Optimal Pricing and Replenishment in a Single-Product Inventory System

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1 Optimal Pricing and Replenishment in a Single-Product Inventory System Hong Chen Cheung Kong Graduate School of Business, China Owen Q. Wu Sauder School of Business, University of British Columbia, Canada David D. Yao IEOR Dept., Columbia University, ew York, USA March 1, 24 Abstract We study an inventory system that supplies price-sensitive demand modeled by Brownian motion, focusing on the optimal pricing and inventory replenishment decisions, under both long-run average and discounted objectives. Analytical solutions are obtained in all cases, and related to or contrasted against previously known results. In addition, we bring out the interplay between the pricing and the replenishment decisions, and the way they react to demand uncertainty. We show that the joint optimization of both decisions may result in significant profit improvement over the traditional way of making the decisions separately or sequentially. We also show that multiple price changes will only result in a limited profit improvement over a single price. Keywords: Joint pricing-replenishment decision, price sensitive demand, Brownian model. Supported in part by University Graduate Fellowship from the University of British Columbia. Supported in part by SF grant DMI Part of this author s research was undertaken while at the Dept of Systems Engineering and Engineering Management, Chinese University of Hong Kong, and supported by HK/RGC Grant CUHK4173/3E. 1

2 1 Introduction We study a single-product continuous-review inventory model with price-sensitive demand. The cumulative demand process is modeled by a Brownian motion with a drift rate that is a function of the price. Replenishment is instantaneous, and demands are satisfied immediately upon arrival. Consequently, the replenishment follows a simple order-up-to policy, with the order-up-to level denoted by S. We allow the pricing decisions to be dynamically adjusted over time. Specifically, we divide S into equal segments, with a given integer. For each segment, there is a price; and all prices are optimally determined, jointly with the replenishment level S, so as to maximize the expected long-run average or discounted profit. There has been a substantial and growing literature on the joint pricing and inventory control. We refer the reader to two recent survey papers, Yano and Gilbert (22), and Elmaghraby and Keskinocak (23). Our review below will focus on those works that relate closely to our study. Whitin (1955), Porteus (1985a), Rajan et al. (1992), among others, study demands that are deterministic functions of prices. Whitin (1955) connects pricing and inventory control in the EOQ (economic order quantity) framework, and Porteus (1985a) provides an explicit solution for the linear demand instance. Rajan et al. (1992) investigates continuous pricing for perishable products for which demands may diminish as products age. Other works study stochastic demand models. Li (1988) considers a make-to-order production system with price-sensitive demand. Both production and demand are modeled by Poisson processes with controllable intensities. The control of demand intensity is through pricing. A barrier policy is shown to be optimal: when the inventory level reaches an upper barrier, the production stops; when the inventory level drops to zero, the demand stops (or the demand is lost). It is also shown that the optimal price is a non-increasing function of the inventory level. Federgruen and Heching (1999) examines a model in which the firm periodically reviews inventory and decides both the replenishment quantity and the price to charge over the period. The replenishment cost is linear, without a fixed setup cost. The prices are changed only at the beginning of each period (as opposed to the continuous pricing scheme in Rajan et al. 1992). It is shown that a base-stock list-price policy is optimal for both average and discounted objectives. Earlier related works include Zabel (1972) and Thowsen (1975). Thomas (1974), Polatoglu and Sahin (2), Chen and Simchi-Levi (23a,b), Feng and 2

3 Chen (23), and Chen et al. (23) extend the above model to include a replenishment setup cost. In the full-backlog setting, it is first conjectured in Thomas (1974), and then proved in Chen and Simchi-Levi (23a), that the (s, S, p) policy is optimal for additive demand (a deterministic demand function plus a random noise) in a finite horizon. Chen and Simchi- Levi (23b) further proves the optimalify of the stationary (s, S, p) policy for general demand in an infinite horizon. Feng and Chen (23) proves the optimality of (s, S, p) policy under more general demand functions, but restricting the prices to a finite set. Assuming lost sales, Polatoglu and Sahin (2) obtains rather involved optimal policies under a general demand model and provides restrictive conditions under which the (s, S, p) policy is optimal. Chen et al. (23) proves that (s, S, p) policy is optimal under additive demand and lost sales. Continuous-review models are studied in Feng and Chen (22), and Chen and Simchi-Levi (23c). Feng and Chen (22) models the demand as a price-sensitive Poisson process. Pricing and replenishment decisions are made upon finishing serving each demand, but the prices are restricted to a given finite set. An (s, S, p) policy is proved optimal, with the optimal prices depending on the inventory level in a rather structured manner. Chen and Simchi-Levi (23c) generalizes this model by considering compound renewal demand process with both the interarrival times and the size of the demand depending on the price. It is shown that the (s, S, p) policy is still optimal. Our work differs from the above papers in two aspects. First, we model the demand process by Brownian motion, with a drift term being a function of the price. The Brownian model, with its continuous path, is appropriate for modeling fast-moving items. It is a natural model when demand forecast involves Gaussian noises. Furthermore, it allows us to bring out explicitly the impact of demand variability in the optimal pricing and replenishment decisions, whereas results along this line are quite limited in the existing literature. (For other works that model production-inventory systems using Brownian motion, we refer to Puterman (1975) and Harrison (1985).) Second, we examine the number of price changes allowed as inventory depletes, and demonstrate that using a small number of prices, optimally determined, is usually good enough. Some of the key findings and new insights from our study include: Demand variability incurs an additional inventory holding cost. As demand variability increases, the optimal price decreases and the optimal replenishment level increases. 3

