Multiple-Item Dynamic Pricing under a Common Pricing Constraint

Transcription

1 Multiple-Item Dynamic Pricing under a Common Pricing Constraint Binbin Liu Joseph Milner Joseph L. Rotman School of Management University of Toronto 105 St. George Street Toronto, ON, Canada M5S 3E6 binbin.liu@rotman.utoronto.ca January 2006

2 Multiple-Item Dynamic Pricing under a Common Pricing Constraint Abstract We study how firms should dynamically price multiple items when by convention they are required to maintain a common price. We define the problem and present the structure of the optimal policy of the problem in its most general form. We then develop the optimal policy to the deterministic version of the problem, which is an n-segment pricing policy. We also study a case where a single price change is allowed for the stochastic problem assuming prices are given and show that a threshold policy holds. These restricted problems give us insight into the optimal solution of the general stochastic problem and enable us to establish heuristics that can effectively solve the problem.

3 1 Introduction We study the problem faced by retailers of fashion goods and other seasonal products of dynamically pricing multiple items under a constraint that they must be priced commonly. Such constraints arise in response to consumer perceptions that despite differences in items, differential pricing among items is, in some sense, inherently unfair and that retailers are opportunistic for altering prices between similar goods. The question then arises as to how the single price that applies to multiple items should be dynamically determined over the course of a selling season. As an example, in fashion retail industry when a single order per season is placed, a retailer may carry the same product (a shirt or dress) in an assortment of sizes and colors. Despite good forecasting and ordering procedures, the actual demand for the variants of the shirt will often differ from the plan, leaving relatively higher inventories of one variant versus another. Theoretically, the retailer would want to differentially price the variants in order to sell the items at the highest expected revenue. However, for several reasons a retailer may resist doing so. If the main difference is in size, a customer may feel that differential pricing is in some way discriminatory and while not illegal, it certainly may engender negative perceptions of the retailer. Similarly, customers may be annoyed if they find out that the color of the item they have chosen is differently priced from another similar item. Therefore, we observe that fashion goods are often priced identically across all the size and color assortments. This can be observed from many fashion retail stores like GAP, Old Navy, and Polo Ralph Lauren. However, during the season such retailers often vary the single price, usually through markdowns as the season progresses. Examples occur in other industries. For example, golf equipment retailers that sell both lefthanded and right-handed clubs. We observe that the common practice is to price both types of clubs similarly. For example, a golf equipment online store sells Callaway Big Bertha 454cc Titanium Drivers for $299. This price applies to any size of men s right-hand or men s left-hand clubs. Further, the same price is extended to a number of other variants (e.g., women s variants and senior s variants of the same club.) Again, dynamic pricing during a season may be observed as the next year s product line is introduced. Finally, the practice can also be observed in the airline industry. Airlines typically offer a single price at the same time for all seats in the same class (first, business or economy). However, as many 1

4 travelers know, not all seats within a class are the same. Seats in exit rows often provide more leg room while aisle seats allow the traveler to exit without disturbing neighbors. With increased flight lengths, travelers are increasingly interested in their particular seat choice and may be willing to pay for the ability to choose a seat (USA Today, April 25, 2005). While airlines may reserve choice seats for preferred customers, they have not differentiated prices based on seat choice within a fare class. The failure to do so may in part be attributed to a need to overbook flights as well as safety concerns regarding restricting disabled or young travelers from certain rows. Thus airlines face the problem of commonly pricing a good with differentiated demand. In this paper we consider the problem faced by a retailer of dynamically pricing multiple items over a finite horizon under a common pricing constraint. We present a model of continuous time pricing with stochastic demand and note some structural properties of its solution. In order to gain insight into the problem, we consider a deterministic version of the problem and present its solution. We then consider a restricted, stochastic version where a single price change (a markdown) is allowed at an arbitrary time. We show that the optimal markdown time is given by a time threshold policy based on multidimensional inventory. We then develop a heuristic solution based on these restricted problems and show that relatively large gains may be made vis-a-vis policies that do not consider the inventory of multiple items. Dynamic pricing of inventory is a very active area of research. Comprehensive reviews of the recent literature can be found in McGill and Van Ryzin (1999), Elmaghraby and Keskinocak (2003), and Bitran and Caldentey (2003). Initial work from the economics and marketing literatures considers reasons for observed dynamic pricing. Lazear (1986) presents a two-period model and illustrates that by reducing prices a retailer can increase total revenue. Pashigian (1988) extends the model to consider competition as a means of explaining data showing an increase in the percentage of markups and frequency of sales, and argues that dynamic pricing is due to the growing importance of fashion in merchandizing. Pashigian and Bowen (1991) argue that demand uncertainty and price discrimination can better explain markdown pricing practice. The most relevant previous literature to our research discusses continuous time pricing models with stochastic demand and limited non-reorderable inventory. Gallego and Van Ryzin (1994) consider the single-item case. They derive the deterministic optimal solution and prove that it provides a comprehensive guide to seating variants. 2

