Optimal Pricing and Inventory Control Policy in Periodic-Review Systems with Fixed Ordering Cost and Lost Sales

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1 Optimal Pricing and Inventory Control Policy in Periodic-Review Systems with Fixed Ordering Cost and Lost Sales Youhua (Frank) Chen, 1 Saibal Ray, Yuyue Song 1 Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Hong Kong Faculty of Management, McGill University, Montreal, H3A 1G5, Canada Received 8 August 3; revised 6 August 4; accepted September 5 DOI 1.1/nav.17 Published online 1 December 5 in Wiley InterScience ( Abstract: This paper studies a periodic-review pricing and inventory control problem for a retailer, which faces stochastic pricesensitive demand, under quite general modeling assumptions. Any unsatisfied demand is lost, and any leftover inventory at the end of the finite selling horizon has a salvage value. The cost component for the retailer includes holding, shortage, and both variable and fixed ordering costs. The retailer s objective is to maximize its discounted expected profit over the selling horizon by dynamically deciding on the optimal pricing and replenishment policy for each period. We show that, under a mild assumption on the additive demand function, at the beginning of each period an (s, S) policy is optimal for replenishment, and the value of the optimal price depends on the inventory level after the replenishment decision has been done. Our numerical study also suggests that for a sufficiently long selling horizon, the optimal policy is almost stationary. Furthermore, the fixed ordering cost (K) plays a significant role in our modeling framework. Specifically, any increase in K results in lower s and higher S. On the other hand, the profit impact of dynamically changing the retail price, contrasted with a single fixed price throughout the selling horizon, also increases with K. We demonstrate that using the optimal policy values from a model with backordering of unmet demands as approximations in our model might result in significant profit penalty. 5 Wiley Periodicals, Inc. Naval Research Logistics 53: , 6 Keywords: fixed cost pricing; stochastic demand; joint pricing and inventory control; dynamic programming; (s, S) policy; lost sales; 1. INTRODUCTION The basic setting in this paper is a retailer dealing in a single item and facing random, price-sensitive demand. Any of these demands that the retailer cannot satisfy readily from its stock is lost to eager competitors. The retailer follows a periodic review policy for its inventory control. This implies that at each review epoch (e.g., every Friday or every 15 days) the retailer must decide whether to procure any inventory, and, if yes, then how much. Each replenishment (order) is associated with two costs: a fixed cost (e.g., transportation cost, unloading cost) and a variable portion proportional to the quantity ordered (purchase cost). Obviously, it is then imperative for the retailer to trade off the holding (for any unsold items), ordering, and lost sales costs while deciding on its optimal inventory control policy. Correspondence to: Y. Chen (yhchen@se.cuhk.edu.hk); S. Ray (saibal.ray@mcgill.ca); Y. Song (yuyue.song@mcgill.ca) Although traditionally retailers have charged a relatively fixed price for a product throughout its life cycle, recently more and more retailers are resorting to dynamic pricing strategy [7, 1]. The most common such pricing strategy involves at each review epoch retailers deciding not only on their replenishment plan, but also on the price for the next period [1]. One of the reasons behind the popularity of dynamic pricing is the considerable ease with which prices can nowadays be changed in a retail environment due to advances in information technology. Obviously, customer demand for the following period then, in addition to being random, depends on the selected price. This strategy is followed in a number of retail sectors like grocery [1], furniture ( office products superstore like Staples ( and fashion products ( A similar strategy is even followed in some hi-tech industries, including Dell []. Since replenishment and pricing decisions are done simultaneously, it is natural that there are considerable interactions between the two decisions. For example, if the price for a 5 Wiley Periodicals, Inc.

2 118 Naval Research Logistics, Vol. 53 (6) period is set at a high level, then the customer demand would most probably be low. In that case, at the end of the period the retailer might be left with substantial inventory and so the replenishment quantity for the next period would be low (maybe even zero). On the other hand, a large quantity of unsold inventory at the end of a period might induce the retailer to reduce the price for the following period. Also, since most retail products have a relatively short life cycle, retailers have a finite number of periods in which to use the two control levers pricing and stocking level to maximize their profits, taking all relevant costs and benefits into account. The objective of this paper is to develop and analyze a jointly optimal dynamic pricing and inventory control (replenishment) policy for a retailer over a finite discrete selling horizon.. MODEL FRAMEWORK AND RELATED LITERATURE In our model, the finite selling horizon for the retailer is divided into T periods. At the beginning of each period inventory is reviewed and (if required) replenished, and at the same time the unit selling price for the period is decided. Demands in consecutive periods are independent, identically distributed, non-negative random variables and are sensitive to the price set for the particular period. More specifically, we assume that the demand in any period can be represented by the demand function D(p, ɛ) = d(p) + ɛ, where d(p) is a decreasing, deterministic function of the selling price p for the period, and ɛ is a non-negative, continuous random variable defined in the range (A, B) with E(ɛ) = µ. Without loss of generality, we assume A =. The density and the cumulative distribution functions of ɛ are denoted by f (u)(> for any u (, B)) and F (u), respectively. For expository reasons, all cost and demand parameters are assumed to be stationary over the entire selling horizon. We