4 The traditional way of separating the pricing and replenishment decisions could result in significant profit loss, as compared with the joint decision. Multiple price changes will only result in a limited profit improvement over a single price (when both are optimally determined). The relative improvement, however, becomes more significant in applications where the profit margin is low. The rest of the paper is organized as follows. In Section 2, we present a formal description of our model, in terms of the demand and inventory processes, the cost functions, and the pricing and replenishment decisions. In Section 3, we study the optimal pricing and replenishment decisions under the long-run average objective. We start from making these decisions separately, so as to highlight the comparisons against prior studies and known results, and then present the joint optimization model and demonstrate the profit improvement. In Section 4, we present analogous results under the discounted objective, emphasizing the contrasts against the average objective case. We conclude the paper pointing out possible extensions in Section 5. 2 Model Description and Preliminary Results We consider a continuous-review inventory model with a price-sensitive demand. The objective is to determine the inventory replenishment and pricing decisions that strike a balance between the sales revenue and the cost for holding and replenishing inventory over time, so as to maximize the expected long-run average or discounted profit. The specifics of the demand model, the cost parameters and the control policy are elaborated in the following three subsections. 2.1 The Demand Model The subject of our study is a single-product inventory system supplying a price-sensitive demand stream. The cumulative demand up to time t is denoted as D(t), and modeled by a diffusion process: D(t) = t λ(p u )du + σb(t), t, (1) where p t is the price charged at time t; λ(p t ) is the demand rate at time t, which is a decreasing function of p t ; B(t) denotes the standard Brownian motion; and σ is a positive constant measuring the variability of the demand (or the error of demand forecast). This Brownian demand model can be related to other discrete (i.e., integer-valued) demand processes through strong approximation (Csörgő and Horváth 1993). Consider, for instance, a 4

5 Poisson demand process with an instantaneous rate λ(p t ), and write the cumulative demand as A ( t λ(p u )du ), where A( ) denotes the Poisson process with a unit rate. Then the strong approximation implies that a version of A on a suitable probability space satisfies sup t T A ( t λ(p u )du ) t λ(p u )du B ( t λ(p u )du ) = O(log(T )), where B is a standard Brownian motion defined on the same probability space. If the process A follows a more general probability law (i.e., not necessarily Poisson), then the order of approximation will be O(T 1/r ) for a constant r (2, 4). Using Brownian model to approximate discrete (point) process has been a standard approach in many other applications, stochastic networks in particular; refer to, e.g., Harrison (1988, 23), and Chen and Yao (21). Let P and L denote, respectively, the domain and the range of the demand rate function λ( ). Both are assumed to be intervals of R + (the set of nonnegative real numbers); in addition, L. Assumption 1 (on demand rate) The demand rate λ(p) and its inverse p(λ) are both positivevalued, strictly decreasing, and twice continuously differentiable in the interior of P and L, respectively. The revenue rate r(λ) = p(λ)λ is strictly concave in λ. Many commonly-used demand functions satisfy the above assumption, including the following examples, where the parameters α, β and δ are all positive: The linear demand function λ = α βp, p [, α/β]: r(λ) = α β λ 1 β λ2 is strictly concave; The exponential demand function λ = αe βp, p : r(λ) = 1 β λ log(λ/α) is strictly concave; The power demand function λ = βp δ, p : r(λ) = λ 1 δ +1 β 1 δ δ 1. is strictly concave if 2.2 Cost Parameters Let h be the cost for holding one unit of inventory for one unit of time. Let c(s) be the cost to replenish S units of inventory. Assumption 2 (on replenishment cost) The replenishment cost function c(s) is twice continuously differentiable and increasing in S for S (, ). The average cost a(s) = c(s)/s is strictly convex in S, and a(s), as S. 5

6 Consider a special case: c(s) = K + cs δ, S >, where K, c, δ >. When δ = 1, this is the most commonly used linear function with a setup cost K. The average cost function, a(s) = K S + csδ 1 is convex if δ (, 1] [2, ]. Furthermore, as we shall demonstrate below, when S satisfies a (S), which is the primary case of interest, a(s) is convex for all δ >. To see this, consider δ (1, 2), and note that and when a (S), we have a (S) = 1 S 2 ( K + c(δ 1)Sδ ); a (S) = 2 δ ( S δ K + c(δ 1)Sδ) a (S)(2 δ). S 2.3 Pricing and Replenishment Policies Assume replenishment is instantaneous, i.e., with zero leadtime. We further assume that all orders (demand) will be supplied immediately upon arrival; i.e., no back-order is allowed, or there is an infinite back-order cost penalty. (Our results extend readily to the back-order case; refer to Section 5.) The replenishment follows a continuous-review, order-up-to policy. Specifically, whenever the inventory level drops to zero, it is brought up to S instantaneously via a replenishment, where S is a decision variable. We shall refer to the time between two consecutive replenishments as a cycle. We adopt the following dynamic pricing strategy. Let 1 be a given integer, and let S = S > S 1 > > S 1 > S =. Immediately after a replenishment at the beginning of a cycle, price p 1 is charged until the inventory drops to S 1 ; price p 2 is then charged until the inventory drops to S 2 ;...; and finally when the inventory level drops to S 1, price p is charged until the inventory drops to S =, when another cycle begins. The same pricing strategy applies to all cycles. For simplicity, we set S n = S( n)/. That is, we divide the full inventory of S units into equal segments, and price each segment with a different price as the inventory is depleted by demand. In summary, the decision variables are: (S, p), where S R +, and p = (p 1,..., p ) P. Within a cycle, we shall refer to the time when the price p n is applied as period n. 2.4 The Inventory Process Without loss of generality, suppose at time zero the inventory is filled up to S. We focus on the first cycle which ends at the time when inventory reaches zero. Let T =, and let T n be 6

7 the first time when inventory drops to S n : T n := inf{t : D(t) = ns/}, n = 1, 2,...,. The length of period n is therefore τ n := T n T n 1. Since T n s are stopping times, by the strong Markov property of Brownian motion, τ n is just the time during which S/ units of demand has occurred under the price p n. That is, τ n dist. = inf{t : λ(p n )t + σb(t) = S/}. (2) Let X(t) denote the inventory-level at t. We have, X(t) = S D(t), t [, T ). Since our replenishment-pricing policy is stationary, X(t) is a regenerative process with the replenishment epochs being its regenerative points. We conclude this section with a lemma, which gives the first two moments and the generating function of the stopping time τ n. (The proof is in the appendix.) Lemma 2.1 For the stopping time τ n in (2), we have E(τ n ) = S λ n, (3) E(τn) 2 = σ2 λ 2 E(τ n ) + E 2 (τ n ) = σ2 S n λ 3 n E(e γτn ) = exp[ where λ n = λ(p n ) >, and γ > is a parameter. 3 Long-Run Average Objective λ 2 n +2σ 2 γ λ n σ 2 + S2 2 λ 2, (4) n S ], (5) To optimize the long-run average profit, thanks to the regenerative structure of the inventory process, it suffices for us to focus on the first cycle. Recall that period n refers to the period in which the price p n applies, and the inventory drops from S n 1 = ( n+1)s to S n = ( n)s. We first derive the total inventory over this period. Applying integration by parts, and recognizing dx(t) = dd(t), X(T n 1 ) = S n 1, X(T n ) = S n, 7

