Inventory, Periodic Discounts, and the Timing Effect

Transcription

1 Inventory, Periodic Discounts, and the Timing Effect Hyun-soo Ahn Mehmet Gümüş Philip Kaminsky Department of Operations and Management Science, Ross School of Business, University of Michigan, Ann Arbor MI Desautels Faculty of Management, McGill University, Montreal, Quebec H3A 1G5 Department of Industrial Engineering and Operations Research, University of California, Berkeley CA We consider the joint pricing and inventory control problem for a single product whose demand distribution in each period is determined both by whether or not a sale price is offered in the current period, and the number of periods since the last time the sale price was offered. We show that optimal inventory ordering policy is a state dependent base stock policy; however, the optimal pricing policy can be quite complicated due to both the value and cost of holding inventory and delaying sales. We conduct a computational study to explore the effect of various cost and demand parameters on optimal pricing, ordering, and markdown decisions, propose a simple threshold type policy suggested by relaxation of the model to a make-to-order system, and computationally evaluate the performance of this heuristic. 1. Introduction and Literature Review The success of dynamic pricing and revenue management in the air travel, hotel, and car rental industries has naturally led to the desire to extend these concepts to other industries, including those with non-perishable products. Indeed, dynamic pricing is not new to these industries, but in the past it has primarily been utilized in the form of markdowns and promotions to eliminate excess inventory. However, as sophisticated data processing technology has spread in various retail industries, and as the internet and e-commerce have grown, dynamic pricing approaches have become more feasible as tools to help retailers match supply with demand more effectively and increase operating profit. There is a growing literature in operations management that explores dynamic pricing models in inventory ordering or production planning settings, modifying the assumptions, models, and results of traditional revenue management settings for these sometimes very different environments. In an environment in which inventory can be replenished and customers have multiple opportunities 1

2 to buy the same product, for example, intertemporal demand interaction can significantly impact effective ordering and pricing strategies. In this context, intertemporal demand interaction refers to the sensitivity of current demand not only to current pricing, but also to past pricing decisions. A quick review of the circulars in the Sunday paper suggests that retailers use price reductions for more than just eliminating excess inventory of out-of-season products these retailers are instead attempting to benefit from intertemporal demand interactions in order to increase profits. Indeed, in a recent empirical study, Pesendorfer (2002) analyzed pricing data for selected products in a supermarket chain, where prices were changed weekly. He used the scanner data from three ketchup products, and observed several intertemporal price effects that drive demand in each period. The level effect refers to the effect of the previous week s price on the current week s demand. For a given current price, the current demand will be stochastically higher when the previous week s price was high than when the previous week s price was low. For a product that is periodically put on sale, the timing effect refers to the observation that the demand at a given sale price is stochastically increasing in the time since the last sale. Ahn et al. (2007) explore the level effect; in this paper, we explore the timing effect. Other marketing researchers as well as economists have modeled similar issues; however, most of this literature has focused on the effects of intertemporal demand interactions on pricing decisions ignoring inventory considerations. Also, demand is primarily deterministic in these models. Conlisk et al. (1984) and Sobel (1984, 1991) consider two kinds of customers who differ in their valuation for a product. With this demand model, they show that firms engage in cyclic pricing behavior as a traditional skimming strategy in order to distinguish between high and low valuation customers. Assuncao and Meyer (1993) consider stockpiling behavior of consumers who are uncertain about future prices, and investigate how frequency of price reductions affects consumers purchase decision. Their analysis suggests that, as in our model, sales volume during promotional periods is larger than sales volume during regular periods. Slade (1998,1999) proposes a demand model in which consumers goodwill increases (decreases) as the firm continues charging low (high) prices, and analyzes the resulting optimal pricing strategy. However, for a dynamic pricing strategy to be effective in practice, it typically has to be aligned with inventory ordering decisions, and there are several growing streams of research focusing on joint pricing and inventory ordering models in the operations management literature. The first stream focuses on the coordination of pricing and production decisions 2

3 for perishable or seasonal goods this is typically referred to as revenue management. For these products, it is reasonable to assume that no replenishment (production) is allowed after the initial period, and thus most existing work focuses on analyzing the trajectory of the optimal pricing policy as a function of remaining inventory and time. Gallego and van Ryzin (1994) and Bitran and Mondschein (1997) consider a finite horizon dynamic pricing problem with uncertain demand and show that the optimal price increases in the number of periods remaining and decreases in inventory level. Subrahmanyan and Shoemaker (1996) consider a similar problem, but allow the retailer to determine the initial stocking quantity. Review papers by McGill and van Ryzin (1999), Petruzzi and Dada (1999) and Bitran and Caldentey (2003) provide comprehensive surveys of the area. Although there are a few recent papers that deal with more sophisticated consumer behavior (e.g., Aviv and Pazgal (2007), Elmaghraby et al. (2007), Su (2007), and Zhou et al. (2007)), all of them assume no replenishment during the selling season, and thus do not consider inventory/production factors. In the second stream, researchers consider the coordination of pricing and inventory control with independent demand. In this stream of research, demand is a random variable that depends only on current price. Under the assumption that unsatisfied demand in each period is fully backlogged, Federgruen and Heching (1999) and Chen and Simchi-Levi (2004a, 2004b) consider periodic review models with both finite and infinite horizons and characterize the form of optimal inventory ordering policies. Polatoglu and Sahin (2000), Chen et al. (2006) and Song et al. (2007) analyze various lost sales models. For continuous review models, Feng and Chen (2003) model demand using a Poisson process with pricesensitive intensity, while Chen et al. (2007) model the demand process as a Brownian motion with price sensitive drift rate. Researchers have also explored the coordination of pricing and inventory control with Markovian demand, where demand is a random variable that depends on both the pricing decision and the state of the world. Gayon et al. (2007) consider a demand model that is generated by a Markov modulated Poisson process and obtain structural results for the optimal pricing and inventory ordering policies. Yin and Rajaram (2007) extend Chen and Simchi-Levi s model to the Markovian demand case and characterize the form of the optimal policy. Several review papers provide comprehensive surveys of research milestones and future opportunities for joint production-pricing problems; see Eliashberg and Steinberg (1993) for classical models and Elmaghraby and Keskinocak (2003), Yano and Gilbert (2003), Chan et al. (2004), and the references therein for recent 3