discuss the case of non-stationary demand pattern in Section 6. Each replenishment incurs a fixed ordering (setup) cost K( ) and a variable part proportional to the procurement quantity, where c(> ) is the per unit variable cost. The replenishment lead time is assumed to be zero. There are also two other inventory-related costs. At the end of each period, any non-negative inventory level I is charged a holding cost proportional to the remaining inventory quantity, hi +, where h(> ) represents the holding cost per unit per period (in this paper, I + represents max(, I) and I represents max(, I)). The entire leftover inventory (I + )is carried over from one period to the next, unless the particular period is the last one of the selling horizon. On the other hand, if the demand in a period is more than the inventory level at the beginning of the period, then all unmet demands are lost. A shortage cost bi is charged proportional to the shortage level, where b(> ) denotes the shortage cost per unit per period. Any excess inventory left at the end of the last period (T ) is salvaged. For ease of analysis, we assume the salvage value to be c per unit. Before formulating the problem we make the following mild assumptions about d(p) and ɛ. ASSUMPTION 1: The deterministic demand function d(p) is decreasing, concave, and 3d + pd for p [c, P u ], where P u is the highest possible price such that d(p). In the literature additive demand functions are normally associated with a linear d(p)both in single- and multi-period settings (see [9, 13, 16] and references therein). Our d(p) is substantially less restrictive and holds true for any function of the form d(p) = a βp γ, γ 1 (linear demand is a special case of this function when γ = 1), ln[(a βp) γ ], γ, and a e (γ p), γ. Although Federgruen and Heching [5], Chen and Simchi-Levi [, 3], and Feng and Chen [6] assume a more general decreasing concave/quasi-concave d(p), note that their models are based on backordering of excess demands. ASSUMPTION : Let r(u) = f (u)/(1 F (u)) be the failure rate function of ɛ s distribution. The failure rate function satisfies r (u) + [r(u)] > for any u (, B). As is well known, this assumption is met when ɛ s distribution has an increasing failure rate (IFR). Most of the theoretical distributions in the literature belong to the IFR class, e.g., Uniform, Normal (as well as truncated Normal at zero), Exponential, Gamma (with shape parameter s 1), Beta (with both parameters 1), and Weibull distribution (with shape parameter s 1) (refer to [13]). Evidently, both of the above assumptions are quite general from modeling and realism perspectives. Let α( 1) be the discount factor, y be the stock level after the replenishment decision at the beginning of a period (i.e., order-up-to level), and z = y d(p) be the riskless leftovers at the end of the period. The single period expected profit for the retailer, given that the initial inventory level is zero before the ordering decision, is then given by G(z, p) = + z z (p[d(p) + u] h[z u])f (u) du (p[d(p) + z] b[u z])f (u) du c[d(p) + z]+αc z (z u)f (u) du. (1)

3 Chen, Ray, and Song: Pricing and Inventory Control 119 The first term of the above expression represents the expected revenue and holding cost in case of excess inventory, the second term represents the expected revenue and shortage cost with unmet demands, the third term denotes the expected procurement cost, and the last term corresponds to the expected salvage value. For any y, the feasible range for z and p will be given by {(z, p) z + d(p) = y, p [c, P u ]}. Next we focus on formulating the multi-period profit function. Note that in this paper x represents the initial inventory level at the beginning of a period before ordering has been done, implying y x. For any x( ) at the beginning of period t, we define V t (x) as the maximum total discounted expected profit from period t until the end of the selling horizon (period T ), less the procurement cost of x. Hence, based on our assumptions, V T +1 (x) = for any x [, + ). Thus, we have the dynamic equation V t (x) = max {y x,z+d(p)=y,p [c,p u ]} { Kδ(y x) + G(z, p) + αj t (z)} () for t = T,..., 1, where z J t (z) = V t+1 (z u)f (u) du + V t+1 ()f (u) du, z (3) for any t(1 t T)and z. For convenience, let V t (x) = V t () for any x< (recall that we assume all unmet demands are lost). While J t and V t+1 are related, there is a fundamental difference. With an order-up-to inventory level of y at the beginning of period t, the retailer will be left with z riskless inventory units at the end of the period. However, when the randomness of demand is taken into account, the retailer would actually be left with a quantity less than or equal to z (since ɛ ). Depending on the actual amount left, J t (z) denotes the expected profit for the retailer from period t + 1 until period T, taking into account all possible values of V t+1 and their respective probabilities. The objective of the retailer is to maximize the total discounted expected profit over the finite selling horizon by optimally deciding on: (i) how much (if any) inventory to order in each period and (ii) the retail selling price to charge in each period. Our modeling framework builds on (and generalizes) a number of related joint pricing and inventory control models present in the literature. These papers can be broadly categorized into two streams: deterministic and stochastic. The earliest deterministic model is the one by Whitin [19], which proposes a link between inventory control and pricing within the traditional economic order quantity framework through a linear price-demand relation. Subsequent studies in this category have recently been surveyed by Yano and Gilbert []. Models in the stochastic category can be further subdivided based on whether they are single period (newsvendor) or multi-period in scope. One of the most significant issues in modeling uncertain demand when price is endogenous is whether the uncertainty is additive or multiplicative in nature. The additive demand is modeled as the sum of a deterministic, price-sensitive component and a random component that is independent of price (like in this paper). The multiplicative demand is similar to the additive demand in spirit, but combines the two