8 we have Tn T n 1 X(t)dt = T n S n T n 1 S n 1 Tn T n 1 tdx(t) Tn Tn = T n S n T n 1 S n 1 T n 1 dx(t) + (t T n 1 )dd(t) T n 1 T n 1 Tn = τ n S n + (t T n 1 ) [ λ(p n )dt + σdb(t) ]. T n 1 A simple change of variable yields Tn T n 1 (t T n 1 )λ(p n )dt = τn uλ(p n )du = 1 2 λ(p n)τ 2 n; whereas [ Tn ] E (t T n 1 )db(t) T n 1 [ Tn ] = E tdb(t) E [ ] [ T n 1 E B(Tn ) B(T n 1 ) ] =, T n 1 follows from the martingale property of B(t) and the optional stopping theorem. Let v n (S, p n ) denote the expected profit (sales revenue minus inventory holding cost) during period n. Then, making use of the above derivation, along with Lemma 2.1, we have v n (S, p n ) = p [ Tn ] ns E hx(t)dt T n 1 = p ns he[τ n]s n 1 2 hλ(p n)e[τn] 2 = p ns hs2 ( n ) 2 hσ2 S λ(p n ) 2λ(p n ) 2, (6) ote in the above expression, the first term is the sales revenue from period n, the second term is the inventory holding cost attributed to the deterministic part of the demand (i.e., the drift part of the Brownian motion), and the last term is the additional holding cost incurred by demand uncertainty. For ease of analysis, below we shall often use {µ n = λ(p n ) 1, n = 1,..., } as decision variables and denote µ = (µ 1,..., µ ) M and M = {λ(p) 1 : p P}. Then, the long-run average objective can be written as follows: n=1 V (S, µ) = v n(s, p n ) c(s) S n=1 µ = n n=1 [p( 1 µ n ) hs ( n )µ n hσ2 n=1 µ n ] 2 µ2 n a(s) The additional holding cost due to demand uncertainty is represented by hσ2 n=1 µ2 n 2. n=1 µn 8. (7)

9 For the special case when = 1, the price and the demand rate are both constants in a cycle (and hence constant throughout the horizon). The objective function (7) takes a simpler form. For comparison with some classical work, we use λ as the decision variable and denote the long-run average profit under single-price policy as V (S, λ). Then, V (S, λ) = v 1(S, p(λ)) c(s) S/λ = r(λ) hs 2 λa(s) hσ2 2λ. (8) The classical EOQ model only consists of the second and third terms in (8), which are the total cost if the demand is deterministic with a constant rate. Whitin (1955) and Porteus (1985a) considered the price-sensitive EOQ model, which involves the first three terms in (8). Our model gives rise to an additional holding cost due to demand variability. 3.1 Optimal Replenishment with Fixed Prices In this case, the set of prices p, or equivalently, µ is given, and the firm s problem is max S> V (S, µ). Under Assumption 1, V (S, µ) is strictly concave in S. ote that V (S, µ) as S or S (the latter is due to Assumption 2 that a(s) as S ). Thus, the unique optimal replenishment level is determined by the first-order condition: ( S = a 1 h 2 ( n + 1 ) 2 )µ n, (9) where a 1 is well-defined since a( ) is strictly convex and a ( ) is strictly increasing under Assumption 2. In practice, the average cost as a function of quantity is usually first decreasing (due to economy of scale) and then increasing (due to capacity or other technological restrictions). However, at the optimal replenishment level, we have a (S ) < for any fixed prices. This observation helps to reduce the search space when the replenishment level is optimized jointly with pricing decisions (see Section 3.3). The way that the demand variability and the holding cost impact the optimal replenishment level can be readily derived from (9). Proposition 3.1 With prices fixed, (i) the optimal replenishment level S is independent of σ, and decreasing in h and in p n (for any n); (ii) the optimal profit is decreasing in σ and h. n=1 9

10 Proof. Part (i) is obvious from (9). (ote, in particular, that the additional holding cost due to demand variability is independent of the replenishment level.) For (ii), consider σ 1 σ 2, and denote the corresponding maximizers as S1 and S 2 and the objective values as V (S 1, µ, σ 1) and V (S2, µ, σ 1), respectively. We have V (S1, µ, σ 1) V (S2, µ, σ 1) V (S2, µ, σ 2), where the first inequality is due to the maximality of S 1 and the second one follows immediately from the objective function in (7). The case of decreasing in h is completely analogous. Example 3.1 Consider the linear replenishment cost: c(s) = K + cs, where K, c >. The first order condition in (9) leads to the familiar EOQ formula: S = 2Kλa ( h, where λ 2 2n + 1 ) 1 a = 2 µ n. (1) n=1 In the standard EOQ model, the demand rate is taken as the average demand per time unit, which in our setting becomes λ = ( 1 ) 1. µ n (11) n=1 Suppose that the fixed prices satisfy p 1 p 2 p (or µ 1 µ 2 µ ), i.e., a lower price is charged when the inventory level is higher (we will see this price pattern in Proposition 3.3). Higher weights are given to smaller µ n s in (1), while µ n s are equally weighted in (11). Hence, λ λ a. This implies that the standard EOQ with a time-average demand rate may lead to a replenishment level lower than the optimally desirable. Example 3.2 Let c(s) = 1 + 5S, λ(p) = 5 p, h = 1 and = 1. In Figure 1(a), the EOQ quantity is shown as a function of the single fixed price, S = 1 1 2p. Figure 1(b) illustrates the optimal profit corresponding to various levels of demand variability. The peak of each V curve is reached when both the price and the replenishment level are jointly optimized. When the price deviates far away from the optimum, the profit falls dramatically. Figure 1: Optimal replenishment with a fixed price 1

