Integration of Inventory and Pricing Decisions with Costly Price Adjustments

Transcription

1 OPERATIONS RESEARCH Vol. 59, No. 5, September October 2011, pp iss X eiss INFORMS Itegratio of Ivetory ad Pricig Decisios with Costly Price Adjustmets Xi Che Idustrial Eterprise ad Systems Egieerig, Uiversity of Illiois at Urbaa Champaig, Urbaa, Illiois 61801, Sea X. Zhou, Youhua (Frak) Che Departmet of Systems Egieerig ad Egieerig Maagemet, The Chiese Uiversity of Hog Kog, Hog Kog Motivated by the widespread adoptio of dyamic pricig i idustry ad the empirical evidece of costly price adjustmets, i this paper we cosider a periodic-review ivetory model with price adjustmet costs that cosist of both fixed ad variable compoets. I each period, demad is stochastic ad price-depedet. The firm eeds to coordiate the pricig ad ivetory repleishmet decisios i each period to maximize its total discouted profit over a fiite plaig horizo. We develop the geeral model ad characterize the optimal policies for two special scearios, amely, a model with ivetory carryover ad o fixed price-chage costs ad a model with fixed price-chage costs ad o ivetory carryover. Fially, we propose a ituitive heuristic policy to tackle the geeral system whose optimal policy is expected to be very complicated. Our umerical studies show that this heuristic policy performs well. Subject classificatios: ivetory system; pricig; price adjustmet costs; fixed costs; base-stock. Area of review: Reveue Maagemet. History: Received March 2008; revisios received March 2009, December 2009, July 2010; accepted Jauary Itroductio The past decade has witessed the proliferatio of dyamic pricig practice i various idustries (see McGill ad va Ryzi 1999, Elmaghraby ad Keskiocok 2003). Facilitated by sophisticated iformatio techologies such as eterprise resource plaig (ERP) systems ad electroic tags, price chages are becomig easier. However, these chages are ot costless. Ideed, two major types of price adjustmet costs are idetified i the ecoomics literature: maagerial costs ad physical costs (or meu costs i the ecoomics literature). Maagerial costs are directly related to the time ad attetio required of maagers to gather the relevat iformatio ad to make ad implemet decisios (Berge et al. 2003). Typically, these costs arise withi a firm as a result of iformatio gatherig, decisio makig, ad commuicatios. Physical costs are icurred for retailers such as Best Buy ad Target through the labor costs that result from maually chagig thousads of shelf prices withi their stores. Other physical costs that firms such as 3M, Ericsso, ad Bed Bath ad Beyod experiece are the costs associated with producig, pritig, ad distributig their price books or catalogs. May empirical studies have show that both maagerial costs ad physical costs are sigificat i retailig ad other idustries (Rotemberg 1982b, Levy et al. 1997, Slade 1998, Aguirregabiria 1999, Berge et al. 2003, Zbaracki et al. 2004, Kao 2006). I particular, Levy et al. (1997) ad Slade (1998) fid through empirical studies that physical costs play a crucial role i the price-settig behavior of retail supermarkets. I a relatively recet study, Berge et al. (2003) estimate the maagerial costs icurred by firms whe they chage prices to be more tha six times the magitude of the physical costs. Ideed, as Berge et al. (2003) further assert, the physical costs of chagig prices have bee reduced because of advaces i iformatio techology, whereas the maagerial costs of doig so might actually have icreased due to the added complexity of dealig with o-lie ad i-store pricig, the added data from customers buyig through web sites, ad the additioal kowledge ad systems required to uderstad the e-busiess. I additio to these empirical fidigs, a umber of quatitative models for aalyzig optimal pricig strategies have also bee developed i the ecoomics literature. With a fixed price adjustmet cost ad stochastic iflatio, Sheshiski ad Weiss (1977) show that s S-type pricig policies are optimal for a cotiuous-time model with determiistic price-depedet demad. Capli ad Spulber (1987) cosider multiple firms that adopt a s S pricig policy ad show that the aggregate price of idividual firms chages with moey supply i the market over time ad thus aggregate price stickiess disappears. See Sheshiski ad Weiss (1993) ad refereces therei for further discussio o costly price adjustmet ad s S pricig policies. 1144