components in a product form. As a result, the sensitivity of demand variance to price changes differs in the two forms (refer to [13] for more details). As our focus in this paper is on multi-period models, we leave the reader to Lau and Lau [9], Petruzzi and Dada [13], and Yano and Gilbert [] for various versions and extensions of the newsvendor problem. The early multi-period integrated pricing-inventory control models incorporating demand uncertainty are the ones by Mills [11], Zabel [1, ], and Thowsen [16]. However, fixed ordering cost, which is a defining element of this paper, is not a part of any of these modeling frameworks. If indeed the fixed ordering cost is zero (and unmet demands are backordered), then it is now known that the base-stock list price is optimal for the combined additive and multiplicative demand function (refer to [5]). That is, in each period the optimal policy is characterized by a critical order-up-to level and a price that depends on the initial inventory level. The paper by Thomas [18] is the first one to introduce fixed ordering costs into multi-period models with stochastic demands (still assuming backordering of unmet demands). This paper conjectures that an (s, S, P) policy will perform well if the feasible price range is assumed to be continuous, but show that such a policy is not necessarily optimal when there are only a few feasible discrete prices that are far apart. Over a quarter-century later, Polatoglu and Sahin [15] analyze a similar framework under the assumption that all unmet demands are lost. In their model, demand in each period is a continuous random variable distributed over two price-dependent bounds. They specifically assume a stochastic ordering of demands with respect to prices. However, the structure of their optimal policy is very complicated, e.g., there may be multiple order-up-to levels, and their proof is abstruse. Although some sufficient conditions are provided under which the simple (s, S, P)policy is optimal, these conditions are difficult to verify and non-intuitive. Even when we plug in a special form of demand function (e.g., linear) into Polatoglu and Sahin s setting, it is still not clear (unless we carry out extensive and cumbersome numerical tests) whether the required conditions are met. Compared to their work, our demand model is more specific, which allows us to obtain more comprehensible optimal policy structure. The paper that is perhaps most relevant to ours is a recent one by Chen and Simchi-Levi []. They also analyze a

4 1 Naval Research Logistics, Vol. 53 (6) Table 1. Single period models with lost sales assumption. Demand form Riskless part Random part Profit function Zabel [1] Multiplicative General General Not clear Polatoglu [14] Additive Linear Uniform Unimodal Polatoglu [14] Multiplicative Linear Exponential Unimodal Petruzzi and Dada [13] Additive Linear General Unimodal Petruzzi and Dada [13] Multiplicative Power General Unimodal This paper Additive General General Unimodal framework similar to Thomas and show that while for the additive demand model an (s, S, P) policy is optimal, for more general demand functions this policy might not result in maximum profits. The authors characterize the optimal policy for the latter case using an elegant concept called symmetric K-convexity. Although backordering could be a realistic assumption in some manufacturing settings, it is quite restrictive in a retail environment. Technically, also, there is a significant difference between the two assumptions: while the expected sales in a period is the expected demand when backorders are allowed, it is the expected value of the maximum of the random demand and the beginning-of-period inventory level when unmet demands are lost. This difference gives rise to some intrinsic analytical difficulties for lost sales problems in the inventory control literature. Since the product of the expected sales and the prevailing price is the expected revenue, it is relatively easy to show that the expected revenue is jointly concave in price and inventory level in the context of backordering, but concavity fails even when the demand function is assumed to be linear for the lost sales case [5]. Concavity is an important requirement for profit maximization analysis. We can get around this problem because we only use the unimodality property of the profit function in our research. Another related recent research is the one by Monahan Petruzzi, and Zhao [1], who consider a dynamic pricing problem over a discrete finite selling horizon. However, in their framework there is only one replenishment opportunity at the beginning of the first period, in contrast to our dynamic inventory control assumption. They characterize the structure of the optimal policy and also provide an efficient algorithm to compute the parameter values of the optimal policy. As evident from the above discussion, the most distinguishing feature of our model is its generality. Our framework is especially suited to a retail environment where the issues of fixed ordering cost, lost sales, and finite selling horizon play very significant roles. Some lost sales models that are most related to this paper are summarized in Tables 1 and. In this paper, we have been able to characterize the structure of an optimal joint pricing and inventory control policy for the multi-period problem in the form of an (s t, S t, P t ) policy for period t. That is, we can determine two inventory levels, s t and S t (s t S t ), for any period t. If the inventory level at the beginning of the period is less than or equal to s t, then an order should be placed to increase the inventory level to S t, or else, if the level is greater than s t, then there is no need to order. The optimal price P t can be set based on the inventory level after the ordering decision has been made. Our numerical study also enables us to understand the behavior of the optimal policy values. Specifically, for an infinite horizon problem the optimal policy seems to be almost stationary. Moreover, the optimal values as well as the extent of profit improvement from dynamically changing prices are affected by the fixed ordering cost. We are also able to show that under certain circumstances using the optimal policy values