5 is asymptotically optimal for the stochastic case. One approach to the continuous time pricing problem is to limit the number of price changes. Feng and Gallego (1995) study the problem of a single price change, i.e., determining the optimal time to discount from a given higher price to a given lower price or vice-versa, and show that a time threshold policy based on the inventory is optimal. Feng and Xiao (2000a, 2000b) generalize the problem to allow multiple price changes among predetermined prices and prove the optimality of threshold policies. Bitran and Mondschein (1997) formulate the periodic review pricing problem as a dynamic program and show that the errors between periodic pricing and continuous pricing are small when appropriate number of reviews are chosen. Additional research has considered the coordination of inventory and pricing policy (Rajan et al. (1992), Federgruen and Heching (1999), and Chen and Simchi-Levi (2004a, 2004b)) as well as non-homogenous demand cases (Feng and Gallego (2000) and Zhao and Zheng (2000)). Multiple-item dynamic pricing has been studied in several papers. Gallego and Van Ryzin (1997) study how to dynamically set prices for individual items in order to maximize revenue when production is subject to a set of common resource constraints. They derive the optimality conditions for the stochastic case and propose two asymptotically optimal heuristics. Maglaras and Meissner (2005) consider a similar dynamic pricing problem for multiple-item that share production capacity. They derive the optimal control policy for the deterministic case and propose three asymptotically optimal heuristics. Neither of these papers considers the common pricing constraint that is the focus of this study. More closely related are Smith and Achabal (1998), and Bitran et al. (1998). Smith and Achabal (1998) study dynamic pricing of an assortment of goods in a deterministic environment. The sales rate is assumed to be dependant on the price, inventory level and seasonal variations. They incorporate the effect of depleting an assortment of items by assuming the sales rate decreases when the inventory level falls below the minimum required inventory level for a full presentation. Bitran et al. (1998) consider a single-item pricing problem for multiple retail stores. In the paper, each store sells to a different customer class but all stores must use a common price. The case in which inventory cannot be transferred between stores may be considered analogous to our problem by translating multiple stores as multiple items. Our paper differs from theirs in that we consider a continuous time stochastic model and take an analytical approach, while most of their results are based on simulation. We present the optimal policy under two restricted cases, and then provide an efficient heuristic for the solution to the common pricing problem. 3

6 Our paper contributes to the literature by proposing the common pricing problem for multiple goods and presenting optimal policies for restricted versions of the problem, specifically the deterministic and limited price change versions. Further, we show that the deterministic problem provides a set of proposed markdown times. These times are then used to determine a set of prices for the stochastic problem with a limited number of price changes. Finally, we show that using these prices a multidimensional threshold policy can be derived as a heuristic solution to the original problem which is asymptotically optimal. We demonstrate that while modest gains are achieved over simple heuristics when demand forecasts are accurate, large improvements are possible when forecasts prove inaccurate and inventory ordering policies cannot adjust to meet unexpected demand. The structure of this paper is as follows. In Section 2, we present the model, derive its optimality conditions, and discuss structural properties of the optimal solutions. In Section 3, we present the two-item, deterministic version of the problem and derive the optimal solution. After analyzing the solution, we extend the problem to the multiple-item case. In Section 4 we consider a restricted version of the n-item, stochastic problem where only single price reduction is allowed. We show that the optimal switching time between two prices is given by a time threshold policy. In Section 5, we discuss four heuristics and show their asymptotic optimality. We then present simulation results for these heuristics. We draw conclusions in Section 6. 2 The Multiple-Item Common Pricing Problem We consider the problem of dynamically determining a common price, p s, at time s for n goods over a period of length T in order to maximize the total revenue of a retailer. Let Q i be the initial inventory of item i, i = 1,..., n. Let N i s be the total number of customers for item i up to time s. At a given time s, we assume customers for item i arrive according to a price-dependent Poisson process with rate λ i (p s ) for 0 s T. In particular, we assume that the total arrival rate for all items, λ, is a price dependent function of p s, i.e., λ(p s ), and customers choose item i according to a probability of w i with i wi = 1 so that λ i (p s ) = w i λ(p s ) for i = 1,..., n. Thus N i s is a Poisson process with non-homogenous demand rate λ i (p s ) and N i s is independent of N j s for i j. We note our assumption implies that all items have a common stochastic price elasticity as the elasticity dλ i dp p λ i = w i dλ(p) dp p λ i = dλ(p) dp p λ(p) is independent of i. Demand is lost when a customer s choice is not 4

7 available. We assume the inverse function, p(λ), exists. Let r(λ) = λp(λ) be the instantaneous revenue function; we assume r(λ) to be continuous, bounded, and concave. Let ˆλ be the minimum instantaneous revenue maximizing λ, i.e., ˆλ = min{λ : r(λ) = max λ>0 r(λ)}. Let M i t be the total number of item i sold up to time t, then M i t = min[n i t, Q i ]. Let U to be the class of all non-anticipating pricing policies that satisfies T dms i Q i for all i = 1,..., n, and t2 t 1 dm i s 0 t2 t 1 dn i s for all t 1 < t 2, t 1, t 2 [0, T ], i = 1,..., n. Let Q {Q i } i=1,...,n be the vector of inventory for the n items, and I(Q) be the set of items with Q i > 0. Let V u (Q, t) be the expected revenue to go under policy u U given the inventory vector of Q and t periods to go. That is, V u (Q, t) = E u [ i where V u (Q, 0) = 0 for all Q and V u (0, t) = 0 for all t. t 0 p s (dm i s)], Let V (Q, t) be the maximum revenue over the period [0, T ] among all possible policies u U, that is, V (Q, t) = sup V u (Q, t). (1) u U The multiple-item common pricing problem is to determine a common pricing policy u that gives V (Q, T ). Define Q i = {Q 1, Q 2,..., Q i 1, Q i 1, Q i+1,..., Q n }. We can derive the Hamilton-Jacobi sufficient condition informally by considering the dynamic programming formulation (noting t designating the time remaining in the horizen) [ V (Q, t) = sup λ i I(Q) +(1 i I(Q) ( ) w i λ t p(λ) + V (Q i, t t) w i λ t)v (Q, t t) + o( t) Rearranging, letting r(λ) = λp(λ) and letting t 0, we find the Hamilton-Jacobi equation as: V (Q, t) = sup ( t))) w i r(λ) + λ(v (Q i, t) V (Q,. (2) λ i I(Q) 5 ].