4 developments in this area. In most of these models, however, the current demand (or demand intensity) is assumed to be independent of price history. In contrast to this research, we explicitly model the relationship between current demand and price history. In this respect, our model is most closely related to Cheng and Sethi (1999), who consider an (endogenous) Markovian demand model where the current demand is influenced by a state variable, which in turn is controlled by the pricing decision. However, in our model, the demand distribution is a function of the state variable and the current price, whereas in Cheng and Sethi (1999), the demand distribution is solely determined by the state variable. This distinction allows us to capture the timing effect discussed above. This paper is also related to Ahn et al. (2007), in which the authors analyze the level effect of intertemporal demand interaction in a deterministic setting. In this paper, in contrast, we focus on capturing the timing effect in a stochastic setting. See Gümüş (2007) for a detailed comparison of the models in Ahn et al. (2007) and those in this paper. In this paper, we present a stylized model that captures the impact of pricing decisions in one period on demands in others, and characterize the optimal inventory policy under this demand model. In particular, we explicitly capture the impact of the time since the last time a sale price was offered on the demand at a given sale price. We show that that the presence of inventory has a non-trivial effects on dynamic pricing decisions. To develop insight into these effects, we analyze a make-to-order version of our model, and building on this analysis, we propose a simple threshold-type policy for the pricing strategy. Finally, we conduct an extensive computational study to illustrate the impact of various cost and demand parameters on pricing and inventory ordering decisions and to evaluate the performance of our proposed threshold policy. In the next section, we introduce our model, in which the distribution of demand in sale periods depends on both the current price and the price history. 2. Model Formulation We consider a discrete time, T -period finite horizon, single item stochastic demand model where at each period the retailer offers one of two possible prices, a regular price or a sale price, denoted p r and p s (p r > p s ), respectively. At the start of each period (t = 1,..., T ), the retailer makes inventory ordering (i.e., how much to order) and pricing decisions (sale or regular) based on the current inventory level and the number of periods since the last sale. 4

5 Once decisions are made, demand is realized and holding or penalty cost is incurred. Let x t represent the inventory level at the beginning of period t before ordering, k t be the number of periods since the last sale before period t (where k t = 1 if there was a sale in period t 1), y t be the inventory level in period t after ordering, and p t be the retail price in period t. Our objective is to determine the optimal pricing and inventory plan that maximizes the expected (discounted) profit. There is no fixed ordering cost, no maximum ordering capacity and no delivery lead time (orders are received instantaneously). For the bulk of the paper, we assume that all unsatisfied demand is lost. We briefly consider the backorder case towards the end of the paper. To model the reaction of customers to a sale, we divide potential customers into two groups: consumers who will purchase independent of whether or not the product is on sale or not and consumers who will purchase only if the product is on sale. We call the first cohort regular customers and the second cohort sale customers. When the retailer charges the regular price, only regular customers will purchase the product. Some sale customers will remain and wait for a sale while others will leave. On the other hand, when the retailer offers a sale in period t, regular and sale customers who enter the market in period t as well as sale customers who have entered the system and remained since the last sale will buy the product (at the sale price). Hence, demand at the sale price is likely to be higher than demand at the regular price, and demand at the sale price is likely to be higher when more periods have passed since the last sale than when fewer periods have passed. Demand in period t, t = 1,..., T is therefore a function of the price in that period, p t and the number of periods since the last sale, k t. To capture this, let ξ t (p t, k t ) be a random variable representing the demand in period t when the retail price is p t and the number of periods since the last sales is k t. We assume that ξ t (p t, k t ) is a non-negative continuous random variable and denote its distribution and density Φ t (ξ p t, k t ) and φ t (ξ p t, k t ), respectively. Given p t and k t, ξ t (p t, k t ) is defined as follows: ξ r if p t = p r ; ξ t (p t, k t ) ξk s, if p t = p s and k t = k. where ξ r and ξ s k are both non-negative continuous random variables for regular and sale demand when k periods have elapsed since the last sale. Also, let Φ r (ξ) and φ r (ξ) be the c.d.f. and p.d.f. of demand when the regular price is offered. Similarly, let Φ s k (ξ) and φs k (ξ) be the c.d.f. and p.d.f. of demand when the sale price is offered and k periods have passed 5

6 since the last sale. Let µ r = E[ξ r ] and µ s k = E[ξs k ]. To capture the consumer behavior discussed above, we make the following assumption. Assumption 1. ξ r ST ξ s 1 and ξ s k is stochastically increasing in k. Although one could argue that regular demand should be also be effected by the frequency of sales, we do not model this possibility in order to keep the model tractable while still capturing relevant intertemporal demand effects. Consider the state transitions in this lost sales model. For each period t = 1,..., T, suppose (x t, k t ) represent the inventory level at the start of period t and the number of periods since the last sale. If the retailer orders y t x t units and sets its retail price at p t, then the inventory level and the number of periods since the last sale at the start of period t + 1 are described as follows: x t+1 = max[y t ξ t (p t, k t ), 0] = [y t ξ t (p t, k t )] +, t = 1,..., T and { kt + 1, if p k t+1 = t = p r ; 1, if p t = p s. Throughout the paper, we define [u] + = max[u, 0] and [u] = max[ u, 0] for any real number u. For a given state (x t, k t ) and retailer action (y t, p t ), the retailer realizes the following revenue and costs. p t min[y t, ξ t (p t, k t )] is the revenue in period t. c(y t x t ) = c (y t x t ) is the cost of raising inventory from x t to y t in period t, where c is the variable cost. h(y t ξ t (p t, k t )) = h + [y t ξ t (p t, k t )] + + h [y t ξ t (p t, k t )] is the inventory and penalty cost at the end of period t, where h + is the per-unit holding cost and h is the per-unit stock-out penalty cost. Note that since we are already capturing the loss of revenue in our revenue term, h represents the loss of goodwill cost or any other lost-sales related cost that is not directly related to the current revenue. (When we consider the backorder case, the same cost function applies except that h represents the per-unit backorder cost.) f T +1 (x T +1 ) = c x T +1 is the salvage value function for the terminal inventory. This is a standard assumption in many inventory finite horizon inventory models (see, e.g., Porteus (2002)) that will facilitate subsequent analysis. 6