2 Che, Zhou, ad Che: Itegratio of Ivetory ad Pricig Decisios Operatios Research 59(5), pp , 2011 INFORMS 1145 Eve though the ecoomics literature illustrates that price adjustmet costs do exist ad play a crucial role i shapig firms pricig strategies (e.g., Sheshiski ad Weiss 1993), the literature o ivetory ad pricig coordiatio by the operatios maagemet commuity has largely igored them. The purpose of this paper is to fill this gap by icorporatig price adjustmet costs ito itegrated ivetory ad pricig models ad ivestigatig the structural properties of optimal ivetory ad pricig policies. More specifically, we cosider a firm that maages a sigle product, periodic-review fiite-horizo ivetory system with stochastic price-depedet demad ad costly price adjustmets. Similar to Federgrue ad Hechig (1999), at the begiig of each period, a orderig decisio is made ad a liear orderig cost is icurred. The sellig price of the period is also determied. The demad i each period is radom ad depeds o the sellig price set at the begiig of the period. However, differet from Federgrue ad Hechig (1999) ad other papers o ivetory ad pricig coordiatio, a price-adjustmet cost is icurred whe the price i oe period is set to be differet from that i the previous period. This might ivolve a fixedcost compoet that is idepedet of the magitude of the price chages ad a variable-cost compoet that depeds o their magitude. The objective of the firm is to coordiate the pricig ad ivetory repleishmet decisios i each period to maximize its expected total discouted profit over a fiite plaig horizo. It turs out that the structure of the optimal policy i our model allowig for the full geerality is rather complicated. Thus, we cosider two special cases of our model: a model with ivetory carryover ad o fixed price-chage costs ad a model with fixed price-chage costs ad o ivetory carryover. For the first model with a geeral covex price adjustmet cost, we prove that a state-depedet base-stock ivetory ad list-price policy is optimal. I this policy, the base-stock level is oicreasig i the price of the previous period. If the iitial ivetory level is below the base-stock level, the the list price is charged; otherwise, a discout is offered. Both the list price ad the discout deped o the price of the previous period. Whe the variable cost is piece-wise liear (with two liear pieces) ad covex, we ca further prove that the optimal price follows a two-sided threshold-type policy, i which the threshold levels are oicreasig i the iitial ivetory level ad odecreasig i the price of the previous period. For the secod model, we employ Scarf s (1960) cocepts of K-cocavity ad the symmetric K-cocavity proposed i Che ad Simchi-Levi (2004a) to show that the optimal pricig strategy is the oe-sided s S policy if the price chage is ui-directioal ad the two-sided s S A policy if it is bi-directioal. For the first model, we assume that usatisfied demad is backlogged, whereas for the secod, we assume that it is filled by a emergecy order or alteratively, by a order at the begiig of the ext period. Uder additioal coditios, all our structural results hold for models with lost sales. Fially, for the geeral problem with fixed plus liear variable price-adjustmet costs, we develop a ituitive heuristic policy a base-stock ivetory ad two-sided s S pricig policy to maage ivetory ad set sellig prices. We provide a approach to computig the policy parameters. Compared with the optimal policy, which is difficult to compute, our heuristic policy is ameable to practical implemetatio oce the cotrol parameters have bee calculated. I additio, the umerical study demostrates that the heuristic is quite effective. Our paper falls withi the growig research stream o ivetory ad pricig coordiatio that started with Whiti (1955). For a review of this literature, readers are referred to Elmaghraby ad Keskiocak (2003), Cha et al. (2004), ad Che ad Simchi-Levi (2011). Sigificat progress has bee made recetly i aalyzig itegrated ivetory ad pricig models with a fixed orderig cost ad stochastic demad for both backlog (see Chao ad Zhou 2006; Che ad Simchi-Levi 2004a, b, 2006; Huh ad Jaakirama 2008) ad lost sales (see Che et al. 2005, Huh ad Jaakirama 2008, Sog et al. 2008) cases. The previous studies, however, predomiatly assume that the price adjustmet is costless. To the best of our kowledge, there are oly two papers that are closely related to ours: Aguirregabiria (1999) ad Celik et al. (2009). Motivated by the pheomeo i practice of large periods without omial price chages ad short periods with low prices, Aguirregabiria (1999) proposes a ivetory ad pricig model that icorporates both fixed orderig costs ad fixed price-adjustmet costs but focuses more o empirical studies. Celik et al. (2009) focus o a cotiuous-time reveue maagemet problem with costly price chages. They characterize the optimal pricig polices for the case of ample ivetory ad develop several heuristics based o a correspodig fluid model. However, their model does ot take ito accout the ivetory repleishmet decisio. It is also appropriate to metio here that Netessie s (2006) paper, which formulates ad aalyzes a determiistic model to optimize the timig of price chages, also recogizes the importace of the impact of these costs o ivetory ad pricig decisios. The rest of this paper is orgaized as follows. I 2 we preset the geeral model, itroduce the otatio, ad defie a class of geeral ivetory ad pricig policies. I 3 we cosider the case without fixed costs. I 4 we aalyze the model without ivetory carryover. I 5 we provide the coditios uder which the results of the backlog case ca be exteded to the lost sales case. We develop a heuristic policy for the geeral system i 6 ad coduct a umerical study. We also exted the heuristic to the case with ivetory depedet price adjustmet costs. Fially, i 7 we coclude the paper with a discussio of possible extesios.