8 We can show the following structural results of the optimal solution λ (Q, t) and V (Q, t). Theorem 1 V (Q, t) is strictly increasing in Q i and t, and concave in t [0, T ]. Furthermore, there exists a unique optimal intensity λ (Q, t) for t [0, T ] (resp., price p (Q, t)) that is strictly decreasing (resp., increasing) in t. (All proofs appear in the Appendix.) The theorem supports the intuitive idea that expected revenues should increase with stock and time. Further, it implies that the price decreases over time for a given level of inventory. However, in contrast to the single-item model, we do not show that the price increases at a given point in time as the inventory decreases. It is quite possible that the price will decrease as the inventory decreases. To observe this, consider the following two period, discrete time example: Example: Consider the case of finding a common price for two items in each of two periods. Suppose that the total demand in each period is λ(p) = 200 2p with probability 0.5 and 0 with probability 0.5. Let p 1 denote the price charged in the first period. Note that the second period pricing decision can be made after observing the actual demand in the first period. Let p 2 denote the second period price if the demand is 200 2pin the first period, and p 2 denote the second period price if no customers show up in first period. Assume w 1 = 0.9 and w 2 = 0.1. As above, we assume no salvage value for excess inventory. Define S to be the set of items. In our example, S = {1, 2}. The problem is then to determine ( V = max 0.5 p 1 min[w i (200 2p 1 ), Q i ] + 0.5p p 1,p 2,p 2 min[w i (200 2p 2), Q i ] 2 i S i S ) +0.5p 2 min[w i (200 2p 2 ), (Q i w i (200 2p 1 )) + ]. i S This abstracted problem can be solved as a nonlinear programming. Letting initial inventory Q 2 = 50 and varying the initial Q 1, we present several solutions of p 1 as illustrated in Table 2. Q p Table 1: An Example of Price Values for Various Q 1 We observe that as Q 1 increases in the first period, the price initially increases and subsequently 6

9 decreases. As the item mix changes, the price initially increases to reflect the greater supply of the higher demand item. A lower price is not required to raise revenues; rather one should switch prices in concert with the demand for item 1. However once there is sufficient inventory of item 1, the price decreases to reflect the need to sell both items. This points to a difficulty in solving the continuous time problem, namely that the price behaves differently near the non-negativity boundary of the inventory space and away from it. Formally, for the continuous time model, the single-item case serves as a boundary solution for the multiple-item case when only a single item remains in stock. Let V0 (Q, T ) be the maximum expected revenue for a single-item problem (Q is a scalar) with demand function λ(p) and let p 0 (Q, T ) be the optimal price policy. Then as above, V 0 (Q, t) We have the following relationship. = sup [r(λ) + λ (V 0 (Q 1), t) V 0 (Q, t))]. (3) λ Theorem 2 If only a single item remains, i.e., Q = {Q i > 0, Q i = 0} for some i, then V (Q, T ) = V 0 (Qi, w i T ), and p (Q, t) = p 0 (Qi, w i t) for 0 t T. The theorem implies that the optimal policy when a single item remains in stock is identical to that of the problem with a single-item with the time adjusted to reflect the reduced demand rate for the single item. As discussed in Gallego and Van Ryzin (1994), it is difficult to solve the Hamilton-Jacobi equation analytically even for the single-item case except for the case of an exponential demand function. However, this case does not translate easily into the multiple-item case because of the non-monotonic behavior of the price near the boundaries. In order to approach the problem, we therefore consider the deterministic problem next. 3 The Deterministic Common Pricing Problem We next consider the problem assuming the demand rate is a deterministic function of the price. That is, we assume the demand at time s is λ (p(s)). For simplicity of exposition and to gain insights into the problem, we first consider the case of two-item (n = 2) with inventory vector Q = {Q 1, Q 2 }. We assume that demand can be for fractional units, i.e., each unit of demand is for w 1 units of item 1 and w 2 units of item 2 with w 1 + w 2 = 1. The any time s, the demand rate for 7

10 item i is λ i (s) = w i λ(p(s)). Since there is a one to one correspondence between p and λ, the problem can be formulated as a stochastic control problem with the demand rate λ(s) for s (0, T ) as the control parameter. Let V D (Q 1, Q 2, T ) be the maximal revenue for the deterministic case. Then V D (Q 1, Q 2, T ) is given by the following control problem: V D (Q 1, Q 2, T ) = max s.t. with the boundary conditions λ(s) T 0 T 0 T 0 p(λ(s))(λ 1 (s) + λ 2 (s))ds λ 1 (s)ds Q 1 λ 2 (s)ds Q 2 λ 1 (s) w 1 λ(s) s (0, T ) λ 2 (s) w 2 λ(s) s (0, T ) V D (x, y, 0) = 0 for all x, y 0; V D (0, 0, t) = 0 for all t [0, T ]. The deterministic two-item problem can be solved easily by considering two cases: (i) the case where the inventories for the two items are proportional to their demand rates and (ii) the case where only one item remains in stock. These are treated in the following lemmas. Recall ˆλ is the instantaneous revenue maximizing demand rate. Lemma 1 When Q 1 /w 1 = Q 2 /w 2 = k, the optimal demand rate is λ(s) = λ D = min[ˆλ, k T ] for all s [0, T ], and the maximum revenue is V D (Q 1, Q 2, T ) = T r(λ D ). Since p = p(λ) is a decreasing function of λ, the optimal price is a single price p (s) = max[p(ˆλ), p(k/t )]. The lemma implies that when the inventory of the two items is proportional to their demand rates a single price should be employed; if enough inventory is available, price so that the instantaneous revenue is maximized (leading to excess inventory at the end of the horizon), otherwise, price to deplete the inventory evenly over the horizon. Lemma 2 If Q 1 = 0, and Q 2 > 0, the optimal demand rate is λ(s) = λ D = min[ˆλ, Q2 ] for all w 2 T s (0, T ) and the maximum revenue is V D (0, Q 2, T ) = w 2 T r(λ D ) (and similarly for Q 2 = 0, Q 1 > 0). 8