7 Let V t (x t, k t ) be the expected discounted profit-to-go function under the optimal policy starting from state (x t, k t ) and J t (y t, p t ; x t, k t ) be the expected profit-to-go function if the retailer chooses to offer price p t and raise the inventory to level y t, and then continues optimally afterwards. Then, the retailer s problem can be expressed as a stochastic dynamic program satisfying the following recursive relation: { } V t (x t, k t ) = max max J t (y t, p t ; x t, k t ) and p t {p r,p s } y t x t [ ] pt min[y t, ξ] h + [y t ξ] + h [y t ξ] J t (y t, p t ; x t, k t ) = c(y t x t ) + φ t (ξ p t, k t )dξ 0 +αv t+1 ([y t ξ] +, k t+1 ) where V T +1 (x T +1, k T +1 ) = f T +1 (x T +1 ), and α is the discount rate. For notational simplicity, we drop time indices from decision and state variables when references are obvious. In many inventory problems, it is useful to work with transformed versions of value functions. We let Ĵt(y, p ; k) = J t (y, p; x, k) cx and ˆV t (x, k) = V t (x, k) cx. Then, the optimality equations can be rewritten as follows: { } ˆV t (x, k) = max max Ĵ t (y, p; k) p {p r,p s } y x where Ĵt(y, p; k) satisfies: Ĵ t (y, p ; k) = 0 (2.1) [p min{y, ξ} cy + αc[y ξ] + h + [y ξ] + h [y ξ] + α ˆV t+1 ([y ξ] +, k t+1 )]φ t (ξ p, k)dξ. In this transformed formulation, condition for the value function becomes ˆV T +1 ( ) = 0. Ĵ t (y, p ; k) does not depend on x and the terminal Note that we have two decision variables and two state variables, and thus, in general the optimal decisions are functions of the two state variables, y t (x, k) and p t (x, k). In the next section, we analyze the form of these optimal decisions and characterize the structural relationships between the optimal inventory ordering/pricing decisions and the state variables. In Section 4, we explore a make-to-stock version of our model, in Section 5, we use the insight from this exploration to develop an effective heuristic for our model and present the results of our computational study, and in Section 6, we present several extensions of our model and conclude. 3. Optimal Policy In order to characterize the structure of the optimal inventory ordering policy, we need to make some additional assumptions. 7

8 Assumption 2. The regular and sale demand distributions (i.e., the distributions of ξ r and ξk s, k = 1, 2,...) are strongly unimodal. Generally, the convolution of a unimodal distribution with a unimodal function is not unimodal. In other words, the unimodality is not preserved by the class of unimodal distributions. defined it as follows: Ibragimov (1965) introduced the class of strongly unimodal distributions and Definition 1. A distribution of a random variable is said to be strongly unimodal if its convolution with any unimodal function is unimodal. Ibragimov (1965) proved that a strongly unimodal distribution is unimodal and showed that the class of non-degenerate strongly unimodal distributions coincides with the class of distributions with log-concave density. We utilize this property to show that a base stock policy is optimal for our model. Typically, to prove the optimality of a base stock policy in a periodic review inventory model, researchers will show that the value function is concave (convex under minimization) in inventory order-up-to level. Since concavity of functions is preserved under the expectation operator, an induction-based argument can be used to demonstrate that a base stock policy is optimal for all time periods. Unfortunately, the single period value function of this model is not concave in inventory order-up-to levels. However, it can be shown to be unimodal, which is well known to also be a sufficient condition for the optimality of the base stock policy, and thus we need Assumption 2 to preserve unimodality under the expectation operator. Although this assumption is somewhat technical, a number of important random variables are indeed strongly unimodal. gamma and uniform distributions, among the others. Examples include normal, truncated normal, exponential, See Dharmadhikari and Joag-Dev (1988) for examples, results, and applications of strongly unimodal distributions. Using this assumption, we now show that the optimal ordering policy is a state-dependent base stock policy: Theorem 1. Suppose that Assumption 2 holds. In each time period t, for each p and k, there exists an optimal s t (p, k) such that if the starting inventory level x t < s t (p, k), it is optimal to raise the inventory level to s t (p, k), and otherwise it is optimal to do nothing. Proof. For notational simplicity, we omit the subscripts from state and decision variables. Let k be a positive integer and p {p r, p s }. It is sufficient to show that Ĵt(y, k, p) is unimodal in y for 8