11 3.2 Optimal Pricing with a Fixed Replenishment Level The Single-Price Problem Here = 1. We want to optimize the single price, or equivalently, the demand rate, as follows: max λ L V (S, λ) = r(λ) hs 2 λa(s) hσ2 2λ. Under Assumptions 1 and 2, V (S, λ) is strictly concave in λ. Assuming an interior solution (which is rather innocuous, since the last two terms in the objective forbid λ to be extreme), the unique optimal λ follows from the first-order condition: We have the following proposition. r (λ) + hσ2 2λ 2 = a(s). (12) Proposition 3.2 Suppose a single price is optimized while the replenishment level is held fixed. (i) If the optimal price p is in the interior, then the marginal revenue at p is no larger than the average replenishment cost, i.e., r (λ(p )) a(s). (ii) The optimal price p is decreasing in σ and h. (iii) The optimal price p is increasing in a( ); if a(s) is decreasing in S, then p is decreasing in S. (iv) The optimal profit is decreasing in σ and h. Proof. The statement in (i) follows immediately from (12). For (ii) and (iii), note that 2 V λ σ >, 2 V λ h = σ2 >, and 2 V 2λ 2 λ (a(s)) = 1 <. That is, the objective function V is supermodular in (λ, σ), supermodular in (λ, h) and submodular in (λ, a(s)). Hence, (ii) = hσ λ 2 and (iii) follow from standard results for super/submodular functions (Topkis 1978). For (iv), follow a similar argument to the one that proves Proposition 3.1 (ii). Example 3.3 If λ(p) = βp 1, c(s) = K + cs, where β, K, c >, then r(λ) = β, and the first-order condition (12) becomes leading to the optimal price hσ 2 2λ 2 = K S + c, p = β λ = β σ which is clearly decreasing in σ, h and S. 2( K S + c), h 11

12 Example 3.4 If λ(p) = α βp, where α, β >, then r(λ) = (α λ)λ/β, and the first-order condition in (12) becomes which can be simplified to α 2λ β + hσ2 2λ 2 = a(s), f(λ) 4λ 3 + 2[βa(S) α]λ 2 hβσ 2 =. ote that f() <, f(+ ) = +, f (λ) = 4λ[3λ + βa(s) α]; hence, f(λ) is either strictly increasing or first decreasing and then increasing. In either case, the above equation has a unique solution λ in (, ). If λ α, it is the optimal demand rate, and the optimal price is p = (α λ )/β. If λ > α, it can be shown that V is increasing in λ for λ [, λ ], and therefore the optimal demand rate is α and the optimal price is zero. If σ or h increases or a( ) decreases, then f(λ) decreases for any λ; and consequently, λ increases. This illustrates the monotonicity in (ii) and (iii) of Proposition 3.2. Example 3.5 Under the same setting as Example 3.2, Figure 2 plots the optimal price and the corresponding profit against various levels of replenishment and demand variability. The results depict the qualitative trends in Proposition 3.2. The peak of each V curve is reached when the price and inventory are jointly optimized. In contrast to Figure 1(b), the profit here appears less sensitive to the replenishment level, as long as the pricing decision is optimized. Figure 2: Optimal price with a fixed replenishment level The -Price Problem In this case, the decision variables are the set of prices p, or equivalently, µ. Specifically, the problem is max V (S, µ) = µ M n=1 [p( 1 µ n ) hs ( n )µ n hσ2 n=1 µ n ] 2 µ2 n a(s). (13) The strict concavity of the revenue function r(λ) (Assumption 1) implies that the function p( 1 µ ) is strictly concave in µ. This is because r (λ) = 2p (λ) + λp (λ) and d 2 p( 1 µ )/dµ2 = 1 (2p ( 1 µ 3 µ )+ 1 µ p ( 1 µ )) have the same sign. Thus, the numerator of the objective is strictly concave in µ. The ratio of a positive concave function over a positive linear function is known to be 12

13 pseudo-concave (Mangasarian 197). Hence, V (S, µ) is pseudo-concave in µ. Furthermore, it is strictly pseudo-concave in the sense that µ V (S, µ 1 )(µ 2 µ 1 ) V (S, µ 2 ) < V (S, µ 1 ). (14) Hence, the optimal µ must be unique. (This follows from a straightforward extension of Mangasarian (197) from pseudo-concavity to strict pseudo-concavity.) Suppose the optimality is achieved at an interior point. Then the optimal µ is uniquely determined by the first-order condition: ṽ n (S, µ n ) µ n = k=1 ṽk(s, µ k ) c(s) k=1 µ, n = 1,...,, (15) k where ṽ n (S, µ n ) = v n (S, p( 1 µ n )). ote that the right side of the above does not depend on n. This implies that the optimal pricing must be such that the marginal profit is the same across all periods (assuming an interior optimum). The following proposition describes the basic optimal pricing pattern. Proposition 3.3 For any replenishment quantity S, the optimal prices are increasing over the periods, i.e., p 1 p 2 p. Proof. Since p n = p( 1 µ n ) is increasing in µ n, it suffices to show that the optimal µ n is increasing in n. But this follows immediately from (13), since given any (µ 1,..., µ ), rearranging these variables (but not changing their values) in increasing order will maximize the term n=1 nµ n in the numerator, while all other terms remain unchanged. Hence, within each cycle, the optimal prices are increasing over the periods. The following lemma gives an inequality to bound the price increments. The inequality can be derived from the equal marginal profit property, but it still holds without the interior optimum assumption, and its proof is in the appendix. Lemma 3.1 For fixed S, let µ be the optimal prices. Then for 1 m < n, µ n µ m S(n m) ( σ 2 + B ) h S(n m) σ 2, where B = inf { d2 p( 1 µ ) dµ 2 : µ [µ 1, µ ] }. ext we explore the impact of the parameters, σ, h and S on the optimal prices. We shall write µ (σ), µ (h) and µ (S) to emphasize the dependence of the maximizer of (13) on these parameters. The proof of the next lemma is also in the appendix. 13