11 As in the previous case, the demand rate for the single-item case is set to either deplete the inventory at the end of the period or at the revenue maximizing rate. We next show that the solution to the general deterministic problem is a combination of the two cases above. We define a two-segment pricing policy (τ, p 1, p 2 ) such that p(t) = p 1 on [0, τ] and p(t) = p 2 on [τ, T ]. We show such a policy is optimal with Lemma 1 applying in the first segment and Lemma 2 in the second. We assume without loss of generality that Q 1 /w 1 < Q 2 /w 2. (This holds for the remaining discussion of the two-item case.) Theorem 3 Let f(τ) = V D (Q 1, Q 1 w2 w 1, τ) + V D (0, Q 2 Q 1 w 2 /w 1, T τ), then f(τ) is concave and there exists an optimal two-segment pricing policy where V D (Q 1, Q 2, T ) = max τ [0,T ] f(τ), p 1 Q1 = p(min[, ˆλ]) and p w 1 τ 2 = p(min[ Q2 /w 2 Q 1 /w 1 T τ, ˆλ]), with τ = arg max τ [0,T ] f(τ). The optimal pricing policy implies item 1 should be depleted at price p 1 and then item 2 should be depleted at price p 2 revenue maximizing rate. Special Case: Linear Demand unless either of the items inventory exceeds the amount needed for the Consider the case of a linear demand function p = α βλ so that r(λ) = λ(α βλ). To simplify the notation, define τ L = Q1 w 1ˆλ, and τ H = T Q2 /w 2 Q 1 /w 1. τ ˆλ L is the time that item 1 would be depleted at the revenue maximizing rate. T τ H is the time required to deplete the excess item 2 (Q 2 Q 1 w 2 /w 1 ) at the revenue maximizing rate. In this case ˆλ = α 2β and r(ˆλ) = α2 4β. Then it is straightforward to show from Theorem 3 that the optimal price switching time τ = 2β α 2β α Q 1 Q 1 w 1 w 2 w 1 w T 2β 2 α (Q2 w 2 + Q 1 w 1 w 2 T if τ H τ T or equivalently Q1 w 1 w 2 (1 w 2 )). Similarly τ = Q 2 +Q 1 /(w 1 w 2 )(1 if τ w 2 L τ τ H, which holds if T ( 2β ) α w 2 )). Finally, as in Theorem 3, if T < 2β α (Q1 /(w 1 w 2 )), τ = T. )( Q2 w 2 + Q1 w 1 (1 w 2 Thus letting T L = 2β α Q 1 w 1 w 2 and T H = 2β α ( Q2 + Q1 w 2 w 1 (1 w 2 )), we have w 2 9

12 τ = T if T < T L T L if T L T < T H T L T H T if T T H. Let T s = ( 2β α )(Q1 /w 1 ). By substituting τ into V D, we have the maximal revenue α 2 4β T if T T s r( Q1 V D w = 1 T )T if T s T < T L r( w 2 / 2β α )T L + w a α2 4β (T T L) if T L T < T H r( Q2 + Q1 w 2 T w 1 T (1 w 2 )) T L T H T + w 2 r( Q2 /w 2 Q 1 /w 1 )(1 T L T H )T if T T H. (1 T L TH )T (4) (5) This linear case illustrates that the time allocated to item 1, τ, increases linearly with the supply of the item. With high quantities of both items or a relatively shorter selling season (when T T s or equivalently Q 1 α 2β w1 T and Q 2 α 2β w2 T ), the supply is sold at the revenue maximizing rate, i.e., p 1 = p(ˆλ). When there is slightly less item 1 (T s T T L or equivalently α 2β w1 w 2 T Q 1 α 2β w1 T ), the price is set a little higher to deplete item 1 exactly at time T. When there is even less item 1 and still enough item 2 (T L T T H or equivalently Q 1 < α 2β w1 T and Q 2 > α 2β w2 T Q1 w 2 (1 w w 2 )), then until time τ, inventory is rationed 1 at a even higher price p 1 so that item 1 is depleted at time τ, and after time τ the revenue maximizing rate is used. Finally when there is insufficient item 1 and 2 (T T H or equivalently Q 2 < α 2β w2 T Q 1 w 2 (1 w w 2 ) and Q 1 Q 2 w 1 /w 2 ), prices are further raised, and in each segment the 1 inventory is rationed so that item 1 is depleted at time τ and item 2 is depleted at time T. As a numerical example, suppose a retailer has 80 blue jackets (item 1) and 30 white ones (item 2), and a 10-week selling season. The customers Poisson arrival rate is 100 p per week for any given price p. Each customer purchases 0.8 units of item 1 and 0.2 units of item 2. In this case, α = 100, β = 1, Q 1 = 80, Q 2 = 30, T = 10, w 1 = 0.8, and w 2 = 0.2. Then T L and T H can be calculated as T L = 4.47 and T H = Since T > T H, according to equation (4), τ = T L T H = Plugging into 10

13 the T > T H case in equation (5), V D (80, 30, 10) = Correspondingly, the pricing strategy is: p 1 = p( Q1 ) = and p w 1 τ 2 = p( Q2 /w 2 Q 1 /w 1 T τ ) = That is, the optimal pricing policy is to apply the price $87.76 from the beginning of the season until time 8.17, and then discount the price to $72.64 until the end of the season. 3.1 Properties We now discuss several properties of the optimal policy. First we can bound τ as follows: Proposition 1 τ min[q 1 /(w 1ˆλ), T ], and if Q 2 /w 2 = Q 1 /w 1, τ = T. The proposition implies that if Q 1 > w 1ˆλT (and therefore Q 2 > w 2ˆλT ), τ = T. That is, items are not depleted if there is sufficient inventory to satisfy the demand at the revenue maximizing rate. Note that when the items are balanced (Q 2 /w 2 = Q 1 /w 1 ), there is only a single period. In this case, either both or neither are depleted at time T under a single price. Next we consider some comparative statics of the optimal policy for the deterministic problem. Proposition 2 V D (Q 1, Q 2, T )is non-decreasing in Q 1, Q 2, and T. τ is non-increasing in Q 2 and non-decreasing in Q 1 and T. Proposition 2 confirms the expectation that the run-out time for the good with less inventory (relative to its demand rate) will increase (weakly) as the supply of that goods and time increase and will decrease as the supply of the other goods increases. That is, when the retailer has more popular items, it is in a better position to postpone discounting. But if it has more inventory of the unpopular items, it should discount earlier to capture more revenue from the less popular item. Also we observe that as the selling season becomes shorter, the retailer will start discounting earlier to attract more customers in order to maximize the revenue of that season. For example, if a shorter winter season than usual is forecasted, winter coats and cold weather accessories would be put on sale sooner than usual to realize as much revenue as possible. Next we confirm the intuition that the price should decrease in the second period. Proposition 3 If τ < T, then there are two segments with only one item remaining in the second segment with p 2 p 1 ; p 1 and p 2 are non-increasing in Q 2 and Q 1 and non-decreasing in T. 11