9 given p and k. After some algebraic manipulation, we can rewrite Ĵt(y, p; k) as a convolution of two functions, the second of which is strongly unimodal by Assumption 2: where G t (w; p, k) is as follows: Ĵ t (y, p; k) = 0 G t (y ξ; p, k)φ t (ξ p, k)dξ G t (w; p, k) = (p c)e[ξ t (p, k)] p[w] (1 α)c[w] + + c[w] h + [w] + h [w] + α ˆV t+1 ([w] +, k t+1 ) Let π t (p, k) = (p c)e[ξ t (p, k)]. Noting that E[ξ t (p, k)] = µ r if p = p r and that E[ξ t (p, k)] = µ s k if p = p s, we have { (p π t (p, k) = r c)µ r, if p = p r, (p s c)µ s k, if p = ps. Replacing (p c)e[ξ t (p, k)] with π t (p, k) and rearranging the terms, we have G t (w; p, k) = π t (p, k) (p + h c)[w] (h + + (1 α)c)[w] + + α ˆV t+1 ([w] +, k t+1 ) It is sufficient to show that for a given p and k, G t (w; p, k) is unimodal in w since the convolution of a unimodal function with a strongly unimodal density is also unimodal (c.f., Ibragimov (1956) or Theorem 1.10 in Dharmadhikari and Joag-Dev (1988)). We show the unimodality of G t (w; p, k) in w by induction on t. Suppose t = T. The first term π(p, k) is constant in w. Since p + h c 0, the sum of the second and third terms is a unimodal function and has a unique maximizer at w = 0. Since V T +1 (w, k) = cw for all w 0, we have ˆV T +1 (w, k) = 0. Hence, G T (w; p, k) is unimodal in w because the sum of functions unimodal in w that have the same unique maximizer must also be unimodal in w with the same maximizer. Now, we show that if G t+1 (w, p, k) is unimodal, G t (w, k, p) is also unimodal. The first three terms are the same as in G T (w, p, k), and therefore the sum of these three terms is unimodal in w with a unique maximizer at w = 0. To complete the proof, we need to show that the final term (i.e. α ˆV t+1 ([w] +, k t+1 )) is non-increasing in w and that its maximum is achieved at w = 0. Note that from induction hypothesis, G t+1 (w, p, k) is unimodal in w, so Ĵt+1(y, p; k) is unimodal in y. Hence, max y x Ĵ t+1 (y, p; k) is non-increasing in x 0 as increasing x decreases the region over which the function is maximized. This implies that { } ˆV t+1 ([w] +, k) = max max Ĵ t+1 (y, p r ; k), max Ĵ t (y, p s ; k) y [w] + y [w] + is also non-increasing in w and achieves the maximum at w 0 since [w] + = 0 for w 0 and V t+1 ([w] +, k) is the maximum of two non-increasing functions. Hence, G t (w; p, k) is unimodal in w because the sum of functions unimodal in w that have same unique maximizer must also be unimodal in w. Theorem 1 characterizes the optimal order quantity for a given retail price for a retailer with starting inventory level x and k periods since the last sale. Recall that s t (p, k) is the unconstrained order-up-to level in period t when the offered price is p and the number of periods since the last sale is k, so s t (p, k) = arg max Ĵ t (y, p; k). y 0 9

10 This optimal order-up-to level critically depends on two factors, the difference between the regular and sale prices and the number of periods since the last sale. The sale demand increases in the number of periods since the last sales, k, and this drives the desired orderup-to level up. On the other hand, the lower retail price during the sale makes the product less attractive and drives the order-up-to level down. Depending on which force is more significant, the order-up-to level under the sale price can be larger (or smaller) than that under the regular price. For many realistic scenarios, however, it is reasonable to think that the retailer will try to sell more under the sale price than the regular price. In order to to capture this behavior, we make the following assumption: Assumption 3. The order-up-to level of a single period newsvendor problem with the regular and sale price demand distributions of our model is decreasing in price. That is, ( ) ( ) Φ s( 1) p s + h c 1 Φ r( 1) p r + h c. p s + h + + h αc p r + h + + h αc Assumption 3 enables us to further characterize the structure of the optimal policy. Theorem 2. Under Assumptions 1-3: (i) When the regular price is offered, the base-stock level is independent of the number of periods since the last sale and the number of remaining periods in the planning horizon. That is, there exists a constant s r > 0 such that s t (p r, k) = s r for all t and k. (ii) The base-stock level under the sale price is always larger than the base-stock level under the regular price: s t (p s, k) s r for all k and t. (iii) For a given k 1, ˆV t (x, k) is constant in x [0, s r ] for all t. Proof. We prove the results by induction. First, let t = T. (i) Since ˆV T +1 (x, k) = 0 for all x 0, Ĵ T (y, p r ; k) = π T (p r, k) + = π T (p r, k) 0 y { (p + h c)[y ξ] (h + + (1 α)c)[y ξ] +} φ(ξ p r, k)dξ (p + h c)(ξ y)φ(ξ p r, k)dξ y 0 (h + + (1 α)c)(y ξ)φ(ξ p r, k)dξ Define Ĵ T (y, pr ; k) to be the derivative of ĴT (y, p r ; k) with respect to y. Applying Leibniz s rule, Ĵ T (y, pr ; k) can be written as follows: Ĵ T (y, p r ; k) = y (p r + h c)φ(ξ p r, k)dξ y = (p + h c) Φ r (y p r, k)(p + h + + h αc) 0 (h + + (1 α)c)φ(ξ p r, k)dξ 10

11 where Φ r (y) is the cumulative distribution of regular demand. Since ĴT (y, p r ; k) is unimodal in y by Theorem 1, we find the maximizer (i.e. the base stock level under the regular price) employing the standard newsvendor solution: ( s T (p r, k) = (Φ r ) 1 p r + h ) c p r + h + + h αc pr, k where (Φ r ) 1 ( p r, k) is the inverse function of the cumulative distribution of regular demand p and r +h c p r +h + +h αc is the critical fractile associated with regular price. Since Φr (y) does not depend on k, s T (p r, k) is constant in k. We denote this base stock level s r. (ii) s T (p s, k) s r immediately follows from Assumption 3 and ξ r st ξ s k, k 1. Thus, ˆV T +1 (x, k) = 0 for all x 0. (iii) For given p and k, Ĵ T (y, p; k) is a unimodal function in y. Furthermore, from part (ii) s r = arg max y 0 Ĵ T (y, p r ; k) arg max Ĵ T (y, p s ; k) = s T (p s, k) for all k 1. y 0 Hence, for any x s r, { } ˆV T (x, k) = max max Ĵ T (y, p; k) p {p r,p s } y x ] = max [ĴT (s r, p r ; k), ĴT (s T (p s, k), p s ; k) Therefore, ˆV T (x, k) is a constant for x s r. On the other hand, for x > s r, we have ] ˆV T (x, k) = max [ĴT (x, p r ; k), ĴT (max[x, s T (p s, k)], p s ; k) Since both ĴT (x, p r ; k) and ĴT (max[x, s T (p s, k)], p s ; k) are continuous and non-increasing in x, ˆV T (x, k) is also continuous and non-increasing in x. Now, assume the results hold for t + 1. We prove that they hold for t: (i) Since Ĵt(y, p r ; k) is unimodal in y, it is sufficient to show that s r satisfies the first order condition. From the induction hypothesis, ˆV t+1 (x; k) is constant for x s r and non-increasing beyond s r. Hence ˆV t+1 (x; k) = 0 for x s r. Taking the derivative of Ĵt(y, p r ; k) with respect to y yields Ĵ t (y, p r ; k) = (p + h c) Φ r (y)(p + h + + h αc) for y s r. Solving for y, we have ( s t (p r, k) = s r = (Φ r ) 1 p r + h ) c p r + h + + h αc pr, k for all k. From the induction hypothesis and the fact that Ĵt(y, p r ; k) is unimodal in y, Ĵ t (y, p r ; k) achieves the maximum at y = s r. (ii) Note that for any y s r, w = [y ξ(p, k)] + s r. Thus, V t+1 (w, k) remains constant for all w s r and Ĵ t (y, p s ; k) = (p s + h c) Φ s k (y)(ps + h + + h αc) for y s r. 11