from a backordering model (e.g., []) as approximations in our lost sales model can result in significant profit penalty. The remainder of this paper is organized as follows. In the next section we characterize the optimal policy for a slightly generalized version of the existing single period newsvendor model. In Section 4, we investigate the optimal policy for the multi-period version under the assumption that there is a non-negative fixed ordering cost associated with each replenishment. Section 5 deals with our numerical study, while Section 6 discusses the technical issues related to some possible generalizations of the basic model. Our concluding remarks are presented in Section 7. Table. Multi-period models with lost sales assumption. Demand form Riskless/random Fixed cost Optimal policy Zabel [] Additive General/uniform Zero Partial base-stock Thowsen [16] Additive General/general Zero Base-stock Polatoglu and Sahin [15] Additive General/zero Positive (s, S, P)policy Polatoglu and Sahin [15] General General/general Positive Not clear This paper Additive General/general Positive (s, S, P)policy

5 Chen, Ray, and Song: Pricing and Inventory Control THE SINGLE PERIOD NEWSVENDOR PROBLEM WITH SALVAGE VALUE In this section we characterize the optimal joint pricing and inventory control policy (both structure and values) for a single-period newsvendor problem Analysis of the Profit Function in Terms of Riskless Leftovers at the End of the Period (z) The aim of this subsection is to show that the single-period profit function G(z, p) of (1) has a unique maximizing policy for the retailer. In order to achieve this objective we first prove the uniqueness of the profit-maximizing price for a given z, i.e., p(z), then establish the behavior of p(z), and subsequently show that there is also a unique z, Z, that maximizes G(z, p(z)). Some of the results in this section are similar to those of Petruzzi and Dada [13]. However, note that our model includes a salvage value and a more general d(p) compared to their paper. Suppose that the retailer has no inventory at the beginning of the period, i.e., x =, and the focus is on determining the jointly optimal purchase quantity and price. For simplicity, define (z) = z to be the expected leftovers and (z) = z (z u)f (u) du (u z)f (u) du to be the expected shortages. The expected single period profit in (1) can then be rewritten as G(z, p) = (p c)[d(p) + µ] [(c + h) (z) + (p + b c) (z)] + αc z (z u)f (u) du, (4) where (p c)(d(p)+µ) is the riskless profit, [(c+h) (z)+ (p + b c) (z)] represents the expected costs incurred as a result of the presence of uncertainty, and the last term is the discounted expected salvage value of the leftovers at the end of the period. Differentiating G(z, p) with respect to p,weget and G(z, p) p = d(p) + µ + (p c)d (p) (z) (5) G(z, p) p = d (p) + (p c)d <. (6) From Assumption 1 and the above equations it is obvious that for any given z, there is a unique p(z) [c, P u ] that maximizes G(z, p). For this p(z),ifp(z) (c, P u ),wehave and d(p) + µ + (p c)d (p) (z) =, (7) [d (p) + (p c)d ]p = F(z) 1, (8) [3d + (p c)d ]p +[d (p) + (p c)d ]p = f(z). (9) For the linear demand model d(p) = a bp (a >, b>, a c), as in the work of Petruzzi and Dada [13], the b optimal price is uniquely determined, for any given z, bythe expression p(z) = p (z) b, where p = a+bc+µ can be b viewed as the optimal riskless price. Rearranging (8) we can conclude that p(z) is increasing in z for p(z) (c, P u ). Let Z l be such that p(z l ) = c. In that case, d(c) + µ (Z l ) = (refer to (7)), implying Z l = d(c) <. If there exists any z [Z l, + ) such that p(z) = P u, then from (7) we know that µ + (P u c)d (P u ) (z) = (note that d(p u ) = ), and we term that z as Z u. Otherwise, let Z u =+. Based on (8) and (9) and our assumption about d(p), Lemma 1 follows. LEMMA 1: p(z)is increasing and concave for z [Z l, Z u ), where Z l = d(c) < and Z u, and p(z) = P u for z [Z u, + ). The feasible optimal price function in terms of z can then be expressed as { p(z), z [Z l, Z u ) p(z) = P u, z [Z u, + ). In the following, we characterize G(z, p(z))for z [Z l, + ). Using the facts that G(z,p) p {p=p(z)} = for z (Z l, Z u ), dp(z) = in the range z (Z u, + ), and G(z,p) z {p=p(z)} is continuous for z [Z l, + ), and taking derivative with respect to z, weget dg(z, p(z)) G(z, p) = z = (c + h) {p=p(z)} + (p(z) + b + h)[1 F(z)]+αcF(z), (1) d G(z, p(z)) =[1 F(z)]p (p(z) αc + b + h)f (z), (11) and d 3 G(z, p(z)) 3 =[1 F(z)]p f(z)p (p αc + b + h)f (z). (1)

6 1 Naval Research Logistics, Vol. 53 (6) Therefore, if z (, B), we have (using (7) (9)) d 3 G(z, p(z)) 3 { d G(z,p(z)) = + }= [F(z) 1]3 [3d + (p c)d ] [d + (p c)d ] 3 1 F(z) d + (p c)d (r(z) + r (z)) <. This means that at a local maximum or minimum, dg(z,p(z)) is concave. It is easy to verify that dg(z,p(z)) {z B} = (c + h) + αc < dg(z,p(z)), {z=} = (c + h) + (p() + b + h) > (Z l < and p(z) is increasing = p() > c) and dg(z,p(z)) {z=z l } = (c + h) + (p(z l ) + b + h)(1 ) = (c + h) + (c + b + h) = b. It follows that dg(z,p(z)) is unimodal for z (, B). Also, d G(z,p(z)) = for z [B, + ), and as d G(z,p(z)) = p in the range z [Z l dg(z,p(z)),], is nonnegative for z [Z l,]. Note that there is a unique z = Z( (, B)), below which dg(z,p(z)) and above which dg(z,p(z)) <. This unique z = Z clearly is the maximizer of G(z, p(z)), z [Z l, + ). Furthermore, as dg(z,p(z)) is unimodal and increasing at z =, it is non-increasing for z [Z, + ). Based on the above discussion, we have the following lemma. LEMMA : If Assumptions 1 and hold, then there exists a unique Z (, B), which maximizes G(z, p(z)). G(z, p(z)) is non-decreasing for z [Z l, Z] and non-increasing for z [Z, + ). The maximizer Z enables us to determine the optimal price p(z) and the optimal stocking quantity (order-up-to level) y(z) = Z + d(p(z)). 