14 This reflects the practice of many retailers. For example, fashion retailers often set a high price at the beginning of the selling season. As time goes on, they may begin running out of some items and face decreasing demand. At that time, they may place the remaining items on sale. The more inventory they carry, regardless of the assortment, the more pressure they would feel to lower the price. The time remaining is also an important factor in the pricing decision. As the end of the selling season approaches, retailers need to clear inventories through end-of-season sales. Next we observe the deterministic problem solution provides an upper bound on the solution value to the stochastic problem. Proposition 4 V (Q 1, Q 2, T ) V D (Q 1, Q 2, T ). We will use this upper bound to determine the effectiveness of a two-price heuristic for the stochastic problem considered in Section 4. We next consider the case of multiple-item. 3.2 Multiple-Item Case We now generalize the previous results to the n-item case. Given a vector Q of length n, and a set of relative demand weights w i, i = 1,..., n with n i=1 w i = 1, we want to determine the price process p t for t (0, T ) or equivalently λ(t) to maximize the revenue. That is, we generalize the control problem as ( T n ) V D (Q, T ) = max p(λ(s)) λ i (s) ds (6) λ(s) 0 i=1 s.t. T λ i (s) Q i for i = 1,..., n 0 λ i (s) w i λ(s) for i = 1,..., n, and s (0, T ), with boundary conditions V D (Q, 0) = 0 for all Q Z n and V D (0, t) = 0 for all t (0, T ). Note that 0 is a vector of 0 of length n. We assume without loss of generality that Q i /w i is increasing in i. Also we assume the ratios Q i /w i are distinct; if not and Q i /w i = Q j /w j, j i, then we can pool their inventory and demand ratios together as in Proposition 1. Since we assume the demand weights are constant with respect to price, we know that the items will be depleted in order (i = 1,..., n). Then the multiple period problem is reduced to determining a set of times τ = {τ 1,..., τ n } to deplete the n items 12

15 with n i=1 τ i = T. Here τ i represents the time between when the (i 1) th item is depleted and when the i th item is depleted or time runs out. In particular, τ 1 is the time when item 1 is depleted or time runs out. Note we show that if τ i > 0 for i = 1,..., j and j i=1 τ i = T so that τ j+1 = τ j+2 =... = τ n = 0, then item 1 to j 1 are depleted, item j, item j + 1,..., item n are not depleted and the instantaneous revenue maximizing price is used from the time j 1 i=1 τ i to T. Let {0} k be a vector of zeroes of length k and Vj D(Q, τ j) be the revenue over the period from the moment item j 1 runs out to the moment item j runs out or time runs out. The length of this period is τ j according to the definition, and ( where {0} 0 = φ. Vj D (Q, τ j ) = V D 1,..., j 1 and Q k = w k ( Qj w j {{0} j 1, {w k ( Qj w j Qj 1 w j 1 )} k=j,...,n}, τ j That is, V D j is the value of the deterministic problem when Q i = 0 for i = Qj 1 w j 1 ) for k = j... n. ) Observe in this subproblem Vj D(Q, τ j), all of the items k = j,..., n have inventory levels that are proportional to their demand ratios (Q k /w k = Q j /w j Q j 1 /w j 1 ) so that one is depleted at time t if and only if all are depleted at time t. The following theorem shows that the problem as formulated in (6) can be reduced to a concave optimization problem over the values of τ = {τ 1,..., τ n } utilizing the values of Vj D(Q, τ j). Theorem 4 There exists an optimal n-segment pricing policy where we divide the total horizon T into n segments, and charge the price p j on the i th segment with the length τ j, j = 1,..., n. Let f(q, τ ) = n j=1 V D j (Q, τ j), then f(q, τ ) is a concave function of τ. V D (Q, τ ) = max τ τ n=t f(q, τ ), with the optimal price in period j given by ( p j = p min[(q j /w j Q j 1 /w j 1 )/τ j, ˆλ] ), j = 1,..., n, and p 1 p 2... p n. Figure 1: n-segment Optimal Pricing Strategy This theorem reduces the revenue maximization problem into finding the optimal division of the horizon into n segments to maximize a concave function as depicted in Figure 1. The optimal price 13

16 in each segment is determined similarly as in Theorem 3, and the optimal price is still decreasing over time. 4 The Stochastic Problem with a Single Markdown As noted in Section 2, the stochastic problem is very difficult, though as seen in Section 3, the value of the solution can be bounded above by the solution to the deterministic problem. In this section we consider the n-item case and develop a lower bound of the maximum revenue by allowing a single price markdown from p 1 to p 2 (p 1 > p 2 ). The question is to determine when the price change should be made. We show that for each inventory vector q, 0 q Q, there is a time threshold X q that determines if the price should be changed. To determine the threshold policy we first define the problem and present sufficient conditions for optimality. We then discuss how to calculate the thresholds. Finally we discuss how to determine p 1 and p 2. Let J(Q, T ; τ) be the expected revenue given by using p 1 from time 0 to τ and p 2 from τ to T for some τ T. Let N j 1 (τ) be the total number of arrivals for item j before changing the price, and N j 2 (t τ) be the total number for item j after changing the price. As before arrivals are Poisson. Then, n J(Q, T ; τ) = (p 1 p 2 )E min[q j, N j 1 (τ)] (7) j=1 n +p 2 E min[q j, N j 1 (τ) + N j 2 (T τ)]. j=1 Let J(Q, T ) = sup E[J(Q, T ; τ)] (8) τ T be the expected revenue under an optimal policy where T is the set of stopping times with respect to the sigma field σ{n1 i (s), i = 1,..., n : 0 s t}. Note that by considering the set of stopping times T, we allow all policies including those based on observed demand. Let G(Q, t) be the marginal gain per unit time for postponing changing the price for a small amount of time ( t) given current inventory Q with t time units to go. This ignores the possible extra value this discounting option might bring when exercised in the future and simply assumes the discounting option is exercised right after a very short period of time ( t). Let G j (Q j, t) be 14