12 From Assumption 3 and the fact that ξ r st ξk s, Ĵ t (y, p s ; k) = (ps + h c) Φ s y=s r k (sr )(p s + h + + h αc) (p r + h c) Φ r (s r )(p r + h + + h αc) = Ĵ T (y, p r ; k) = 0. y=s r Therefore, s t (p s ; k) must be greater than or equal to s r for all k. (iii) Since s r s t (p s, k), the result follows from the same argument employed when t = T. By combining the results of previous two theorems, we can begin to characterize the optimal pricing and inventory policy. For this purpose, in addition to the regular base stock and sale base stock levels, we define a third critical level s t (k) that represents the smallest starting inventory level such that offering the sale price is optimal when k periods have passed since the last sale. In other words, if the starting inventory level is below s t (k), it is optimal to charge the regular price. Note that s t (k) can range from zero if offering the sale price is more profitable for any starting inventory level (i.e., max y 0 Ĵ T (y, p r ; k) max y 0 Ĵ T (y, p s ; k)), to a value greater than the regular or sale price base stock level. The structure of optimal policy depends on the relative ordering of s t (k) with respect to s r and s t (p s, k): Theorem 3. Suppose Assumptions 1 3 hold. If there have been k periods since the last sale, the optimal pricing and inventory policy in period t takes one of the following three forms depending on the starting inventory level at the beginning of period t, x, relative to the threshold s t (k): (i) s t (k) = 0 : If x s t (p s, k), it is optimal to order up to s t (p s, k) and offer the sale price p s. Otherwise, it is optimal to not order and charge a state dependent price, p t (x, k) = arg max p {p r,p s } Ĵt(x, p; k) (see Figure 1). (ii) s r < s t (k) s t (p s, k): If x [0, s r ), it is optimal to order up to s r and sell at the regular price p r. If x [s r, s t (k)), it is optimal not to order, and to sell the product at the regular price p r. If x [ s t (k), s t (p s, k)), it is optimal to order up to s t (p s, k) and sell at the sale price p s. If x s t (p s, k), it is optimal to not order, and to follow the state dependent pricing policy, p t (x, k) = arg max p {p r,p s } Ĵt(x, p; k) (see Figure 2). (iii) s t (p s, k) s t (k): If x s r, it is optimal to order up to s r and sell at the regular price p r. If x > s r, it is optimal to not order and to offer the regular price p r for 12

13 x (s r, s t (k)), and the state-dependent price p t (x, k) = arg max p {p r,p s } Ĵt(x, p; k) if x t s t (k) (see Figure 3). Proof. Notice that from equation (2.1), { ˆV t (x, k) = max max y x } Ĵ t (y, p r ; k), max Ĵ t (y, p r ; k). y x Define Ĵ t (x, p r ; k) = max y x Ĵ t (y, p r ; k) and Ĵ t (x, p s ; k) = max y x Ĵ t (y, p s ; k), respectively. Both are non-increasing in x by Theorem 1. Specifically, from Theorems 1 and 2, Ĵ t (x, p r ; k) = Ĵ t (s r, p r ; k) for x s r and Ĵ t (x, p r ; k) = Ĵt(x, p r ; k) otherwise, and Ĵ t (x, p s ; k) = Ĵt(s t (p s ; k), p s ; k) for x s t (p s ; k) and Ĵ t (x, p r ; k) = Ĵt(x, p r ; k) otherwise. Hence, the structure of the optimal policy is determined by how these two non-increasing functions behave with respect to starting inventory level x; we explore three different cases: Case 1: Ĵ t (s r, p r ; k) Ĵt(s t (p s ; k), p s ; k) The fact that s t (p s ; k) s r (Theorem 2.(ii)) and the fact that Ĵt(y, p; k) is unimodal in y imply that Ĵt (x, p r ; k) = Ĵt(s r, p r ; k) Ĵ t (x, p s ; k) = Ĵt(s t (p s ; k), p s ; k) for x s r and, Ĵt (x, p r ; k) Ĵt(s r, p r ; k) Ĵ t (x, p s ; k) = Ĵt(s t (p s ; k), p s ; k) for s r < x s t (p s ; k). Hence, as long as the starting inventory x is below s t (p s ; k), it is optimal to raise the inventory level to s t (p s ; k) and offer the sale, hence s t (k) = 0. For x > s t (p s ; k), notice that the unimodality of Ĵ t (y, p; k) in y implies that Ĵt (x, p r ; k) = Ĵt(x, p r ; k) and Ĵt (x, p s ; k) = Ĵt(x, p s ; k), thus it is optimal to charge a state-dependent price p t (x, k) = arg max p {p r,p s } Ĵt(x, p; k) and not order. Case 2: Ĵ t (s r, p r ; k) > Ĵt(s t (p s ; k), p s ; k) and Ĵt(s t (p s ; k), p r ; k) Ĵt(s t (p s ; k), p s ; k) Applying the condition and using the fact that s t (p s ; k) s r, we have Ĵ t (x, p r ; k) = Ĵt(s r, p r ; k) > Ĵ t (x, p s ; k) = Ĵ t (s t (p s ; k), p s ; k) for x s r. For x s r, ordering up to s r and selling at the regular price p r is therefore optimal. For x (s r, s t (p s ; k)], note that Ĵ t (x, p r ; k) = Ĵt(x, p r ; k), which is (weakly) decreasing while Ĵ t (x, p s ; k) remains to be Ĵ t (s t (p s ; k), p s ; k). Since Ĵt(y, p r ; k) is continuous in y, there must be a s t (k) (s r, s t (p s ; k)] such that s t (k) = min{x (s r, s t (p s ; k)] Ĵ t (x, p r ; k) = Ĵ t (s t (p s ; k), p s ; k)}. It follows from the definition of s t (k) and the unimodality of Ĵt(y, p r ; k) that Ĵt (x, p r ; k) = Ĵt(x, p r ; k) > Ĵt(s t (p s ; k), p s ; k) for s r x < s t (k) and Ĵt (x, p r ; k) = Ĵt(x, p r ; k) Ĵ t (x, p s ; k) = Ĵ t (s t (p s ; k), p s ; k) for s t (k) x s t (p s ; k). Hence, when s r x < s t (k), it is optimal to sell the existing inventory at the regular price, p r and not order. On the other hand, if s t (k) x s t (x, k), then it is optimal to raise the inventory to s t (x, k) and sell at the sale price, p s. For x > s t (p s ; k), Ĵ t (x, p r ; k) = Ĵ t (x, p r ; k) and Ĵ t (x, p s ; k) = Ĵt(x, p s ; k). Hence, it is optimal to follow a state-dependent price p t (x, k) = arg max p {p r,p s } Ĵt(x, p; k) and not order. 13