3.. Analysis of the Profit Function in Terms of the Order-up-to Level Although we generalized the additive model of Petruzzi and Dada [13], most of the above results are quite similar. In this subsection we analyze a different version of the singleperiod problem. Suppose the retailer has already in stock the order-up-to inventory level of y( ), not necessarily optimal, procured at a cost of cy at the beginning of the period. What is then the optimal price that the retailer should charge to its customers? We term this optimal price P(y). Of course, if the retailer s initial stock level happens to be y(z), then from Lemma we know that P(y = y(z)) = p(z). In the following, we characterize P(y = y(z)). We first show that for any given order-up-to inventory level y at the beginning of the period there is a unique profit-maximizing retail price P(y). We then show that the expected profit function solely in terms of y, i.e., G(y) = G(Z(y), P(y)) where Z(y) = y d(p(y)), is unimodal. Figure 1. Relationship between p(z) and p y (z). Hence, we conclude that the optimal replenishment policy, given that there is a fixed cost K( ) for any replenishment, is an (s, S) policy and the optimal retail price can be set based on the optimal order-up-to inventory level. As will be shown in Section 4, the technique used for solving the oneperiod problem can indeed be generalized to characterize the optimal policy of a multi-period problem. Before going on to the detailed proof we introduce the notation p y (z) to denote the function representing p in terms of z, for any given y( ), such that {(z, p) z + d(p) = y, p [c, P u ]}, i.e., p y (z) = d 1 (y z). In Fig. 1, we can see the relationship between p(z) and p y (z) for any given y( ). Let z(y, p) = y d(p) for any y. Then the expected profit in terms of y and p can be expressed as G(z(y, p), p). Before analyzing G(z(y, p), p) we show that for any given y there exists a Z(y), which maximizes G(z, p y (z)).as G(z, p) z and we get dg(z, p y (z)) = (c + h) + (p + b + h)[1 F(z)]+αcF(z) G(z, p) p = d + µ (z) + (p c)d, { ( G(z, p) G(z, p) = + 1 )} z p d {p=py (z)} { = (p + h) + (p + b + h)[1 F(z)] + αcf(z) d + µ (z) d } {p=p y (z)}

7 Chen, Ray, and Song: Pricing and Inventory Control 13 and d { G(z, p y (z)) F(z) = (p αc + b + h)f (z) d d + µ (z) } d. d 3 {p=p y (z)} If y =, then z for any (z, p) {(z, p) z + d(p) =, p [c, P u ]}. Hence, F(z) = and dg(z,p y(z)) = b (note that d + µ (z) = ). Clearly, y > and z < implies d(p y (z)) + µ (z) = y >, while y > and z implies d(p y (z)) + µ (z) d(p y (z)) + z z f (u) du >. Hence, if y >, then we always have (d +µ (z)) {p=py (z)} >. Therefore, for any given y, d G(z, p y (z)). We thus see, for any y( ), that there exists (Z(y), P(y)) {(z, p) z + d(p) = y, p [c, P u ]}, which maximizes G(z, p y (z)) where P(y) = p y (Z(y)).If(Z(y), P(y)) is not unique for some y( ), then we choose Z(y) as the one with the largest value among all the maximizers of G(z, p y (z)). Z(y) is then either an interior point or a boundary point on the feasible region, i.e., Z(y) is either given by the solution to dg(z,p y(z)) = orz(y) = y, and the optimal price P(y) = p y (Z(y)). Note that P(y) [c, P u ] for any y [, + ). Therefore, P(y) = c on some set of open intervals, P(y) = P u on some set of open intervals, and c<p(y)<p u on some set of open intervals. We can now present the following property about Z(y) and P(y), the detailed proof of which is provided in the Appendix. LEMMA 3: Let Z(y) = y d(p(y)). Under Assumptions 1 and, both P(y) and Z(y) are continuous for y [, + ). Furthermore, Z(y) is non-negative and increasing for y [, + ). From (13) and Lemma, we obtain the following important theorem about the single-period expected profit function. It says that y(z) maximizes G(P (y), Z(y)), which enables us to substitute this y(z)in P(y)and Z(y) to obtain the overall optimal order-up-to level and the corresponding optimal price. THEOREM 1: Under Assumptions 1 and, P(y) p(z) for y y(z)and P(y) p(z)for y>y(z), where y = z + d(p(z)). Furthermore, G(Z(y), P(y)) is unimodal for y [, + ) and there exists a unique y(z) that maximizes G(Z(y), P(y)). PROOF: First we consider the case y y(z). For any y 1 [, y(z)], let (z, p ) {(z, p) p = p(z), z [Z l, Z]} {(z, p) z + d(p) = y 1, p [c, P u ]}. By Lemma, G(z,p) z {p=p,z=z } = dg(z,p(z)) {p=p,z=z } and {p=p,z=z }. Hence, G(z,p) p dg(z, p y1 (z)) { G(z, p) = + z {p=p,z=z } G(z, p) p ( 1 d )} {p=p,z=z }. On the other hand, we know that for any given y 1 ( y 1 y(z)), G(z, p y1 (z)) is concave. To make proper comparison, we further define Z 1 (= Z(y 1 )) as the unique maximizer of G(z, p y1 (z)) (if there are multiple maximizers, then we define Z 1 as the one with the largest value among them) and P 1 = p y1 (Z 1 ). Clearly, Z 1 should be larger than or equal to z.it is also obvious that (Z 1, P 1 ) is above the curve of p = p(z) (refer to Fig. ). Similarly, for y>y(z), we can show that P(y) p(z), which completes the proof of the first part. The increasing property of Z(y) implies that the higher the order-up-to level y at the beginning of a period, the higher the optimal riskless leftovers Z(y) at the end of the period. Substituting P(y)and Z(y) in G it is possible to express the profit function only in terms of the initial order-up-to level y, i.e., G(Z(y), P(y)). Keeping in mind the facts that either G(z(y,p),p) dp (y) p {p=p(y)} = or = and that both P(y)and dy Z(y) are continuous for y [, + ),weget dg(z(y), P(y)) dy = G(z, p) z = (c + h) {p=p(y),z=z(y)} + (P + b + h)[1 F(Z(y))]+αcF(Z(y)). (13) Figure. and P(y). Optimal policy values and the relationship between p(z)

8 14 Naval Research Logistics, Vol. 53 (6) From (13) we have, dg(z(y), P(y)) dy = dg(z, p y 1 (z)) {y=y1 } = {z=z 1 } G(z, p) z G(z, p) p {z=z 1,p=P 1 } ( 1 ). d {z=z 1,p=P 1 } The concavity of G(z, p y1 (z)) and the definition of Z 1 implies dg(z,p y1 (z)) {z=z 1 }. As P 1 is above the curve of p(z) where y 1 = z + d(p(z)), we get G(z,p) p {z=z 1,p=P 1 }. Thus, for any y 1 y(z), dg(z(y), P(y)) dy. {y=y1 } Hence, G(Z(y), P(y)) is non-decreasing for y [, y(z)]. We can follow a similar logic to also prove that G(Z(y), P(y)) is non-increasing for y [y(z), ]. Hence, there exists a unique y(z) that maximizes G(Z(y), P(y)). Note the relation between the results in Sections 3.1 and 3.. The optimal y(z)in Section 3. is identical to that of Section 3.1. Also, P(y = y(z)) of Section 3. equals p(z) of Section 3.1, and Z(y = y(z)) of Section 3. equals Z of Section 3.1. Hence, at optimality our model is equivalent to that of Petruzzi and Dada [13]; however, ours is more general since we also characterize the profit function at any arbitrary order-up-to inventory level y( ). Some of the results are pictorially depicted in Fig.. The above characterization helps us understand the optimal single-period ordering policy when the retailer has an initial stock level x>at the beginning of the period and there is a fixed cost K( ) associated with any procurement. Since G(Z(y), P(y))is unimodal in y, we can convince ourselves that if x is below a threshold y at which the value of G(Z(y), P(y)) G(Z, p(z)) K, then it is optimal to order-up-to y(z). For initial stock level more than the threshold y, it is optimal not to order anything. 