3 Che, Zhou, ad Che: Itegratio of Ivetory ad Pricig Decisios 1146 Operatios Research 59(5), pp , 2011 INFORMS 2. The Geeral Model Settig Cosider a firm that maages a sigle-product, periodicreview ivetory system with price-depedet demad i a N -period plaig horizo. I each period, the firm eeds to make both ivetory repleishmet ad pricig decisios so as to maximize the total expected discouted profit over the plaig horizo. The time sequece of evets is as follows. At the begiig of each period, the firm first reviews its ivetory level ad makes a repleishmet decisio. The order the firm places is received immediately (leadtime 0) ad icurs a variable cost of c per uit. The, based o the ivetory level after repleishmet ad the sellig price i the previous period, the firm decides whether to adjust the price ad, if so, by how much. Durig this period, the radom demad is realized, ad all the reveues ad costs are calculated at the ed of the period. Let p be the sellig price of period. For tractability, we assume that demad i period, deoted by D p, depeds liearly o a stochastic compoet,, ad the sellig price, p. Specifically, D p = p where ad are two positive radom variables, ad =. For simplicity, are assumed to be idepedet of each other across time. The liear demad assumptio is made to esure that the oe-period expected profit (excludig possible fixed cost compoets) is joitly cocave i ivetory ad price. It should be oted that liear demad fuctios are commoly used i the literature ad i practice (e.g., Petruzzi ad Dada 1999, Simchi-Levi et al. 2005). Let p ad p be the lower ad upper bouds of sellig price p. We assume D p to be oegative for ay p p p ad ay realizatio of. (Therefore, if is draw from a positively valued rage, the will take values i 0 / p, which esures that D p is oegative.) Let D p = ED p. It is clear that the expected reveue of period, give as R p = Ep D p = p E E p is cocave i p. If the price-adjustmet cost ivolves oly a fixed compoet that is idepedet of the price-adjustmet magitude, the results i this paper hold for a more geeral demad fuctio D p = f p, where f f p f p is a time-idepedet strictly icreasig fuctio, ad the expected reveue f 1 de E d (f 1 : f p f p p p is the iverse fuctio of f ) is cocave i d. I this case, we ca use the ivetory level ad d as the primal decisio variables i our aalysis rather tha the ivetory level ad price. Differet from the majority of papers o ivetory ad pricig coordiatio, we assume that a price chage from oe period to the ext is costly. Defie A = 1 if A 0; otherwise, A = 0. The cost of a price adjustmet with magitude i period is deoted by C p, which is give as C p = K + U (1) where the fixed cost K represets the meu cost or physical cost associated with a price chage, ad the variable cost U represets the maagerial or customer cost depedig o the magitude of the price adjustmet. I the ecoomics literature, it is commoly assumed that the variable cost U is covex ad icreases with the size of the price chage because the decisio ad iteral commuicatio costs are higher for larger price chages (see Zbaracki et al for direct evidece from idustrial markets ad detailed aalysis). Several forms of U have bee used i the ecoomics literature, icludig piecewise liear fuctios U = u (Tsiddo 1991, Slade 1998) ad quadratic fuctios U = u 2 (Rotemberg 1982a, b; Roberts 1992). These studies have bee icoclusive about the relative magitude of K ad U, with some suggestig that the meu cost is small ad isigificat (see McCallum 1986, Koieczy 1993), ad others fidig it to be large (see Levy et al. 1997, Berge et al. 2003). Thus, several differet forms of price adjustmet costs have bee used. For istace, Slade (1998) cosiders a model with both a fixed cost ad piecewise liear variable costs ad empirically shows that both exist. Rotemberg (1982a) cosiders a model with a quadratic variable cost aloe ad states that this cost primarily accouts for the implicit cost that results from customers ufavorable reactio to large price chages. Rotemberg (1982b) further fids evidece of a sigificat variable price-adjustmet cost from aggregate U.S. ecoomic data, whereas Aguirregabiria (1999) ad Kao (2006) focus oly o fixed costs. A more geeral priceadjustmet cost is proposed by Celik et al. (2009), who cosider a meu cost K that may deped o the ivetory level i additio to a covex variable price adjustmet cost. Such a cost is reasoable i settigs i which the price tag law requires a price tag for each item. We provide a brief discussio of the difficulty of icludig it i our model ad exted our heuristic to this more geeral settig i 6. Depedig o whether ivetory ca be carried over betwee periods, we aalyze two differet models. The first is a durable product model i which ivetory ca be carried over from oe period to aother. I each period, the edig ivetory icurs a uit holdig cost h, ad ay excess demad is assumed to be backlogged ad to icur a uit shortage cost b. The secod model is a perishable product model that does ot allow ivetory carryover. Ay leftover ivetory at the ed of each period is salvaged at r per uit. I additio, ay usatisfied demad is filled by a emergecy order at cost b per uit. I the ext two sectios, we aalyze these two models ad characterize the optimal pricig ad ivetory repleishmet policies.