14 Case 3: Ĵ t (s r, p r ; k) Ĵt(s t (p s ; k), p r ; k) > Ĵt(s t (p s ; k), p s ; k). Applying the inequality Ĵt(s r, p r ; k) > Ĵt(s t (p s ; k), p s ; k) and the fact that s t (p s ; k) s r again imply that Ĵ t (x, p r ; k) = Ĵt(s r, p r ; k) > Ĵ t (x, p s ; k) = Ĵ t (s t (p s ; k), p s ; k) for x s r. For x s r, ordering up to s r and selling at the regular price p r is therefore optimal. Notice that the condition implies that Ĵ t (s r, p r ; k) Ĵ t (x, p r ; k) = Ĵt(x, p r ; k) > Ĵt(s t (p s ; k), p s ; k) for x (s r, s t (p s, k), so it is optimal to sell the existing inventory at the regular price, p r, and not order. If x > s t (p s, k), then it is optimal to not order in either price. Hence, it is optimal to follow a state-dependent price p t (x, k) = arg max p {p r,p s } Ĵt(x, p; k) and not order. Pricing Policy p s p s p t (x t, k t ) s t (k t ) = 0 s r s t (p s, k t ) x t Ordering Policy s t (k t ) = 0 s r s t (p s, k t ) x t Figure 1: Optimal base stock and pricing policy when s t (k) = 0 Pricing Policy 0 p r s r p r p s s t (k t ) s t (p s, k t ) p t (x t, k t ) x t Ordering Policy 0 s r s t (k t ) s t (p s, k t ) x t Figure 2: Optimal base stock and pricing policy when s t (k) [s r, s t (p, k)) Pricing Policy 0 p r s r p r p r s t (p s, k t ) s t (k t ) p t (x t, k t ) x t Ordering Policy 0 s r s t (p s, k t ) s t (k t ) x t Figure 3: Optimal base stock and pricing policy when s t (k) [s t (p, k), ] In general, when the starting inventory level is higher than the appropriate base stock level (regular or sale base stock as described above), the optimal pricing policy becomes state 14

15 dependent. In fact, the optimal pricing policy can be quite complicated with respect to both state variables, as we illustrate with several examples. First, consider the change in p t (x, k) with respect to x for a fixed k. Intuitively, one might expect that p t (x, k) would decrease in starting inventory x because higher starting inventory level lead to higher inventory holding costs, suggesting that inventory should be liquidated through a sale. However, this intuition is not in general correct. To see this, consider the following simple 2-period example with the following demand function: Example 1. ξ t (p, k) = { [12 + ɛr ] +, if p = p r ; [ k i=1 18(1 β)i 1 + ɛ s ] +, if p = p s. where β = 0.1 and ɛ r and ɛ s are normally distributed random variables with E[ɛ r ] = E[ɛ s ] = 0, σ ɛr = 4 and σ ɛs = 6. Finally, p r = 30, p s = 15, h + = 4, h = 0, c = 10, α = 1 and the planning horizon is T = 2 periods long. It can be shown that Example 1 satisfies Assumptions 1-3. Figure 4 plots max y x Ĵ t (y, k, p r ) and max y x Ĵ t (y, k, p s ) with respect to x for k = 2 in period one, and for k = 1 and k = 3 in period two. Recall that the optimal pricing decision at period t is determined by the point-wise maximization of max y x Ĵ t (y, k, p r ) and max y x Ĵ t (y, k, p s ): p t (x, k) = arg max {max Ĵ t (y, k, p)} p {p r,p s } y x and that if k = 2 in period one, it will either equal 1 or 3 in period two. Observe that in period one (the left graph) the plots of max y x Ĵ 1 (y, 2, p r ) and max y x Ĵ 1 (y, 2, p s ) cross each other both at point x 1 and point x 2, implying that the optimal pricing decision at in period one first switches from regular to sale price at x 1 and then back to regular price at x 2 as starting inventory increases. Why is this the case? From Figure 4, observe that of the four possible pricing strategies, two dominate. Offer regular price in the first period and sale price (call this strategy-rs) and offer sale price in the first period and regular price in the second (strategy-sr). When initial inventory is low, RS dominates, but as initial inventory increases, the leftover inventory and thus holding cost in period one begins to increase, until SR becomes preferable. However, as starting inventory continues to increase, at some point the leftover inventory at the end of period two becomes significant, and so RS again dominates, since it eliminates additional inventory in period two. 15