4. THE MULTI-PERIOD PROBLEM WITH A FIXED COST FOR ORDERING For the single-period problem, we have characterized the jointly optimal pricing and inventory policy structure in Subsection 3.. In this section, using the same technique as in the proof for the single-period problem, we characterize the optimal policy for the multi-period problem by induction on the remaining periods until the end of the selling horizon. In the following, P t (y) is used to represent the optimal price and Z t (y) the optimal riskless leftover for a given orderup-to level y at the beginning of period t. More specifically, for notational simplicity, assume that for a given y i ( ) at the beginning of period t, Pt i represents Z t (y = y i ), and represents P t (y = y i ), Z i t H t (y i ) = max {(z,p) z+d(p)=yi,p [c,p u ]}{G(z, p) + αj t (z)} = G ( Zt i, P t i ) ( ) + αjt Z i t (14) represents the optimal profit from period t until T. As indicated in (14), (Zt i, P t i) satisfies both Zi t + d(pt i)) = y i and Pt i [c, P u ]. In order to characterize the optimal multi-period policy, we first need to prove three important lemmas, whose proofs are provided in the Appendix. The results in these lemmas will considerably simplify the proof of our main optimality theorem. In the first of the three lemmas (Lemma 4) we develop an obvious, but important, relation between the order-up-to inventory level at the beginning of a period and the corresponding leftover inventories at the end of that period. Suppose that the actual order-up-to inventory level y at the beginning of period t is selected to be larger than or equal to y(z), the optimal order-up-to level for the single-period problem. If the price is set optimally corresponding to y, then the riskless leftovers Zt at the end of period t will be greater than or equal to Z, the optimal riskless leftover for the single-period problem. Formally, LEMMA 4: For any given y y(z), H t (y ) = G(Zt, Pt )+αj t(zt ).IfJ t(z) is non-decreasing for z [, Z], then we have Zt Z. PROOF OF LEMMA 4: We need to study the sign of dg(z, p y (z)) {z=z t } { ( G(z, p) G(z, p) = + 1 )}. z p d {p=p t,z=zt } Let ỹ = Zt + d ( p(zt )).IfZt <Z, then it is obvious that ỹ<y and pỹ(zt )>p y (Zt ) (refer to Fig. 3). As dg(z,p y(z)) is decreasing in terms of p y (z), weget dg(z, p y (z)) { } > dg(z, p ỹ(z)) z=z { } t z=zt { ( G(z, p) G(z, p) = + 1 )} z p d { }. p=pỹ(z),z=zt Note that pỹ(zt ) = p(z t ), G(z,p) z {p=p(z t ),z=zt } and G(z,p) p {p=p(z t ),z=zt }. Thus, dg(z, p y (z)) G(z, p) > z. {p=p(z t ),z=zt } {z=z t }

9 Chen, Ray, and Song: Pricing and Inventory Control 15 THEOREM : Under Assumptions 1 and, for any t(1 t T), the following are true: Figure 3. Relationship between y and ỹ. As we assume that Zt <Zand J t (z) is non-decreasing for z [, Z], G(z, p y (z)) + αj t (z) is increasing at z = Zt. This is a contradiction with the definition of Zt. Hence, the desired result follows. In the following lemma (Lemma 5), for any given riskless leftovers z at the end of period t and the optimal replenishment policy for period t + 1, we characterize the expected total profit from period t + 1 till the end of the selling horizon. Let us define s t+1 and S t+1 as inventory levels for period t + 1 (S t+1 s t+1 ), and assume that (s t+1, S t+1 ) is the optimal inventory replenishment policy at the beginning of period t + 1. Thus, it is optimal for the retailer to order-up-to S t+1 at the beginning of period t + 1 if the initial inventory level is less than or equal to s t+1 and not to order anything otherwise. With this assumption, we can obtain the following property about J t (z) (refer to definition of J t (z) in Section ), where z is the riskless leftovers at the end of period t. LEMMA 5: If (s t+1, S t+1 ) is the optimal inventory replenishment policy for period t + 1, then J t (z ) K + J t (z 1 ) for any z 1 z. The last of the three lemmas (Lemma 6) is actually an extension of Lemma 5. If J t (z 1 ) K + J t (z ) for any z 1 z (as shown in the previous lemma), then Lemma 6 allows us to infer that for initial inventory level x y(z)at the beginning of period t, we do not need to order anything. LEMMA 6: If J t (z 1 ) K + J t (z ) for any z 1 z and J t (z) is non-decreasing in the range z [, Z], then H t (y 1 )> K + H t (y ) for any y(z) y 1 <y. Based on Lemmas 4, 5, and 6 and Theorem 1 for the singleperiod case, we can prove the following by induction on t. 1. J t (z) is non-decreasing and continuous for z [, y(z)] and J t (z 1 ) K + J t (z ) for any z 1 z.. For any given order-up-to inventory level y at the beginning of period t, there exists a P t (y) ( [c, P u ]) that maximizes G(z(y, p), p) + αj t (z(y, p)). Furthermore, P t (y) and Z t (y)(= y d(p t (y))) are continuous. 3. Let H t (y) = G(Z t (y), P t (y)) + αj t (Z t (y)) for any y. H t (y) is non-decreasing for y [, y(z)] and H t (y 1 )> K + H t (y ) for any y(z) y 1 <y. PROOF: We prove the theorem by induction on t.fort = T, the theorem is true as shown in Section 3. Suppose that it is true for period t + 1( T). Let S t+1 be the maximizer of H t+1 (y) for y [, + ) and s t+1 ( s t+1 S t+1 ) be the largest value such that H t+1 (s t+1 ) K + H t+1 (S t+1 ). If there exists no such s t+1 in the range [, S t+1 ], then we can assign any negative value to s t+1.ash t+1 (y) is non-decreasing and continuous for y [, y(z)] and H t+1 (y 1 )> K + H t+1 (y ) for any y(z) y 1 <y,wegets t+1 y(z), s t+1 y(z), and V t+1 (x) = { K + H t+1 (S t+1 ), x s t+1 H t+1 (x), x>s t+1. Thus, V t+1 (x) is non-decreasing and continuous for x [, y(z)]. Therefore, from the expression of J t (z),itisobvious that J t (z) is non-decreasing in the range z [, y(z)], and (s t+1, S t+1 ) is the optimal inventory replenishment policy at the beginning of