4 Che, Zhou, ad Che: Itegratio of Ivetory ad Pricig Decisios Operatios Research 59(5), pp , 2011 INFORMS 1147 At the begiig of each period, give startig ivetory level x ad the sellig price of the previous period, p 1, let v x p 1 deote the total discouted optimal profit from period to N mius c x ad be the discout factor, 0 1. The dyamic programmig formulatio of this problem is v x p 1 = max yx pp p Cp p p 1 + L y p (2) where L y p = R p + G y p + c +1 Ey D p + Ev +1 y D p p (3) i which if excess demad is backlogged ad ivetory carryover is allowed, the G y p = c y h Ey D p + b ED p y + which is the modified oe-period ivetory cost fuctio (ote that x + = max0 x, ad for = 1, c 1 x is also icluded); if ivetory carryover is ot allowed (or excess demad is lost), G y p ad v +1 will be defied i 4 (or 5). I the defiitio of L y p, the first term is the expected reveue; the secod is the orderig, ivetory holdig, ad shortage costs; ad the last two terms are the optimal expected discouted profit from period + 1 to the ed of the plaig horizo. Note that G is joitly cocave because the demad fuctio is liear, which is critical for our subsequet aalysis. For ease of expositio, we assume that the iitial price p 0 is give ad v N +1 x = 0 without loss of geerality. To ed this sectio, we defie the followig class of ivetory ad pricig policies. We show that most of the policies we are about to discuss belog to this class. Defiitio 1 (Base-Stock Ivetory ad Two-Sided s S Pricig Policy). A ivetory repleishmet ad pricig policy y p is called a base-stock ivetory ad two-sided s S pricig policy if, give the sellig price p 1 of period 1 ad the startig ivetory level x of period, ivetory repleishmet follows a base-stock policy with level ȳp 1, i.e., y = maxȳp 1 x; ad sellig price p is determied by two pairs of parameters sx Sx ad zx Zx with sx Sx Zx zx accordig to the followig: Sx if p 1 sx, p = Zx if p 1 zx, (4) p 1 otherwise. Note that all the parameters ad sets are period-depedet. For brevity, we deote this class of policies by, which icludes several simpler ad commoly see policies as special cases. Whe sx = Sx ad zx = Zx, the it becomes a two-sided threshold policy. Fially, whe oly ui-directioal price adjustmet is allowed, the pricig policy is reduced to a oe-sided policy. 3. Model with Ivetory Carryover I this sectio, we aalyze our first model formulated as (2), i which ivetory ca be carried over from oe period to aother. Whe ivetory ca be carried over ad fixed price-adjustmet costs are icurred, problem (2) becomes very complicated. Ideed, cocepts such as K-covexity ad symmetric K-covexity, which are effective for aalyzig stochastic ivetory models ad itegrated ivetory ad pricig models with ecoomies of scale, are ulikely to be applicable to a two-dimesioal dyamic program (2) with a fixed price-adjustmet cost, ad thus a totally ew covexity related cocept might be eeded. I the rest of the sectio, we therefore focus o the model i which a price adjustmet icurs oly variable costs, i.e., K = 0. As Rotemberg (1982b) observes, i some settigs variable costs might be sigificatly more importat tha fixed costs. We provide a heuristic i 6 for the model with a positive fixed price-adjustmet cost Covex Variable Price-Adjustmet Costs I this sectio, we assume the variable price-adjustmet cost U is covex ad characterize the optimal pricig ad ivetory orderig policy. The followig two lemmas preset some structural properties of the value fuctios, which will help us to characterize the optimal policies. Lemma 1. (a) v x p 1 is oicreasig i x. (b) v x p 1 is joitly cocave i x ad p 1. Proof. For part (a), ote that the objective fuctio i (2) is idepedet of x. The result is immediate because we are dealig with a maximizatio problem, ad the feasible domai of y becomes smaller whe x icreases. We prove part (b) by iductio. Because v N +1 = 0, the result holds for N + 1. Suppose it is true for some + 1, N. For, observe that the first two terms i (3) are cocave i p ad p 1, G y p is joitly cocave i y ad p, ad R p c +1 D p = p c +1 D p is cocave i p as D p is liear ad decreasig i p. The cocavity of the last term i L y p follows from the iductive assumptio ad the liearity of D p. Thus, the objective fuctio i the maximizatio problem (2) is joitly cocave i y, p, ad p 1. This immediately implies that v x p 1 is joitly cocave i x ad p 1 because cocavity is preserved uder maximizatio (see Propositio of Simchi-Levi et al. 2005). Our ext result idicates the submodularity of v x p 1 ad L y p.