16 t = 1 max y x Ĵ 1 (y, k, p r ) max y x Ĵ 1 (y, k, p s ) 400 t = 2 max y x Ĵ 2 (y, k, p r ) max y x Ĵ 2 (y, k, p s ) 450 k= k=2 200 k=1,2,3, k= k=1 250 Offer Regular Price Offer Sale Price x 1 x 2 Offer Regular Price Starting inventory, x Starting inventory, x Figure 4: max y x Ĵ 1 (y, k, p) and max y x Ĵ 2 (y, k, p) for p = p r, p s. Next, consider the relationship between between the optimal pricing decision and the number of periods since the last sale, that is, the sensitivity of p t (x, k) to k for a fixed x. In general, the pricing policy need not be decreasing in k since the optimal pricing decision depends on how the marginal accumulation of sale demand changes with k. For this example, we assume that the expected marginal accumulation in sale demand decreases as k increases, i.e. µ k+1 µ k is positive and decreasing in k. With this restriction, one might expect that if it were optimal to offer the sale price when the last sale was k periods ago, it would still be optimal to offer sale if the last sale is offered more than k periods ago since delaying the sale decision would seem not to increase the demand enough to justify waiting one more period to offer a sale. However, even with this type of restriction on sale demand accumulation, the optimal price is not necessarily monotone in k. To see this, consider another example satisfying Assumptions 1 3, where regular and sale demands are drawn from uniform distributions: Example 2. ξ t (p, k) = { U(µr, v r ), if p = p r ; U(µ k, v k ), if p = p s. 16

17 where U(µ, v) is a uniform random variable that has a support from µ v to µ + v. Let µ r = 6, v r = 3, µ k = 9 k i=1 βi 1 and v k = v r k i=1 β2(i 1) with β = Finally, p r = 30, p s = 15, h + = 5, h = 0, c = 10, α = 1 and the planning horizon is T = 2 periods. t = 1 t = k=4 max y x Ĵ 1 (y, k, p r ) max y x Ĵ 1 (y, k, p s ) 300 max y x Ĵ 2 (y, k, p r ) max y x Ĵ 2 (y, k, p s ) k=3 x k= k=4 k=3 Offer Regular Price Offer Regular Price Offer Sale Price Offer Sale Price k=1,2,3, k=1 k=4 220 x Starting inventory, x Starting inventory, x Figure 5: max y x Ĵ 1 (y, k, p) and max y x Ĵ 2 (y, k, p) for p = p r, p s. As for the previous example, we illustrate the possible strategies in figure 5, and as before, of the four possible pricing strategies, two dominate. Offer regular price in the first period and sale price (strategy-rs) and offer sale price in the first period and regular price in the second (strategy-sr). Observe that in period two, for any starting inventory between x 1 and x 2, for k = 4 RS dominates, but for k = 3, SR dominates. Again, this counterintuitive behavior is caused by two opposite effects. On one hand, as k increases from 3 to 4, the total demand over two periods doesn t change much regardless of whether RS or SR is employed due to the nature of the demand function. On the other hand, RS generates leftover inventory that lasts two periods rather than one, which impacts the overall cost more as k increases from 3 to 4. In both these examples, the future impact of current inventory (we call this the long-term or propagation effect) has a direct and somewhat counterintuitive impact on optimal policy. 17

18 4. The Make-to-order System As we demonstrated in the previous section, the optimal policy of our model is sometimes quite complicated, as pricing and inventory decisions are influenced by intertemporal demand, starting inventory level, and the remaining planning horizon. In general, the optimal pricing policy is not necessarily monotone in starting inventory level or time since the last sale. In order to isolate the intertemporal demand effect on the pricing component of our model, we next consider a simplified version of the model, in which we assume that the retailer places an order after it sees the realized demand- a make-to-order system that is, from the perspective of the retailer, deterministic. In this case, it is optimal to order exactly the realized demand, thus the state is simply k, the number of periods since the last sale. We write the retailer s problem as a deterministic dynamic programming problem: V t (k t ) = max [π(p t, k t ) + αv t+1 (k t+1 )] p t {p r,p s } where π(p t, k t ) = (p r c)µ r if p t = p r ; (p s c)µ s k if p t = p s, and V T +1 (k) = 0 for all k. Noting that the demand at the regular price is independent of the the number of periods since last sale, we omit k in π(p r, k) and write it as π(p r ). We assume that the expected demand under the sale price, E[ξ(p s, k)] = µ s k concave and increasing in k, that is, µ s k+1 µs k > µs k+2 µs k+1 is strictly 0, so that the marginal accumulation of sale demand decreases as k increases. Under this assumption, we show that a threshold policy is indeed optimal: there is a k t regular price, p r, if k k t, and to offer the sale price p s, otherwise. such that it is optimal to charge the Theorem 4. Let p t (k) be the optimal price at period t when k periods have passed since the last sale. Then, (i) p t (k) is non-increasing in k. (ii) V t (k + 1) V t (k) π(p s, k + 1) π(p s, k) for all k = 1,..., t and t = 1,..., T. In Appendix A, we prove this result by induction. For finite horizon problems, finding the optimal pricing policy is quite straight forward, as one can recast the dynamic programming as a shortest path problem. Indeed, this motivates us to explore the effectiveness of this type of threshold policy for our original model, the make-to-order model. We investigate this in the computational section of the paper. 18