period t+1. Combining this and Lemma 5, we get J t (z 1 ) K + J t (z ) for any z 1 z. This completes the proof of part (1) for period t. For any given order-up-to inventory level y ( ), the feasible region of (z, p) is {(z, p) z + d(p) = y, p [c, P u ]}. As G(z, p) + αj t+1 (z) is continuous on the closed compact set {(z, p) z + d(p) = y, p [c, P u ]}, there exists a (Zt, P t ) that maximizes G(z, p) + αj t+1(z). It is obvious that Zt = y d(pt ).IfZ t is not unique, then we assume it to be the one with the largest value among all the possible maximizers of G(z, p) + αj t+1 (z).asg(z, p) + αj t+1 (z) is jointly continuous in its two variables, P t (y) and Z t (y) are also continuous in y. We complete the proof of part () for period t. For any given order-up-to inventory level y( y y(z)), z y(z) for any feasible point (z, p) {(z, p) z + d(p) = y, p [c, P u ]}. AsJ t (z) is non-decreasing in the range z [, y(z)], for any (z, p y (z)) satisfying {(z, p y (z)) Z l z

10 16 Naval Research Logistics, Vol. 53 (6) Z, c p y (z) p(z)}, wehave d(g(z, p y (z)) + αj t (z)) { G(z, p) = + α dj t(z) z ( G(z, p) + 1 )} p d {p=py (z)}. Note that if the derivative of J t (z) does not exist at some point, then dj t (z) is understood as the left derivative of J t (z) at that point. Let Z t (y) be the one with the largest value among all the maximizers of G(z, p y (z)) + αj t (z). Then, P t (y) = p y (Z t (y)) p(z) where y = z + d(p(z)). Using this fact and following the method in the proof of Theorem 1, we can show that { G(z, p) z + α dj } t(z) {z=z t (y),p=p t (y)} for any y [, y(z)]. As either [G(z(y,p),p)+αJ t (z(y,p))] p = or dp t (y) =, dy dh t (y) dy { G(z, p) = + α dj }( ) t(z) z(y, p) z y + [G(z(y, p), p) + αj t(z(y, p))] p { G(z, p) = z {p=pt (y)} {z=z t (y),p=p t (y)} {z=zt (y),p=p t (y)} dp t (y) dy + α dj } t(z). {z=z t (y),p=p t (y)} Thus, H t (y) is non-decreasing for y [, y(z)]. For any y(z) y 1 <y, based on the results in part (1) for period t and Lemma 6, we can conclude that H t (y 1 ) H t (y )> K, which completes the whole proof for period t. Based on the above theorem, we can then generate the following corollaries. COROLLARY 1: Under Assumptions 1 and, if K>, then (s t, S t, P t ) is the optimal joint inventory replenishment and pricing policy at the beginning of period t, where S t is the maximizer of H t (y), as defined in (14), and s t ( S t ) is the largest value such that H t (s t ) H t (S t ) K. If the initial inventory level x before ordering for period t is less than or equal to s t, then the optimal order-up-to inventory level y = S t.ifx > s t, then y = x. The optimal price P t (y) is given by p y (Z t (y)) [c, P u ], where y is the optimal order-up-to level for period t. COROLLARY : Under Assumptions 1 and, if K =, then the base-stock policy with parameter y(z) is optimal at the beginning of each period. The optimal price will be given by p(z) for any given initial inventory level x y(z), and a price P t (x) will be charged for any given initial inventory level x>y(z). Hence, our analysis is able to determine the optimal periodic-review pricing and inventory control policy for the retailer, even in the presence of lost sales and fixed ordering cost. 5. ILLUSTRATIVE NUMERICAL EXAMPLES AND SENSITIVITY ANALYSIS In this section, we present numerical examples to illustrate how changes in the fixed ordering cost affect the optimal policy and profit values and the potential profit loss if one uses backordering approximation for the lost sales model. We also ascertain the magnitude of the profit advantage due to our dynamic pricing policy over a static policy (i.e., a pricing policy that sticks to one price over time). Throughout this section we assume: (i) a linear demand model d(p) = 5 p, which clearly satisfies Assumption 1, and (ii) the random term ɛ to be uniformly distributed in [, ], which satisfies Assumption. The value of the fixed ordering cost is varied to represent different test scenarios (explained below). The other parameters are kept constant at the following levels: h =.4, b = 1.5, v = c =.5, α =.95, and T = Effects of Fixed Ordering Cost (K) onthe Optimal Policy Values and Profits Since one of the major contributions of our model is incorporation of the fixed ordering cost (K), we first focus our attention on understanding how changes in K affect the optimal policy values and profits in our framework. For this we use three values of K:, 15, and 3. In Table 3, we present (s t, S t ) values for periods 1 t 1 for all the three values, while the optimal price and profit functions with respect to the initial inventory levels before ordering are shown in Figs. 4 9 (only for periods t 1). Note that for all the figures the first row represents periods 1 8, with period 1 being the leftmost one and period 8 the rightmost one. A similar order is followed for the second (periods 7 5) and third rows (periods 4 ). For example, consider the optimal price and profit functions for K = 3 in Figs. 4 and 5, respectively. The horizontal axis represents the initial inventory level before ordering (x). We know the (s t, S t ) values from Table 3. For x s t, the optimal order-up-to inventory level (y) iss t. This explains the constant values until s t in Figs. 4 and 5, since all of them are based on y = S t.forx>s t, there is no need to order, and so the optimal order-up-to level is the same as the initial level (i.e., y = x). The maximum value of the profit function