5 Che, Zhou, ad Che: Itegratio of Ivetory ad Pricig Decisios 1148 Operatios Research 59(5), pp , 2011 INFORMS Lemma 2. v x p 1 is submodular i x ad p 1 ad L y p is submodular i y ad p. Proof. To facilitate the aalysis, we defie w x p = v x p ad J y p = L y p. The, w x p 1 = max U p 1 p+ L y p (5) yx p p p with w N +1 = 0, ad J y p=r p+c +1 Ey D p +G y p +Ew +1 y D p p (6) To prove the lemma, it is sufficiet to show that w x p 1 is supermodular i x ad p 1 ad J y p is supermodular i y ad p, which we prove i the followig. We prove that w x p 1 is supermodular i x ad p 1 by iductio. Note that the supermodularity of J y p will be proved simultaeously. First, w N +1 x p N = 0 is obviously supermodular. Assume that w +1 x p is supermodular. We ow prove that w x p 1 is supermodular. Note that the fuctio U p 1 p is a composite of a oedimesioal cocave fuctio with p 1 p ad thus is supermodular i p ad p 1 (see Simchi-Levi et al. 2005, Theorem 2.3.6). The same argumet ca be applied to verify the supermodularity of G y p as D p = + p. Meawhile, R p + c +1 Ey D p is a separable fuctio ad thus is clearly supermodular i y ad p. It remais for us to prove that w +1 y D p p is supermodular for a give sample path of. For ay two pairs y p ad y p with y > y ad p < p, we have w +1 y D p p w +1 y D p p w +1 y D p p w +1 y D p p w +1 y D p p w +1 y D p p where the first iequality follows directly from the assumptio of the supermodularity of w +1 y p, ad the secod iequality follows from the cocavity of w +1 y p i its first compoet. Thus, w +1 y D p p, ad hece Ew +1 y D p p, is supermodular i y ad p, which implies that J y p is also supermodular i y ad p. Therefore, the objective fuctio i the maximizatio problem (5) is supermodular i y, p, ad p 1. I additio, it is easy to see that the feasible set is a lattice. Hece, w x p 1 is supermodular, as the maximizatio of a supermodular fuctio over a lattice remais still supermodular (see Theorem of Topkis 1998, or Propositio of Simchi-Levi et al. 2005). The submodularity of v x p 1 is quite ituitive ad implies that the margial optimal profit with respect to ivetory level x decreases as the previous price p 1 icreases. The ituitio behid this is that whe a firm has a additioal uit of ivetory, it teds to set a lower price; with a higher startig price, the possible cost of price adjustmet is also higher, thus causig a decrease i profit. We ow proceed to show the optimal policy. Let ˆP p 1 y = max arg max U p 1 p + L y p pp p p p 1 y p 1 = max arg max U p 1 p + L y p y pp p The followig theorem characterizes the optimal pricig ad ivetory repleishmet policy. Theorem 1. For a geeral covex price-adjustmet cost, if a bi-directioal price chage is allowed, the (a) y p 1 is oicreasig ad p p 1 is odecreasig i p 1, ad ˆP p 1 x is oicreasig i x but odecreasig p 1. Furthermore, ˆP p 1 x/p 1 1. (b) The optimal ivetory policy y = miy p 1 x, ad the optimal pricig policy p = mi ˆP p 1 x p p 1. Proof. From the defiitio of J y p i the proof of Lemma 2, U p 1 p + J y p is supermodular i p y p 1. Thus, as p p 1 y p 1 = max arg max U p 1 p + J y p p p p y the first half of part (a) follows from Theorem i Topkis (1998), which cocers the mootoicity of the maximizer of a supermodular fuctio. Applyig a similar idea ca show ˆP p 1 x is oicreasig i x but odecreasig p 1. To prove the secod half of part (a), observe that the fuctio U ˆp + L x p 1 ˆp is supermodular i ˆp p 1 for a give x from the cocavity of L x ad is supermodular i ˆp x for a give p 1 from Lemma 2. Thus, p 1 ˆP p 1 x is icreasig i p 1, which implies that ˆP p 1 x/p 1 1. From the joit cocavity of fuctio U p 1 p + L y p, we ca coclude that the optimal solutio to problem (2) is give by p p 1 y p 1 if x y p 1 ad x ˆP p 1 x if x > y p 1. The foregoig theorem implies that whe the startig ivetory level x is less tha y p 1, the firm should order up to y p 1 ad set the price at p p 1 ; otherwise, it should ot order ad should sell the product at ˆP p 1 x. Note that whe the variable price-adjustmet cost takes a geeral covex form, the optimal policy is fully state-depedet. I particular, whe the startig ivetory level is higher tha y p 1, the optimal price that the firm should set depeds o both state variables. Thus, the structural result i this theorem is oly a partial characterizatio of the optimal policy. 1