14 Lemma 3.2 Assuming interior optimum, (i) µ (σ) is differentiable and dµ n (σ) dσ k=1 < if and only if µ k (σ)2 2µ n(σ) µ k (σ) < ; (16) k=1 (ii) µ (h) is differentiable and dµ n (h) dh S < if and only if (n k)µ k + ( µ 2 k σ2 k=1 k=1 2 µ nµ k ) <. (17) (iii) µ (S) is differentiable and dµ n(s) ds < if and only if h 2 (n k)µ k (S) < a (S); (18) k=1 The results provided in the above lemma are local, in the sense that they only hold for parameter values that satisfy the specific conditions in question. evertheless, for µ 1 and µ some of these results can be strengthened; see (i)-(iii) of the following proposition. Proposition 3.4 Assuming interior optimum, (i) The highest price p (σ) is decreasing in σ. (ii) The lowest price p 1 (h) is decreasing in h. (iii) If a (S) <, then p 1 (S) is decreasing in S; (iv) The optimal profit is decreasing in σ and h. Proof. (i) By Lemma 3.2, the inequality in (16) always holds for n =, as µ component of µ following Proposition 3.3. Hence µ is decreasing in σ. is the largest (ii) Denote the left side of (17) by L n. We will show that L 1 + L < and L 1 < L, which then implies L 1 <. L 1 + L = S = S <, ( + 1 2k)µ k + [ ] σ2 µ k 2 (µ 1 + µ )µ k k=1 k=1 [ ( 1)(µ 1 µ ) + ( 3)(µ 2 µ 1) + + (µ 2 µ +3 + σ 2 k=1 [ ] µ k (µ k µ ) µ 1µ k 14 2 ) ]

15 where the last inequality follows from µ n increasing in n. ext, note that L 1 L = [ S(1 ) ] + σ 2 (µ µ 1) µ k, where the last inequality follows from Lemma 3.1. Hence L 1 <, and p 1 (h) is decreasing in h. (iii) By Lemma 3.2, if a (S) < and n = 1, the inequality in (18) always holds. Hence µ 1 is decreasing in S. (iv) The proof is completely analogous to the proof for Proposition 3.1 (ii). The proof of part (i)-(iii) of the above proposition relies on the interior optimum assumption in Lemma 3.2, but we find through numerical tests that these monotonicity results hold even without the interior optimum assumption. The next example illustrates Proposition 3.3 and 3.4 and provides comparison plots for Propositions 3.4 and 3.2. Example 3.6 (a) Let p(λ) = 5 1λ, c(s) = 5 + 5S, S = 1, h = 2, and let σ vary from to 2. For = 4, the optimal prices are plotted in Figure 3 (a). The highest price is decreasing in σ while the others are not. There also exist other instances in which all the prices are decreasing in σ. The demand variability incurs additional holding cost hσ 2 µ 2 n µ. When σ increases, in order to balance out the increase of this cost, intuitively we need to decrease µ and at the same time decrease the spread of µ (i.e., µ µ 1 ). The composite effect is mixed except for µ. (b) Let p(λ) = 5 1λ, c(s) = 5 + 5S, S = 1, σ = 1, and let h vary from to 3. The optimal prices are shown in Figure 3 (b). The lowest price is decreasing in h while the others are not. The spread of the prices increases significantly with the increase in h. (c) Let p(λ) = 5 λ, c(s) = 1 + S, h =.5, σ = 3, and let S vary from to 4. The optimal prices are shown in Figure 3 (c). ote that a (S) < for all S. The lowest price is decreasing in S while the others are not. The spread of the prices increases in S. k=1 Figure 3: Optimal prices and optimal single price All the above examples show p 1 < < p. For comparison, the optimal price for the single-price problem are also plotted as p in Figure 3. Dynamic pricing obviously produces more profit than using a single fixed price, since the latter is a special case of the dynamic pricing strategy. To quantify the potential benefit of dynamic pricing is the purpose of the following proposition. 15

16 Proposition 3.5 For fixed S, let µ be the optimal prices, and let V be the optimal average profit defined in (13). Then, where µ = 1 ( V V1 hs2 1 2) 12 µ ( σ 2 + B ), h n=1 µ n and B = inf { d2 p( 1 µ ) dµ 2 : µ [µ 1, µ ] }. Proof. We consider a feasible single-price policy that charges a price p( µ 1 ), and compare this with the optimal -price policy. We show the latter has a lower average revenue, a higher additional holding cost due to demand variability, and the same average ordering cost. Hence, for the optimal -price policy to yield a higher profit, it must have a lower average holding cost attributed to the deterministic part of the demand. By the same reasoning, the difference between these two average holding-cost terms corresponding to the two policies constitute an upperbound on the improvement from the single-price policy to the -price policy. Specifically, the average profit of charging a single price p( µ 1 ) is We have V V 1 V 1 = p( 1 µ ) hs 2 hσ2 µ 2 µ2 a(s). µ V V 1 ] n=1 [p( 1 µ ) p( 1 µ ) hs [ = n n=1 ( n )µ n 2 µ)] hσ2 2 n=1 µ n The first term in the numerator, [ ] n=1 p( 1 µ ) p( 1 µ ) n [ 1 ] n=1 µ 2 n µ 2, since p( 1 µ ) is concave in µ; and the last term in the numerator 1 n=1 µ 2 n µ 2, which is essentially the Cauchy-Schwartz inequality. Thus, V V1 hs n=1 = hs = hs = hs ( n=1 nµ n n=1 µ n [ ( n )µ n 2 µ)] n=1 µ n ) ( ) 2µ 1 + 4µ µ ( + 1)(µ µ ) 2 n=1 µ n ( ( 1)(µ µ 1 ) + ( 3)(µ 1 µ 2 ) +... ) 16 2 µ

17 where in the last line, the series ends with µ /2+1 µ /2 (µ if is even, and ends with 2 (+3)/2 ) if is odd. µ ( 1)/2 Applying Lemma 3.1 and the identity: we have ( 1) 2 + ( 3) ( ) 2 = 2 ( 1)( + 1), 6 ( ) V V1 hs 2 ( 1) 2 µ ( + ( 3) ( σ 2 + B ) 2 )2 2 3 h hs 2 = µ ( ( 1)( + 1) σ 2 + B ) 12 3 ( h = hs2 1 2) 12 µ ( σ 2 + B ). h The bound in the above proposition indicates that the -price policy cannot improve much over the single-price policy when the replenishment quantity and the holding cost are low and the demand variability is high. Also note that this bound does not rely on the interior optimum assumption. It will be used in the next section to derive a bound on improvement for the joint pricing and replenishment strategy. 3.3 Joint Pricing-Replenishment Optimization One Price In this case, the firm s problem is max S>, λ L V (S, λ) = r(λ) hs 2 λa(s) hσ2 2λ. (19) The first-order conditions are given by (9) and (12). The difficulty is, V (S, λ) may not be concave in (S, λ), and it may even have multiple local maxima (see Example 3.7 below). However, under linear demand and linear replenishment cost, a solution satisfying the first order conditions and yielding positive profit must be the global optimum, as demonstrated below. Proposition 3.6 Suppose λ(p) = α βp, c(s) = K + cs, where α, β, K, c >. If S > and λ (, α] satisfy the first-order conditions in (9) and (12), and V (S, λ ) >, then (S, λ ) solves problem (19). 17