11 Chen, Ray, and Song: Pricing and Inventory Control 17 Table 3. Optimal policies for the three test scenarios. t = 1 t = 9 t = 8 t = 7 t = 6 K = (6.17, 6.18) (6.17, 6.18) (6.17, 6.18) (6.17, 6.18) (6.17, 6.18) K = 15 (17.35, 6.18) (18.43, 44.77) (18.11, 47.6) (18.13, 46.7) (18.17, 46.9) K = 3 (13.9, 6.18) (18.19, 44.77) (16.59, 57.44) (15.8, 65.68) (16.75, 5.1) t = 5 t = 4 t = 3 t = t = 1 K = (6.17, 6.18) (6.17, 6.18) (6.17, 6.18) (6.17, 6.18) (6.17, 6.18) K = 15 (18.13, 46.9) (18.16, 46.85) (18.14, 46.89) (18.15, 46.86) (18.15, 46.88) K = 3 (16.47, 59.8) (16.17, 59.9) (16.5, 55.65) (16.44, 58.56) (16.3, 58.5) is obtained at y = S t. Note that the dynamic programming formulation implies that for period t = 1, our model is basically a single-period newsvendor problem. Analysis of the figures results in the following conclusions. 1. While the price seems to be decreasing for K =, it is clearly non-monotone for K>. In the latter scenario (K >), the optimal price is constant for x s t (because an order will be issued to raise x to S t at which p = P(y = S t )) and then seems to be multimodal. However, for large values of x, the optimal price is almost constant (< P(y = S t )). Also, note that as the length of the selling horizon T, the optimal price function becomes almost stationary.. Since for K =, s t = S t (the difference in those values in Table 3 is because of discretization during computation), so the profit function is initially constant (the maximum value) and then decreasing. For K>, the profit function is again constant until s t, but reaches its maximum at S t >s t. Also, while the profit function seems to be almost linearly decreasing for large values of x, it is non-monotone for s t x S t. Analyzing Table 3 we note that, as K increases, s t decreases, while S t increases t. This result is quite intuitive. As the fixed ordering cost increases, the retailer would like to order as few times as possible. This can be achieved by lowering s t, which reduces the probability of placing an order in period t (recall that an order is placed when the initial inventory level s t ). A higher S t, on the other hand, reduces the chance of an order in period (t + 1), since it will probabilistically result in a higher initial stocking level for period (t + 1) (refer to Lemmas 3 and 4). Our numerical experiments also clearly indicate that as T, the values of s and S become almost constant. For the test scenarios mentioned above, the stationary values of (s, S) are as follows: K = (6.17, 6.18), K = 15 (18.15, 46.87), K = 3 (16.4, 58). 5.. Profit Impact of Dynamic Retail Pricing In this subsection, we numerically investigate the extent of benefit that accrues to the retailer as a result of dynamically changing the retail price, rather than sticking to a single optimal price throughout the selling horizon. In order to do so, first we determine the single-period profit-maximizing retail price for the retailer. Then we calculate the profit for the retailer under the assumption that this retail price will be maintained throughout the selling horizon, but the replenishment action would be taken optimally for each period (i.e., inventory control policy would still be dynamic). We term this fixed pricing strategy as static pricing and denote the profit for this strategy by Profit static pricing (x), for any given initial inventory level x( ) at the beginning of the selling horizon. We then contrast Profit static pricing (x) with the optimal dynamic joint pricing and inventory control profit, Profit dynamic pricing (x), from our model in Section 4. It is obvious that Profit static pricing (x) Profit dynamic pricing (x). We define the profit deviation ratio as Profit dynamic pricing (x) Profit static pricing (x) Profit dynamic pricing (x) 1 and report several examples of the potential profit loss for different values of K and x for t = 1 in Table 4. We can see from Table 4 that the profit impact of dynamic pricing is significant when the fixed ordering cost associated with an order (K) is large, but is rather small when K is not substantial. In fact, when K is large the retailer would be unwilling to order frequently. Instead, it would vary the retail price charged to the customers as the primary lever to maximize its profit. On the other hand, for very low values of K, the retailer would order more frequently, but keep the price as well as the inventory level at the beginning of each period almost at a constant level. This effect is evident from Table 3 and Figs. 4, 6, and 8. Hence, it is not surprising that the profit penalty of static pricing increases with the magnitude of K. Another plausible explanation of the impact of K on the value of dynamic pricing might be that when K increases, the total

12 18 Naval Research Logistics, Vol. 53 (6) Figure 4. Optimal price function in terms of initial inventory level for K = 3. profit decreases; hence, the percentage of profit gained by dynamically changing the retail price becomes larger (evident from comparing Figs. 5, 7, and 9) Backordering Approximation for the Lost Sales Model As we have indicated in Section, lost sales models are acknowledged to be technically more challenging than assuming unmet demands in a period are backordered. Chen and Simchi-Levi [] have already characterized the optimal joint pricing and inventory control policy for a problem similar in structure to this paper under backordering assumption. Hence, it might be interesting to study the profit performance of using the policy generated by such a backordering model as approximation in our lost sales model, under the assumption that the cost of backordering is equal to the penalty cost for lost sales and all other parameter values remain the same. The different test scenarios for this study are generated by changing the value of the fixed ordering cost (K) and the

13 Chen, Ray, and Song: Pricing and Inventory Control 19 Figure 5. Optimal profit function in terms of initial inventory level for K = 3. initial inventory level (x), as indicated in Table 5. For each scenario, we compute the optimal (s, S) policy and the optimal retail price at the beginning of each period under the backordering assumption. Then we plug in this policy into our lost sales model to calculate the corresponding Profit backordering (x) for the particular initial stocking level x( ) at the beginning of the selling horizon, i.e., the beginning of period t = 1. It is again obvious that Profit backordering (x) Profit lost sales (x), the optimal profit based on the model in Section 4. Defining the profit deviation ratio as in the previous subsection, we report several examples of potential profit loss corresponding to different scenarios in Table 5. Also note that for each scenario, the (s, S) policy value at the bottom of Table 5 denotes the optimal inventory policy under backordering/lost sales assumption at the beginning of the selling horizon, i.e., t = 1. From Table 5 it is obvious that the deviations in terms of both profits and policy values might be quite substantial, especially when the fixed ordering cost is large enough and