17 the marginal gain from item j. Then we have the following, Lemma 3 G j (Q j, t) = w j [r 1 r 2 p 2 (λ 1 λ 2 )P (N j 2 (t) Qj )] for all Q j > 0, (9) and G(Q, t) = n 1 {Qj 0}G j (Q j, t) for all Q 0. (10) j=1 We next present a sufficient condition for a function W (Q, T ) to be the optimal revenue function given in (8). It is then used to prove the optimality of the threshold policy for the n-item case. 4.1 Optimality Recall that Q i = {Q 1, Q 2,..., Q i 1, Q i 1, Q i+1,..., Q n } as defined in Section 2. Lemma 4 Let W (Q, t) be any continuous and almost everywhere differentiable function. If W (Q, t) satisfies: W (Q, t) = n i=1 ( )) 1 {Qi 0}w i r 1 λ 1 (W (Q, t) W (Q i, t) (11) on W (Q, t) > J(Q, t; 0), and W (Q, t) n i=1 ( )) 1 {Qi 0}w i r 1 λ 1 (W (Q, t) W (Q i, t) (12) on W (Q, t) = J(Q, t; 0) for all Q, t 0, then W (Q, T ) = J(Q, T ). The lemma presents a sufficient condition of the optimal pricing policy. We will next show that a threshold policy satisfies this sufficient conditions, and therefore is optimal. Theorem 5 There exists time thresholds X q (0 q Q), so that it is optimal to change price once the remaining time t is less than or equal to the threshold X q determined by the current status (q). That is: while J(q, t) = { F (q, t) = J(q, t; 0) if t X q J(q, t; 0) + F (q, t) if t > X q, { 0 if t X q H(q, t) if t > X q, (13) (14) 15

18 and H(q, t) satisfies the following differential equations H(q, t) n = λ 1 H(q, t)( 1 {qj 0}w j ) + L(q, t), (15) where L(q, t) = G(q, t) + n j=1 1 {qj 0}w j λ 1 F (q j, t). X q is determined by and H(q, t)satisfies H(q, X q ) = 0 for any q 0. j=1 X q = inf{t 0 : L(q, t) = 0}, (16) This optimal pricing policy establishes a set of time thresholds defined on the inventory space. For any inventory level, there is a time threshold. If the time remaining is less than the threshold, then it is optimal to switch the price immediately. 4.2 Calculation of Thresholds Theorem 5 shows the structure of the optimal solution, but solving the partial differential equations as defined in equation (15) is very difficult. If the thresholds were monotonically increasing, then the integration over the partial differential equations could be transformed to the summation of a group of easily calculated functions to solve this problem efficiently. But unfortunately, the monotonicity does not hold here. As discussed in Section 2, the optimal price may decrease as the inventory decreases. Similarly in this restricted problem, the time thresholds may increase (the discounting will start earlier) as the inventory decreases. This makes the exact solution hard to find. Therefore, we develop the following two approximations. (a) L = G Approximation. Let Y q 1,q 2 = inf t(t 0; G(q 1, q 2, t) = 0), then use X q 1,q 2 = Y q 1,q 2. Since L(q, t) = G(q, t) + n j=1 1 {qj 0}w j λ 1 F (q j, t), this approximation uses only the gain from postponing the discounts for an extra time unit, G function, and ignores the terms F, where the sum of F represents the future revenue brought by the discounting option, which is a major factor that makes this problem difficult. Theorem 5 implies F should decay at an exponential rate (in the proof we show F decays as such). Therefore, the influence of F on L may be small. Thus a simplified approach based only on G may be appropriate. (b) Monotonicity Approximation. A second approximation is to ignore the fact that the thresholds may not be increasing in inventory (e.g., it may be that X q 1,q 2 < X q 1,q 2 for q 2 > q 2 ), and rather assume that the thresholds are monotonically increasing. Analytical solutions of thresholds 16

19 do not require numerical integration under this assumption. We will use the two-item case to demonstrate the calculation of the thresholds in this approximation. If we assume the thresholds are increasing in inventory, when t > X q 1,q 2, q1, q 2 > 0, it is optimal to keep the price at the level of p 1 for at least t X q 1,q 2 units of time. At the end of this period, if no units are sold, then we would discount to price p 2 immediately. Otherwise, we would reevaluate the time remaining with the threshold determined by the current inventory status. That is, when t > X q 1,q 2, q1, q 2 > 0, we approximate J(q 1, q 2, t) i=0 j=0 ) (p 1 (min[i, q 1 ] + min[j, q 2 ]) + J((q 1 i) +, (q 2 j) +, X q 1,q 2) ) P (N1 1 (t X q 1,q 2) = i, N 1 2 (t X q 1,q 2) = j = J(q 1, q 2, t X q 1,q 2; t X q 1,q2) (17) ( ) + J((q 1 i) +, (q 2 j) +, X q 1,q 2)P N 1 (t X q 1,q 2) = i, N 2 (t X q 1,q 2) = j, i=0 j=0 where N a 1 (s) represents the number of Poisson arrivals for item a within time s under the price p 1, a = 1, 2. Because the thresholds are not monotonically increasing in q 1 and q 2, the optimal policy may not wait until the time threshold determined by current inventory level to switch prices. As such the approximation underestimates the value of J(q 1, q 2, t). From equation (17), we can then approximate F (q 1, q 2, t), the extra expected revenue from postponing discounting for t > x q 1 q2, as ( F (q 1, q 2, t) = J(q 1, q 2, t) J(q 1, q 2, t; 0) ( ) + J(q 1, q 2, t X q 1,q 2; t X q 1,q 2) J(q1, q 2, t; 0) (18) + ( ) + J((q 1 i) +, (q 2 j) +, X q 1,q 2)P N 1 (t X q 1,q 2) = i, N 2 (t X q 1,q 2) = j. i=0 j=0 Then following the reasoning in Theorem 5, we can calculate the 2-dimensional threshold matrix as follows: Initiation: X(0, 0) = 0. Step 1: Solve the 1-dimensional thresholds. Assume q 1 = 0, the situation is reduced to the 1-dimensional case with a demand function of λ 2 = w 2 λ(p). Since monotonicity holds for the single product case as shown by Feng and Gallego (1995), the approach gives the exact solution 17