18 Proof. From Example 3.1, the optimal replenishment level for each λ is S (λ) = 2Kλ/h. We replace S in the objective by S (λ), and then find the optimal λ. Let W (λ) := V (S (λ), λ) = (α λ)λ/β cλ 2Khλ hσ2 2λ. We prove that λ satisfying W (λ ) = and W (λ ) > is the global maximizer. Let f(λ) := (α λ)λ/β cλ 2Khλ. It is easy to see that f() =, f () = and f (λ) = 2 β + hk 8λ 3. Let λ be the inflection point, i.e., f ( λ) =. Then, f(λ) starts decreasing from zero, is convex in [, λ] and concave in [ λ, ). Suppose f ( λ), then by convexity/concavity, f (λ) f ( λ) for all λ. Therefore, W (λ) < f(λ) f() = for all λ, so there exists no maximizer with positive profit in this case. Hence, we must have f ( λ) >. ext, let λ 1 be the local minimizer for f(λ) in the convex part of the function. Obviously, λ 1 < λ. First, λ [, λ 1 ] because in this region W (λ) < f(λ). Second, λ [λ 1, λ] because in this region W (λ) = f (λ) + hσ2 >. Hence, λ [ λ, α]. ( λ < α in order for λ to exist.) 2λ 2 Since W (λ) = f(λ) hσ2 2λ is strictly concave in [ λ, α], λ is the unique maximizer for W (λ) in [ λ, α]. It is the global maximizer since all other local maximizers (if any) must be within [, λ 1 ], the region where W (λ) <. The following example illustrates multiple local maxima and the above proposition. Example 3.7 Consider linear demand λ(p) = 5 p, linear replenishment cost c(s) = 5+2S, σ =.2 and h = 1. The optimal S for any fixed λ is given by S(λ) = 1 1λ. Substituting this EOQ into the first-order condition (12) gives 5 2λ + 1 5λ 2 = λ However, simply solving the above equation yields three stationary points: λ 1 =.1619, λ 2 =.11, and λ 3 = Using the second-order condition, it can be verified that λ 1 and λ 3 are local maxima, while λ 2 is a local minimum. Furthermore, V (S(λ 1 ), λ 1 ) = and V (S(λ 3 ), λ 3 ) = Hence λ = λ 3 = and S = The monotonicity of the joint optimum is explored in the following proposition (compared with Proposition 3.1 and 3.2 where a single decision is optimized). Proposition 3.7 If there is a unique optimal price and replenishment level, then (i) the optimal price is decreasing in σ; 18

19 (ii) the optimal replenishment level is increasing in σ and decreasing in h; (iii) the optimal profit is decreasing in σ and h. Proof. (i) The joint optimization problem can be solved sequentially. We first optimize S for each fixed λ > (same as Section 3.1). From Proposition 3.1, the optimal S is invariant to σ, and we denote it by S (λ). Then, the optimal λ can be found by max λ L Ṽ (λ, σ) := r(λ) hs (λ) 2 λa(s (λ)) hσ2 2λ. ow, Ṽ (λ, σ) is supermodular in (λ, σ) since 2 Ṽ λ σ = hσ λ 2, and therefore λ (σ) is increasing in σ, or equivalently, the optimal price is decreasing in σ. (ii) That S is increasing in σ follows immediately from (i) and Proposition 3.1 (i). To examine the effect of h on S, we first optimize λ for each fixed S > (same as Section 3.2), and denote the maximizer by λ (S, h). Then, the optimal S is determined by max S> Let λ S, λ h and λ Sh Ṽ (S, h) := r(λ (S, h)) hs 2 λ (S, h)a(s) denote the partial derivatives. We have hσ2 2λ (S, h). 2 Ṽ S h = r λ Sλ h + r λ Sh 1 2 aλ Sh a λ h + σ2 2λ 2 λ S hσ2 λ 3 λ Sλ h + hσ2 = λ S (r λ h + σ2 2λ 2 hσ2 ) λ 3 λ h 1 2 a λ h (r + a + hσ2 ) 2λ 2 λ Sh 2λ 2 λ Sh Since λ (S, h) is uniquely determined by (12), the last term in the above is zero, and λ S = a r hσ2 λ 3, λ h = σ2 2λ 2, r hσ2 λ 3 Then, the first term in the above is also zero, and we have 2 Ṽ S h = 1 2 a λ h = a σ2 2λ 2 r hσ2 λ 3 ow we show that the above is less than zero if evaluated at S. This is because S satisfies (9), which implies a (S ) = h 2λ, and 2 Ṽ hσ2 S h = λ 3 r + hσ2 λ 3 < hσ 2 4λ 3 hσ 2 λ 3 < 1 4.

20 Hence, Ṽ (S, h) is submodular in (S, h) in the neighborhood of the optima, and therefore, S (h) is decreasing in h. (iii). Let σ 1 σ 2, and the corresponding maximizers be (S1, λ 1 ) and (S 2, λ 2 ). We have V (S1, λ 1, σ 1) V (S2, λ 2, σ 1) V (S2, λ 2, σ 2), where the first inequality is due to the maximality of (S1, λ 1 ) and the second follows immediately from the objective function. decreasing in h is completely analogous. The proof of Remark: When the optimal solutions are not unique, the results in the above proposition continue to hold (except the monotonicity of the optimal replenishment level in h). In lieu of increasing and decreasing, the relevant properties are ascending and descending, respectively. Refer to Topkis (1978). Intuitively, the higher the unit holding cost h, the less inventory should be held; and the less order quantity means the higher average cost and thus the higher price. But this intuition is only partially correct when the demand is deterministic (σ = ). To see this, note that when σ =, the optimal λ is given by max r(λ) λa(s (h)), i.e., λ depends on h only through λ S. But we know S is increasing in h, while r(λ) λa(s ) is supermodular in (S, λ). Hence, the price is increasing in h. However, when σ >, a lower price may offset the additional holding cost due to demand variability; the composite effect is mixed, as shown in the following example. Example 3.8 Let λ(p) = αe p with α >. Let c(s) = S(K log S) for S e K 1, where K > is given. ote that r(λ) = λ log(α/λ) is strictly concave for λ (, α], c(s) is strictly increasing for S [, e K 1 ] and a(s) = K log S is strictly convex and approaches to infinity as S, so Assumptions 1 and 2 are satisfied. In this case, the first-order conditions (9) and (12) become log α log λ 1 = K log S hσ2 2λ 2 and 1 S = h 2λ, which determine the optimal solution: λ = σ 2h 2 K log(h/2α) and S = 2λ h. (The above is indeed a global optimal solution when λ < α, and S e K 1 ; the latter is satisfied if we choose K large enough.) ow, the partial derivative, (λ 2 ) h = σ2 (K + log(h/2α)) 2(K log(h/2α)) 2 2