6 Che, Zhou, ad Che: Itegratio of Ivetory ad Pricig Decisios Operatios Research 59(5), pp , 2011 INFORMS Piecewise Liear Price-Adjustmet Costs Whe the variable price-adjustmet cost is piecewise liear, i.e., U p p 1 = u p p 1, we are able to derive sharper results ad characterize the optimal policy i greater detail. To characterize the optimal policy with this form of price adjustmet cost, we defie several fuctios ad parameters as follows. Let g x p = p + L x p where is a real umber. Defie P x = arg maxg x p 2 pp p Y = max arg maxg x P x x ad y p = max arg maxl y p y Note that if p p 1 ad = u or if p < p 1 ad = u, the p 1 + g x p is the bracketed fuctio i (2). Let p x = P u x, p + x = P u x, Y + = Y u, Y = Y u, P = p Y +, ad P + = p+ Y. It is straightforward to verify that the foregoig fuctios ad quatities are well defied. Also ote that L x p is joitly cocave due to Lemma 1. Lemma 3. (a) P x is oicreasig i ad x. Thus, p x ad p+ x are oicreasig i x ad p x p + x for ay x. (b) Y + Y, P P +. (c) y p is oicreasig i p, y P = Y + ad y P + = Y. (d) p p + y p if p P + ad p p + y p if p P +. (e) p p y p if p P ad p p y p if p P. Proof. Defie for ay, x, ad p, f x p = p + J x p where J x p is give i (6). Because p is supermodular i ad p ad J x p is supermodular i x ad p from Lemma 2, f x p is supermodular i, x, ad p. Observe that P x = arg maxf x p p p p Y P Y = max arg max x p p p f x p From the mootoicity of a maximizer of a supermodular fuctio (Theorem of Topkis 1998, or Theorem of Simchi-Levi et al. 2005), P x is oicreasig i ad x. As a result, p x ad p+ x are oicreasig i x ad p x p+ x for ay x. I additio, Y P Y is odecreasig i, ad hece Y + Y ad P P +. Thus, parts (a) ad (b) hold. We ow prove part (c). Observe that y p = max arg max f y p y for ay. Hece, y p is oicreasig i p (agai from Theorem of Simchi-Levi et al. 2004). Because y p is idepedet of, we have y P = Y + ad y P + = Y. Thus, part (c) holds. Now cosider part (d). Let be ay elemet i the supergradiet set of the cocave fuctio L y p p. We immediately have that y p p arg max x pp p g x p. Because L y p p is cocave, if p P +, the we have u ; otherwise (i.e., if p > P + ), we have u. I either case, p = arg max f y p p p p p Note that p + y p = arg max f u y p p p p p Thus, from the supermodularity of f x p i ad p, p p + y p for p P + ad p p+ y p for p P +. Hece, part (d) holds. Fially, part (e) ca be proved similarly to part (d). A sketch of the fuctios (y p, p + x, p x) ad parameters (Y + Y P + P ) is provided i Figure 2. The properties of these are importat for characterizig the structure of the optimal policy, which we shall see i the followig. We start with the optimal policies for the cases i which oly sigle-directioal price adjustmets are allowed throughout the horizo. Note that i such cases, the correspodig dyamic recursio (2) is derived uder the costrait p 1 p p i the markup-oly case while uder p p p 1 i the markdow-oly case for each period. Theorem 2. Cosider a system with liear priceadjustmet costs that allows oly a sigle-directioal price adjustmet throughout the horizo. Give the startig ivetory level x ad the price of the previous period p 1 : (a) If oly markup is allowed, the optimal ivetory ad pricig policy y 1 p1 belogs to. Specifically, it follows a base-stock ivetory ad oe-sided threshold pricig policy with parameters ȳp 1 = miy + y p 1 ad Sx = sx = mip p x; (b) If oly markdow is allowed, the optimal ivetory ad pricig policy y 2 p2 belogs to. Specifically, it follows a base-stock ivetory ad oe-sided threshold pricig policy with parameters ȳp 1 = maxy y p 1 ad Zx = zx = mip + p+ x.

