t+1 (Y(q 2, y, D))] if y 1 1 + q 2 > a y i, Consider the following two sub-cases: Sub-case 2a: If y1 2 + q 1 > a y i, then given δ(y1 2 ) 0, we consider the following two situations. When y 1 1 + q 2 a y i, δ(y1) 1 = δ(y1) 2 + (s + h)(y1 1 y1) 2 + αe[r min(q 2 + y1 2 + y i, D) r min(q 2 + y1 1 + y i, D) 0, +θ(y1 1 + q 2 + y i D) + θ(y1 2 + q 2 + y i D) + ] where the last inequality holds because (s + h)y 1 + αre min(q 2 + y 1 + y i, D) αθe(y 1 + q 2 + y i D) +, a concave function in y 1, achieves its maximum at y 1 = a y i q 2, and y 2 1 < y1 1 a y i q 2. 31

When y 1 1 + q 2 > a y i, δ(y1) 1 = δ(y1) 2 + (s + h)(a y i q 2 y1) 2 + αe[r min(q 2 + y1 2 + y i, D) r min(a, D) +θ(a D) + θ(y1 2 + q 2 + y i D) + ] which is negative because a y i q 2 is the maximizer of the concave function (s+h)y 1 + αre min(q 2 + y 1 + y i, D) αθe(y 1 + q 2 + y i D) + in y 1, and y 2 1 < a y i q 2. Sub-case 2b: If y1 2+q 1 a y i, then given δ(y1 2 ) 0, we again consider two situations. When y 1 1 + q 2 a y i, then δ(y 1 1) = δ(y 2 1) + f(q 2, y 2 1) f(q 1, y 2 1) f(q 2, y 1 1) f(q 2, y1) 2 f(q 1, y1) 2 f(q 2, y1) 1 f(q 2, a = 0, y i q 1 ) f(q 1, a y i q 1 ) f(q 2, a y i q 1 ) where f(q, y 1 ) = (s + h)(a q y i y 1 ) + αe[r min(q + y 1 + y i, D) θ(q + y 1 + y i D) + ] αe[r min(a, D) θ(a D) + ]. The second inequality holds because [f(q 2, y 2 1 ) f(q 1, y 2 1 )] y 2 1 = α(r + θ)[φ(y1 2 + q 1 + y i ) Φ(y1 2 + q 2 + y i )] 0, and f(q 2, y 1 1 ) y 1 1 When y 1 1 + q 2 > a y i, then = α(r + θ)[φ(a) Φ(y1 1 + q 2 + y i )] 0. δ(y1) 1 = δ(y1) 2 (s + h)(q 2 q 1 ) αe[θ(y1 2 + q 2 + y i D) + ) + 0, θ(y1 2 + q 1 + y i D) + ) + + r min(q 1 + y1 2 + y i, D) r min(q 2 + y1 2 + y i, D)] because (s + h)q + αre min(q + y 2 1 + y i, D) αθe(y 2 1 + q + y i D) + is concave in q. 32

Case 3. If y 1 1 + q 1 a y i, then δ(y 1 ) = αe[r min(q 1 + i=1 y i, D) r min(q 2 + i=1 y i, D) θ(y 1 (D q 1 y i ) + ) + +θ(y 1 (D q 2 y i ) + ) + + π t+1 (Y(q 1, y, D)) π t+1 (Y(q 2, y, D))] αc(q 1 q 2 ) As δ (y 1 ) = (αr + θ)[φ(q 2 + i=1 y i) Φ(q 1 + i=1 y i)] is negative, δ(y) decreases in y and hence the result holds. We can analogously show that if δ(y 2 1 ) < 0, then δ(y1 1 ) < 0. In summary, u t(ẑ 1 t, q) is single crossing in (q; y 1 ). Let ˆq t (y 1, y 2,..., y ) = arg max q 0 u t (ẑ 1 t, q). Then ˆq t is decreasing in y 1. q t (y) = ˆq t (y 1, z t,2,..., z t, ) is decreasing in y 1 because z t,i is independent of y 1 for i 2. (ii) The result has already been shown in the proof of Theorem 4. (iii) We have shown in Theorem 3(i) that if y i L t,i for any given i 2, then z t,1 = 0. That is, max u t(z 1, q) = max u t((0, y 2,..., y ), q). 0 z 1 y 1,q 0 q 0 We will show further that the optimal q = 0 in the above optimization problem. Recall that we have shown in the proof of Lemma 2 that u t ((0, y 2,..., y ), q) is quasi-concave in q. Hence we can denote the set of y i such that the optimal q = 0 as follows: Suppose y i L t,i, we know that u t ((0, y 2,..., y ), q) q Ω = {y i : u t((0, y 2,..., y ), q) q 0}. q=0 αc + αr(1 Φ(y 2 +... + y )) + α(αc h)φ(y 2 +... + y ) q=0 (s + h) + αr(1 Φ(y i +... + y )) + αsφ(y i +... + y ) (s + h) + αr(1 Φ(L t,i +... + y )) + αsφ(l t,i +... + y ) u t((0,..., y i,..., y ), 0) y i 0. yi =L t,i Hene y i Ω. 33

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