21 varies from negative to positive, as h increases from zero. Thus, λ first decreases and then increases in h, or equivalently, p first increases and then decreases in h. Finally, we compare the joint optimization here with the sequential optimization scheme that is usually followed in practice: the marketing/sales department first makes the pricing decision, and then the purchase department decides the replenishment quantity based on demand projection as a consequence of the pricing decision. For instance, the marketing/sales department solves the problem: λ = arg max {r(λ)}, λ and sets p = p(λ ). Then, the purchase department takes λ as given and solves the problem: S = arg min S { λ a(s) + hs/2 }. Clearly, the sequential decision procedure does not take demand variability into account. The optimal price and inventory level found by the sequential scheme certainly satisfy the first-order condition in (9). But the first-order condition in (12) holds only when the demand variability happens to be σ = λ 2a(S )/h. Corollary 3.8 Comparing to the joint optimization, (i) if σ < σ, then the sequential decision underprices and overstocks; (ii) if σ > σ, then the sequential decision overprices and understocks. The proof is a straightforward application of the monotonicity result in Proposition 3.7, and will be omitted. In general, the sequential optimization is sub-optimal. If the demand variability is far away from σ, the sequential decision can lead to significant profit loss, as illustrated in the following examples. Example 3.9 Let λ(p) = 2 p, c(s) = 1 + 5S and h = 1. Suppose the marketing/sales department sets price at p = 1 in order to maximize the revenue rate. Then, the purchase department minimizes the operating cost using the EOQ model: S = threshold σ = λ 2a(S )/h 38. The Figure 4: Sequential decision vs. joint decision In Figure 4 we compare the sequential decision with the joint decision. The parameter range is chosen such that the best decision will be able to achieve a positive profit. Substantial profit 21

22 loss is observed: when σ =, the sequential decision both underprices and overstocks by 28%, resulting 73% profit loss compared to the joint decision; when σ = 1, the sequential decision underprices by 25% and overstocks by 22%, making almost no profit Prices The problem is max V (S, µ) = S>, µ M n=1 [p( 1 µ n ) hs ( n )µ n hσ2 n=1 µ n ] 2 µ2 n a(s). (2) As in the one-price case, the objective function V (S, µ) may not be jointly concave. Under the interior optimum assumption, the optimal solution must satisfy the first-order conditions in (9) and (15). The result in Proposition 3.3 (which holds for any replenishment level) continues to hold here, i.e., p 1 p 2 p. However, the monotonicity in Proposition 3.4 and Proposition 3.7 need not hold here, as evident from the following example. Example 3.1 Let p(λ) = λ + λ 1, c(s) = 5 + S 2 and h =.2. (ote the term λ 1 in p(λ) only adds a constant to the objective function to ensure that the average profit is positive.) Consider = 2. The optimization problem is: max µ 1,µ 2,S p( 1 µ 1 ) + p( 1 µ 2 ) hs 4 (3µ 1 + µ 2 ) hσ2 2 (µ2 1 + µ2 2 ) 2a(S) µ 1 + µ 2. The optimal solutions are plotted in Figure 5. Two observations emerge from the figure. First, Figure 5: Impact of demand variability on optimal solutions there exist jumps in the optimal objective function, due to multiple local maxima. For example, for σ =.243, there are at least two local maximizers: (S = 4.917, µ 1 =.41, µ 2 = 41.51) and (S = 4.464, µ 1 = 5.637, µ 2 = 43.44). The first yields an objective value V =.2639, which is slightly higher than the second one (V =.2638). For σ =.244, the two local maximizers are slightly different: (S = 4.918, µ 1 =.448, µ 2 = 41.33) and (S = 4.425, µ 1 = 6.248, µ 2 = 43.41). However, the objective value corresponds to the second one (V =.26198) is slightly better than the first (V =.26193). Hence, the optimal objective value exhibits discontinuity when σ varies from.243 to.244. (These numerical results are accurate to all the decimal places used. Furthermore, these phenomena can also be verified analytically.) Our second observation 22

23 is that when the optimal solutions are continuous in σ, there exists a range in which both µ 1 and µ 2 are increasing in σ, while S is decreasing in σ. This is in sharp contrast with the results in the single-price case. ext, we develop a bound on the profit improvement as the number of prices () increases. Proposition 3.9 Let (S, µ ) be the optimal joint pricing-replenishment decision, and V be the corresponding optimal profit in (2). Then, where µ = 1 V V1 hs 2 ( 1 2) ), 12 µ (σ 2 + B h n=1 µ n and B = inf { d2 p( 1 µ ) dµ 2 : µ [µ 1, µ ] }. Proof. Let V (S) denote the optimal profit when S is given (i.e., pricing decision only). If S happens to be fixed at S, then the profit is V, i.e., V = V (S ). Applying Proposition 3.5, we have the desired bound immediately: V V1 = V(S ) V1 (S1) V(S ) V1 (S) hs 2 ( 1 2) ). 12 µ (σ 2 + B h The shortfall of the above bound is that it involves the solution to the -price problem. Heuristically, we can use the solution to the joint single-price and replenishment problem, denoted by (S 1, µ ), in the upperbound. Specifically, replace S by S 1, and let B 1 = d2 p dµ 2 ( 1 µ ). That is, V V 1 heur hs µ ( σ 2 + B 1 h ), (21) where heur means heuristically less than. Example 3.11 Let c(s) = K + cs and p(λ) = a bλ, where a, b, c, K are all positive parameters. The optimization problem in (2) becomes max V (S, µ) = S>, µ [b/a, ) n=1 [a b µ n hs ] ( n )µ n hσ2 2 µ2 n K S c n=1 µ. n 23