7 Che, Zhou, ad Che: Itegratio of Ivetory ad Pricig Decisios 1150 Operatios Research 59(5), pp , 2011 INFORMS Figure 1. The optimal ivetory ad pricig policies for oe-directioal price chages. (a) The optimal ivetory ad pricig policy with markup (b) The optimal ivetory ad pricig policy with markdow x p (x) x + p (x) + Y II III y (p 1 ) III IV I IV y (p 1 ) Y II V I V P p 1 + P p 1 The optimal policy is illustrated i Figure 1. Whe oly a markup is allowed (Figure 1(a)), if the startig state x p 1 of period is i regio I, the it is optimal to order up to Y + ad to mark the price up to P. If the state is i regio II, the it is optimal to order othig ad icrease the price to p x. If the state is i regios III ad IV, the it is optimal to do othig. Fially, if the state lies i regio V, the the price is kept uchaged, ad it is optimal to order up to y p 1. It is oteworthy that p x ad y p 1 have the illustrated relatioship ad cross oly oce at Y + P because y P = Y +, p Y + = P ad because of part (e) of Lemma 3. Whe oly a markdow is allowed (Figure 1(b)), if the iitial state x p 1 lies i regios I ad II, the the price should be kept uchaged, ad it is optimal to order up to y p 1. If the state is i regio III, the othig should be doe. If the state is i regio IV, the the firm should order othig ad mark the price dow to p + x. Fially, if the state is i regio V, the it should order up to Y ad mark the price dow to P +. Agai, the visualized relatioship betwee p+ x ad y p 1 follows from y P + = Y, p+ Y = P +, ad from part (d) of Lemma 3. I geeral, whe a bi-directioal price chage is allowed, with the properties preseted i Lemma 3, the optimal policy is characterized i the followig theorem. Its proof ad the proof of Theorem 2 are provided i Appedix A i the electroic compaio, which is available as part of the olie versio that ca be foud at or.joural.iforms.org. Theorem 3. For a system with liear price-adjustmet costs, give the startig ivetory level x ad the price of the previous period p 1, the optimal ivetory ad the pricig policy y p of period belogs to. Specifically, it follows a base-stock ivetory ad two-sided threshold pricig policy with parameters ȳp 1 = maxmiy p 1 Y + Y, sx = Sx = mip x P ad zx = Zx = mip+ x P +. The optimal policy for the case with bi-directioal price chages is illustrated i Figure 2. If the startig state x p 1 I, the it is optimal to order up to Y + ad to mark the price up to P. If the startig state x p 1 II, the it is optimal to order othig ad to mark the price up to p x. If the startig state x p 1 III V, the it is optimal to order othig ad to fix the price at p 1. If the startig state x p 1 IV VI, the it is optimal to order othig ad to mark the price dow to p + x. If the startig state x p 1 VIII IX, the it is optimal to order up to y p 1 ad to fix the price at p 1. Fially, if the startig state x p 1 VII, the it is optimal to order up to Y ad to mark the price dow to P +. The optimal policy provided i Theorem 3 implies that the optimal price p is odecreasig i p 1 ad oicreasig i x (it also follows from the structural result for the case with a geeral covex price-adjustmet cost). It should be poited out that our structural results for the optimal policy exted those of Federgrue ad Hechig (1999), who illustrate i a correspodig itegrated ivetory ad Figure 2. x Y + Y II p (x) I The optimal ivetory ad pricig policy for bi-directioal price chages. III p+ (x) P IX V y (p 1 ) VIII IV P + VI VII p 1

8 Che, Zhou, ad Che: Itegratio of Ivetory ad Pricig Decisios Operatios Research 59(5), pp , 2011 INFORMS 1151 pricig model with costless price adjustmets that a basestock list price policy is optimal. We ed this sectio with the ext result, which shows how the optimal policy parameters chage with the uit price-adjustmet cost u. Propositio 1. If the variable price adjustmet cost u icreases, the Y P, ad p x decrease ad Y +, P +, ad p + x icrease. However, y p 1 is idepedet of u. Proof. The result follows directly from Lemma 3. We provide some ituitio about the last result. Observe that for a give iitial ivetory level x i period ad sellig price p 1 i period 1, the magitude of the price chage (if ay) decreases if the variable price-adjustmet cost u icreases. Thus, the price raise-up-to levels P ad p x decrease, ad the price decrease-dow-to levels P + ad p + x icrease. At the same time, a lower price P implies greater demad ad thus a higher order-up-to level Y, whereas a higher price P + implies less demad ad thus a lower order-up-to level Y +. It is clear that the optimal policy parameters after period are idepedet of u. However, it is ot clear how the optimal policy parameters before period deped o u. 4. Model with No Ivetory Carryover I this sectio, we cosider the secod model i which o ivetory ca be carried over from oe period to the ext. I each period, we assume that ay usatisfied demad must be fulfilled by a expedited order, which icurs a emergecy orderig cost of b per uit. A alterative iterpretatio of emergecy orders is that ay usatisfied demad is filled by a order i the followig period, ad a pealty is charged. It is worth otig that similar assumptios are made i previous studies of ivetory models (e.g., Eeckhoudt et al. 1995, Simchi-Levi et al. 2005, Che 2009). The usold uits at the ed of each period are salvaged with a per-uit value of r (if it is