19 4.1 The Infinite Horizon Model The threshold levels suggested by Theorem 4 are in general time dependent, so that there exists k t for each period t such that if the number of periods since the last sale is less than k t, it is optimal to offer regular price and otherwise it is optimal offer sale price. In order to better understand the nature of these threshold levels, we next consider the infinite horizon version of this problem in both the discounted and average profit cases. For both cases, in Appendix B, we prove that time invariant optimal threshold levels, k exist for each period t. In other words, Lemma 1. There exists an optimal stationary policy for both discounted and average profit cases. This suggests that there exists a cyclic policy where offering a sale every k periods is optimal. In order to characterize the optimal cycle length (that is, the optimal stationary threshold level), we write the expected profit associated with a policy in which the sale price is offered every k periods and find the optimal cycle length. Let Π α (k) and Π A (k) be the discounted and average expected profit of a k-period cyclic policy with initial state k 0 = 1, respectively. After some algebraic manipulation, we get, for all k 1, Π α (k) = π(pr ) 1 α + αk 1 π(ps, k) π(p r ) 1 α k (Discounted profit), and Π A (k) = π(p r ) + π(ps, k) π(p r ) k (Average profit). The derivations of both expressions above are delegated to the Appendix C. From this, it is easy to see that a periodic sale is better than selling at regular price every period if and only if there exists a k such that π(p s, k) π(p r ) 0. In such case, the optimal cycle length is the one that maximizes the profit: 0}. k α = arg max[α k 1 π(ps, k) π(p r ) 1 α k ] (Discounted profit), and k A = arg max π(ps, k) π(p r ) k (Average profit). To find the optimal cycle length, it suffices to consider k k = min{k π(p s, k) π(p r ) Lemma 2. In the infinite horizon discounted profit problem with discount factor α, for k k, 19

20 (i) the k-period cyclic policy is better than the k + 1-period cyclic policy if and only if π(p s, k + 1) π(p s, k) π(p s, k) π(p r ) 1/α = 1 α 1 k (4.1) i=1 αi 1 α 1 α k (ii) Π α (k) is unimodal in k. Lemma 3. In the infinite horizon average profit problem, for k k, (i) The k-period cyclic policy is better than a k + 1-period cyclic policy if and only if π(p s, k + 1) π(p s, k) π(p s, k) π(p r ) 1 k (4.2) (ii) Π A (k) is unimodal in k. The proofs of of these lemmas are in Appendices D and E. Conditions (4.1) and (4.2) imply that if the marginal benefit of extending the cycle by one period becomes sufficiently small, then it is optimal to not extend the length of a cycle. In fact, both conditions describe precisely the minimum marginal gain required to optimally extend the cycle length by at least one period. Solving for the optimal cycle length is not hard since, in both cases, the profit function is unimodal in k. We employ Lemmas 2 and 3 to characterize the optimal pricing policy for the infinite horizon problem. Theorem 5. In the discounted profit problem, (i) It is optimal to sell at the regular price in every period if and only if π(p r ) π(p s, k) for all k 1. (ii) Otherwise, it is optimal to use the k -period cyclic pricing policy where k is the smallest integer greater than k satisfying condition (4.1). The same results hold for the average profit case with condition (4.2) replacing condition (4.1). The proof can be found in Appendix D (discounted profit case) and Appendix E (average profit case). This result is related to other intertemporal pricing results, such as those in Sobel (1984) and Pesendorfer (2002). Both papers model the demand accumulation assuming customers 20

21 are strategic and characterize the equilibrium pricing policy that consists of periodic price reductions. In contrast to their models, this paper considers more general demand accumulations and fully characterizes the optimal pricing policies for both discounted and average profit cases with myopic customers. This result is also qualitatively similar to the optimality of 2-period cyclic pricing policy proved in Ahn et al. (2007) where customers stay in the market for at most 2 periods. 5. Computational Studies We completed two computational studies, one to assess the impact of system parameters on the operation of the system, and one to assess the the performance of a simple threshold policy for this system in light of the potentially complicated structure of the optimal policy discussed previously. 5.1 The Operational Impact of System Parameters In this set of experiments, we vary the system parameters described below, and assess the impact of these parameters on the optimal number of sales over the time horizon, as well as (for the case of markdown percentage) the impact on expected profit. The parameters we vary can be divided into three categories: Prices We fix the regular price, p r and vary the markdown percentage. Let δ be the percentage markdown with respect to the regular price, i.e. ) = (1 ps 100% Production parameters We use stationary linear production costs and inventory holding costs (where shortage cost is zero), i.e. c t (x) = cx and h t (x) = h + x + for all t = 1,... T. Furthermore, we vary the holding cost as a percentage of production cost, i.e. h = h + c 100%, and the production cost as a percentage of regular price, i.e. p r p r ( ) p r c m = 100% where m is the percentage markup for the regular price. 21

22 Demand scenarios For the demand distributions, we use truncated normal random variables. Let N + (µ, σ) be the positive part of a normal random variable with mean µ and standard deviation σ, then: ξ t (p t, k t ) = { N + (µ r, c v µ r ), if p t = p r ; kt i=1 N + (µ s i, c v µ s i ), if p t = p s. where c v is coefficient of variation (the ratio of the standard deviation to the mean), µ r = µ(p r ) and µ s i = { µ(p s ), if i = 1; µ(p s )γ(1 β) i 1, if i > 1. where 0 < β < 1 and 0 < γ < 1. Note that the expected marginal accumulation of sale demand is controlled by two parameters, β and γ. β controls the decrease in rate of expected marginal accumulation as the time since the last sale increases. For example, if β = 0.50, then the marginal accumulation decreases by half each period that the retailer delays the sale decision. γ controls the ratio of expected marginal accumulation in sale demand to the sale demand in the first period after a sale One Varying Parameter For the initial computational study, we consider a base scenario and change the various parameters one at a time, fixing the other parameters at the base level. Set of Parameters We consider three values for each of the parameters: 1. We fix p r = 20 and change the markdown percentage with respect to regular price by varying : {10%, 20%, 30%}. 2. We assume that µ(p) = 30 p. 3. The rate of decrease in marginal accumulation in sale demand measured by β: {0.05, 0.15, 0.25}. 4. The proportion of expected marginal accumulation sale demand to expected sale demand at k = 1 is varied by varying γ = µs 1 µr. µ s 1 5. The uncertainty in demand as measured by the coefficient of variation c v : {0.20, 0.30, 0.40}. 6. Percentage markup for the regular price m: {50%, 70%, 90%}. 7. Inventory holding cost as a percentage of unit cost h: {10%, 15%, 20%}. 22