24 Applying a change of variables, µ = b µ and S = K S, we can rewrite the above problem as follows: max V ( S, µ) = S>, µ [1/a, ) ] n=1 [a c 1 Khb S µ n ( n ) µ n hb2 σ 2 2 µ 2 n 1 S b n=1 µ n Clearly, the above expression indicates that there are four degrees of freedom in terms of independent parameters: (, a c, Khb, hb 2 σ 2 ). Specifically, the four degrees of freedom are determined by, either a or c, and two from (K, h, b, σ). In the numerical studies reported here and below, we choose to vary (, c, h, σ) while fixing (K, a, b)..(22) Figure 6: The effect of, the number of price changes. Figure 6 shows the optimal replenishment levels (Figure 6(a)), prices (Figure 6(b)) and profit values (Figure 6(c)) corresponding to different values of, the number of price changes. We see that as increases the optimal prices are inter-leaved (e.g., the optimal single price is sandwiched between the two-price solutions, which, in turn, are each sandwiched between a neighboring pair of the three-price solutions). (This inter-leaving property seems to persist in all examples we have studied.) Figure 6(c) shows that has a decreasing marginal effect on profit. Using two prices already achieves most of the potential profit improvement, and beyond = 8, the marginal improvement is essentially nil. Example 3.12 Here we continue with the last example, but focus on comparing the optimal profits under = 1 and = 8. We consider the parameter values c (, 3], h (, 5], σ (, 5], while fixing the others at K = 1, a = 5, b = 1. The results reported in Figure 7 are for c = 1. Similar results are observed for c = 5, 1, 2, 3. Figure 7: Multiple prices vs. a single price Figure 7(a) shows the optimal profit corresponding to a single price. The profit is clearly decreasing in h and σ (Proposition 3.7). Furthermore, the negative effect of σ is larger when h is larger and vice versa, suggesting that the profit function is submodular in (h, σ). Intuitively, since the effect of σ shows up in the profit function through the additional holding cost, the higher the h (resp. σ) value, the more sensitive is the profit to σ (resp. h). Figure 7(b) shows the absolute improvement in profit when using 8 prices (V8 V 1 ). The improvement is increasing in h and decreasing in σ. Intuitively, as the inventory holding cost 24

25 increases, the right trade-off among revenue, holding cost and replenishment cost becomes more important, thus more pricing options over time is more beneficial. However, pricing becomes less effective as demand variability (σ) increases. ote that when the profit under a single price V1 approaches zero, the absolute improvement (V8 V 1 ) does not diminish. In particular, while V 1 decreases in h, the improvement increases in h. Indeed, the relative improvement (= V when V 1 8 V 1 V1 is close to zero, as demonstrated in Figure 7(c) and (d). Figure 8: The bound on profit improvement ) is increasing in h, and approaching infinity To conclude this example, we show in Figure 8 the profit improvement in comparison with the heuristic bound in (21). The latter can be seen in this example as giving a good estimate of the maximum potential improvement The Algorithm: Fractional Programming To solve the fractional optimization problem in (2), we can instead solve the following: [ ( 1 ) max V η := p hs( n ) ] µ n hσ2 S>, µ M µ n 2 µ2 n a(s) η µ n, (23) n=1 where η is a parameter such that when the optimal objective value of the problem in (23) is zero, the corresponding solution is the optimal solution to the original problem in (2), and the corresponding η is the optimal objective value of the original problem. To see the equivalence, write the optimal value in (2) as V = A /B. Then, when η = V, the optimal value of (23) is zero. On the other hand, suppose there exists an η which yields a zero objective value in (23), specifically, A ηb =, then for any feasible A, B, we have A ηb, which implies A/B η = A /B. The algorithm described below takes advantage of the separability of the objective function with respect to µ when S and η are given. Specifically, the sub-problems are: ( 1 ) max g n(µ n ) := p µ n M µ n hs( n ) µ n hσ2 2 µ2 n a(s) ηµ n, n = 1,...,. (24) Under Assumption 1, the objective in (24) is concave in µ n. Algorithm for solving (2) 1. Initialize η = η, S = S, and µ n = µ for n = 1,...,. (For instance, η = ; S = S and µ n = µ for all n, with (S, µ ) being the optimal solution to the single-price problem.) Set ε and δ at small positive values (according to required precision). 25 n=1

26 2. Solve the single-dimensional concave function maximization problems in (24). 3. Update S following a specified stepsize or use equation (9). If the difference between the new and the old S values is smaller than ε, go to step 4; otherwise, return to step If V η ε, stop. Otherwise, if V η > ε, increase η to η + δ; if V η < ε, decrease η to η δ. Go to step 2. In step 2, there are ways to accelerate the search procedure. Firstly, in searching the optimal µ n for n = 1,...,, we can take the advantage of the monotonicity of µ n in n following Proposition 3.3. Secondly, we have g n (µ n) g n+1 (µ n) g n+1 (µ n+1 ), for n = 1,..., 1. Hence, once we find µ 1, if g 1(µ 1 ) > ε/, then we can be sure that V η > ε, and to bypass the rest of the algorithm to increase η directly and proceed to the next loop. Thirdly, we can make use of the monotonicity of µ n(s) in S according to Proposition 3.4 (iii). This is particularly useful when the stepsize for updating S in step 3 is small. In step 3, if we are in the early stage of the algorithm, i.e., when V η is still substantially away from zero, then the updating of S needs to cover a wide range so as not to miss the true optimum. This can be done by using large stepsize first and then reduce them gradually. When we are at the late stage of fine-tuning V η, we can even bypass step 3 and simply use the same S value (when it was last updated). In step 3, we can use (9) as an updating scheme. This is in fact the coordinate assent method: we alternate between optimizing µ for fixed S and optimizing S for fixed µ. This procedure is guaranteed to converge to a local maximum, but not necessary the global maximum (see Luenberger (1984)). So it is useful once the algorithm enters the region containing the true optimum without other local maxima. It can be verified that the optimal objective value in (23) is strictly decreasing in η, so there is a unique zero-crossing point at which V η is zero. Thus, in step 4, we can update η following a standard line search algorithm, such as bi-section or the golden ratio. 4 Discounted Objective We now turn to examining the problems in the last section under a discounted objective. We will focus on the contrasts rather than the similarities between these two classes of models. So as to present the results with minimal distraction, all proofs are relegated to the appendix. 26