20 by replacing q 1 by 0. We can put them in the first row: X(0, q 2 ), q 2 = 1,..., Q 2. Similarly we can obtain the first column of the threshold matrix: X(q 1, 0), q 1 = 1,..., Q 1. At the same time, we have two groups of functions of t, F (0, q 2, t) for all q 2 = 1,..., Q 2 and F (q 1, 0, t) for all q 1 = 1,..., Q 1. For any X 0,q 2, q 2 = 1,..., Q 2, we have the values of J(0, k, X 0,q 2), k = 1,..., q 2. Similarly for anyx q 1,0, q 1 = 1,..., Q 1, we have the values of J(k, 0, X q 1,0), k = 1,..., q 1. Step 2: 2.0 Let q 1 = 1, q 2 = Solve function L(q 1, q 2, t) = G(q 1, q 2, t) + λ 1 [w 1 F (q 1 1, q 2, t) + w 2 F (q 1, q 2 1, t)] = 0 to obtain X q 1,q2 as in (16). 2.2 For X q 1,q 2, solve J(i, j, X q 1,q 2), i = 1,..., q1, j = 1,..., q 2 according equation (17). 2.3 Calculate function F (q 1, q 2, t) according to equation (18). 2.4 If q 2 < Q 2, q 2 = q 2 + 1, go back to step 2.1, otherwise, continue. 2.5 If q 1 < Q 1, q 1 = q 1 + 1, q 2 = 1, go back to step 2.1, otherwise, stop. The above algorithm solves for the threshold matrix by iterating initially over the first column and first row, then subsequently over remaining (Q 1 1) (Q 2 1) matrix, row by row. Example: Suppose a retailer has 10 blue jackets (item 1) and 5 white ones (item 2), and a 10-week selling season. The customers Poisson arrival rate is p per week for any given price p, and they choose blue and white according to equal probabilities. The regular price for the jacket is $99, and the problem for the retailer is when to discount the jackets by 50%. The parameters in this problem are as follows: Q 1 = 10, Q 2 = 5, w 1 = 0.5, w 2 = 0.5, T = 10. λ = p, p 1 = $99, and p 2 = $49. We can derive the 2-dimensional threshold matrix as shown in Table 4.2 using either (a) the L = G approximation, or (b) the monotonicity approximation. We observe the tables are very similar, differing by at most While not a proof of the accuracy of the approximations, it is encouraging that they give similar results. Below we show that a heuristic (Heuristic 1) based on these approximations is very effective in solving the problem optimally. 18

21 q 1, q (a) L = G approximation q 1, q (b) Monotonicity approximation Table 2: Time Thresholds Matrix X q 1,q 2 Observe that the thresholds derived in both tables are not monotonically increasing when the inventories increase from zero to one unit of either item. While the thresholds increase along the first row and first column, coinciding with the expected monotonicity for a single item, and also increase for q 1 1 and q 2 1. We observe that when one of the items is depleted, the time threshold may increase. Thus at the point in time when one item is just depleted, the time remaining may be less than the time threshold, requiring discounting to be initiated immediately. It is this immediate change in the policy which is difficult to capture in the solution given in Theorem 5 that requires the approximation. 4.3 Determining the Prices p 1 and p 2 Observe that the stochastic problem with a single price change considered in this section is dependent on values for p 1 and p 2. The dual to this problem is to determine prices p 1 and p 2 that maximize the expected revenue given a time, τ, when the prices will be switched. That is, the dual problem is to determine V (Q 1, Q 2, T ; τ) [ ] 2 2 = max E (p 1 p 2 ) min[n i p 1,p 2 1(τ), Q i ] + p 2 min[n1(τ) i + N2(T i τ), Q i ], (19) i=1 i=1 where as before N j i (t) is the number of arrivals for item j at price p i in a period of length t, i = 1, 2, j = 1, 2. Under the assumption of Poisson arrivals with price dependent arrival rates, since r(λ) is 19

22 continuous, bounded and concave as before, the problem is a convex program and can be readily solved. Because this dual problem is dependent on a switching time, simultaneously solving for a pricing vector and a threshold policy is difficult. However, as we show in the next section, a heuristic based on the solution to the deterministic problem and the dual problem can be very effective. 5 Heuristics In this section we consider four heuristic solutions to the stochastic common pricing problem and apply them to the two-item case. The first heuristic uses the optimal switching time determined by the deterministic problem as input to the dual problem (19) in order to calculate the prices p 1 and p 2. These prices are then used to determine a threshold policy. We then compare this heuristic, Heuristic 1, to the upper bound provided by the maximal revenue of the deterministic problem and three strawmen heuristics. Heuristic 2 applies the static optimal policy given by a deterministic, pooled inventory problem; Heuristic 3 applies the static policy of the deterministic problem considered in Section 3; and Heuristic 4 applies a threshold policy based on the solution to the single-item stochastic problem where inventory is pooled. We show that for reasonable sized problems, the dynamic multiple-item policy given by Heuristic 1 provides very good solutions, even when the initial inventory ratio is very different from the actual demand ratio. Not surprisingly Heuristic 1 dominates strongly the static Heuristics 2 and 3. Further, we find it provides relatively large improvement over the dynamic threshold policy with pooled inventory in Heuristic 4. We now present the heuristics formally. Heuristic 1 Step 1: Solve the deterministic equivalent problem as in Theorem 3 and let τ D = τ (Q 1, Q 2, T ) as calculated by Theorem 3. If Q 1 /w 2 = Q 1 /w 2, let τ D = τ (Q 1 + 1, Q 2, T ). Step 2: Solve the problem in equation (19) for p 1 and p 2 assuming the switching time τ = τ D. Step 3: Using p 1 and p 2 to solve for the threshold policy X(q1, q 2 ) for 0 q i Q i (i = 1, 2). Two approximations are used to derive the thresholds, one using the L = G approximation in this step and the second using monotonicity approximation. Heuristic 2: 20

23 Let Q = Q 1 + Q 2, p 0 = p(q/t ) and ˆp = p(ˆλ), then choose a static price p = max(p 0, ˆp). Heuristic 3: Solve for τ, p 1 and p 2 for the deterministic equivalent problem and apply prices p 1 and p 2 over periods [0, τ ] and [τ, T ] respectively. Heuristic 4: Step 1: Solve for τ D as in step 1 of Heuristic 1. Step 2: Solve for p 1 and p 2 as in step 2 of Heuristic 1. Step 3: Let Q = Q 1 + Q 2 and solve for the 1-dimensional threshold policy over the single pooled item, X q, for 0 q Q. Apply this policy dynamically over time tracking only the total inventory, q = q 1 + q 2. For these heuristics we have the following: Theorem 6 Heuristics 1 and 3 are asymptotically optimal, and Heuristic 2 and 4 are asymptotically optimal when Q 1 and Q 2 increase to infinity while keeping the ratio of Q 1 /Q 2 = w 1 /w 2. While all of the heuristics are expected to perform well for large initial inventories, their performance does differ for lower values. We present below the relative gap observed between the upper bound on the revenue and revenue given by the four heuristics ( upper bound-heuristic upper bound 100%) for various Q 1 and Q 2. We let w 1 = w 2 = 0.5, and T = 10. Two demand functions are considered, linear demand with λ = 100 p, and exponential demand with λ = 100e 0.1p. We simulate 10, 000 demand sample paths for each (Q 1, Q 2 ). We observe in Table 3 that Heuristic 1 performs well for the case of linear demand, generally with 2% of the upper bound, and fair for the case of exponential demand, generally within approximately 7% of the upper bound when the total supply (Q 1 + Q 2 ) is greater than 20 units. We observe, as would be expected from our discussion above, that there is only minor differences in the performance of the two approximations (the L = G and the monotonicity approximations). We observe that the performance is best near, but not on the diagonal (Q 1 = Q 2 ). Recalling that the demand rates are given by w 1 = w 2 = 0.5, one might expect that the heuristic would perform best on the diagonal when the actual demand is proportional to the initial inventory. However, the deterministic problem 21

24 Q 1,Q (a) Q 1,Q (b) Q 1,Q (c) Q 1,Q (d) Table 3: Relative gap for Heuristic 1 for (a) linear demand using the L = G approximation; (b) linear demand using the monotonicity approximation; (c) exponential demand using the L = G approximation; and (d) exponential demand using the monotonicity approximation. gives no information regarding when the discount should take place if Q 1 /w 1 = Q 2 /w 2 because in this case τ = T and therefore we would find a single price is optimal. However, as there should be a discounted price, we reasonably solve the deterministic problem using (Q 1 + 1, Q 2 ) to find τ and subsequently p 1 and p 2 by the dual problem (19) in this case. The lack of information and somewhat arbitrary solution employed contribute to the reduced performance along the diagonal. In Table 4, we compare Heuristic 1 using the L = G approximation to Heuristics 2, 3 and 4 for the case of linear demand. We observe that both static heuristics, Heuristic 2 using pooled inventory and Heuristic 3 where inventory was not pooled, perform poorly. In comparison, we observe that Heuristic 4 that uses a dynamic policy while pooling inventory performs well, especially along the diagonal. However, off the diagonal, Heuristic 1 performs significantly better than Heuristic 4 in a relative sense. In Table 5 we present the relative improvement of Heuristic 1 over Heuristic 4 for both the linear and exponential demand cases. In the linear demand case, the mean improvement is 35% and the median is 34% with a maximum of 58%. In the exponential demand case, the mean 22

25 Q 1,Q Heuristic 1 Q 1,Q Q 1,Q Heuristic 2 Q 1,Q Heuristic 3 Heuristic 4 Table 4: Comparison of Heuristic 1 (L = G approximation), Heuristics 2, 3 and 4 for linear demand. improvement is 23.5%, the median is 20% and the maximum is 53%. This indicates that significant gains in revenue can be achieved by tracking the inventory of each item, even when a common price must be applied to them. Q 1,Q % 39% 45% 53% 55% 58% 5 41% 15% 33% 45% 54% 57% 9 48% 34% 12% 30% 42% 48% 13 50% 43% 28% 12% 24% 34% 17 57% 54% 38% 24% 8% 17% 20 57% 57% 49% 34% 19% 8% (a) Q 1,Q % 22% 34% 43% 50% 53% 5 25% 6% 19% 31% 42% 47% 9 37% 18% 5% 14% 28% 40% 13 45% 33% 17% 3% 12% 21% 17 50% 43% 29% 13% 4% 8% 20 53% 50% 36% 20% 10% 4% (b) Table 5: Relative Improvement of Heuristic 1 over Heuristic 4 under (a) linear demand and (b) exponential demand. 23