egatively valued, the the salvage value represets the disposal cost). The assumptio that the usold uits of the product caot be held i ivetory for the ext period, but istead yield a certai salvage reveue, is ot as restrictive as it appears i that this assumptio is appropriate to model settigs that ivolve perishable products with a short shelf life. To avoid trivialities, we assume that b > c > r. Although we allow the fixed price-chage cost to be greater tha zero (i.e., K > 0 i (1)) i the curret model, for tractability, we require that U = u. I this settig, we are able to characterize the optimal pricig ad orderig policy with the presece of both fixed ad variable priceadjustmet costs. As we assume that all demads eed to be satisfied ad that the remaiig ivetory is salvaged at the ed of each period, the startig ivetory of every period is zero. Thus, it suffices to use the sellig price of the last period as the sole state variable. The repleishmet decisio is thus simplified to the orderig quatity Q of each period. With a slight abuse of otatio, we still let v p 1 deote the total discouted optimal profit from periods to N. The dyamic programmig formulatio of the problem is v p 1 = max Q0 pp p Cp p p 1 + R p where + G Q p + v +1 p (7) G Q p = c Q b ED p Q + + r EQ D p + models the orderig ad ivetory-related costs. Agai, due to the liearity of the demad fuctio, it is ot difficult to see that G Q p is joitly cocave i Q ad p. Hece, we ca first solve the optimal orderig quatity give the price p, Q p, which is the solutio to b c b r PrQ D p = 0 (8) The followig result illustrates that the optimal orderig quatity i period is oicreasig i the sellig price that is set for the period. Its proof is established by showig the submodularity of G Q p, which follows a routie argumet ad thus is omitted here. Lemma 4. Q p is oicreasig i p. Substitutig Q p ito (7), it is the simplified to v p 1 = max pp p Cp p p 1 + L p (9) where L p = G p + R p + v +1 p ad G p = G Q p p is cocave i p. Note that after optimizig the orderig quatity Q, which is myopic, the problem is reduced to oe with a sigle decisio, ad the foregoig dyamic program recursio is almost idetical to that which correspods to aother classical ivetory model the stochastic cash balace problem with symmetric orderig ad retur costs (see Appedix B for more details). Thus, the approach proposed i Che (2003) ad Che ad Simchi-Levi (2009) for aalysis of the stochastic cash balace problem ca be slightly modified to characterize the optimal pricig policy of the model developed here. I additio, the structure of our model s optimal pricig policy is essetially idetical to that of the optimal policy of the stochastic cash balace problem.

9 Che, Zhou, ad Che: Itegratio of Ivetory ad Pricig Decisios 1152 Operatios Research 59(5), pp , 2011 INFORMS The aalysis i Che (2003) ad Che ad Simchi-Levi (2011) is built upo the followig defiitio of symmetric K-cocavity. Defiitio 2. A real-valued fuctio f a b R is called symmetric K-cocave for K 0 if for ay x 0 x 1 a b with 0 1, f 1 x 0 + x 1 1 f x 0 + f x 1 max 1 K (10) (The defiitio i Che ad Simchi-Levi 2004a assumes that a = ad b =. However, it is straightforward to exted the defiitio by restrictig the fuctio i a bouded iterval a b.) The cocept of symmetric K-cocavity is a geeralizatio of Scarf s (1960) classical cocept of K-cocavity ad is itroduced by Che ad Simchi-Levi (2004a) whe they aalyze a itegrated ivetory ad pricig model with fixed orderig cost ad multiplicative demad. Iterestigly, this cocept has applicatios i the stochastic cash balace problem. We list the properties that are useful to our aalysis i Appedix C. Defie P argmax =mi u p+l p p p p P + argmax =mi u p+l p p p p X =p pp u p+l p K u P +L P X + =p P + pu p+l p K +u P + +L P + Let s = if X if X is oempty, ad otherwise s = p; z = if X + if X+ is oempty, ad otherwise z = p. It ca easily be show that s P P + z from their defiitios. If oly a markup or markdow is allowed throughout the plaig horizo, the the dyamic program recursio is almost idetical to that which correspods to the classical stochastic ivetory model aalyzed i Scarf (1960). I this case, we ca apply the cocept of K-cocavity to prove that a oe-sided s S pricig policy is optimal. It should be poited out that the policy parameters i the followig result have differet values tha those i the case with bi-directioal price chages, eve though they share a commo otatio. Propositio 2. (a) If oly markups are allowed i the etire plaig horizo, the for period = 1 N, the v p fuctio is K-cocave i p p, ad the optimal policy belogs to. Specifically, the optimal ivetory policy is base-stock with parameter ȳp 1 = Q p p 1, whereas the optimal price p is a oe-sided s S pricig policy with sx = s ad Sx = P. (b) If oly markdows are allowed i the etire plaig horizo, the for period = 1 N, the w p = v p fuctio is K-cocave i p p ad the optimal policy belogs to. Specifically, the optimal ivetory policy is base-stock with parameter ȳp 1 = Q p p 1, whereas the optimal price p is a oe-sided s S pricig policy with zx = z ad Zx = P +. Theorem 4. (a) v p ad L p are symmetric K-cocave; ad (b) the optimal ivetory policy is base-stock with parameter ȳp 1 = Q p p 1, ad the optimal price p is determied by P if p 1 s, p 1 P p = p 1 if p 1 s s +P /2, if p 1 s +P /2P + +z /2, p 1 P + if p 1 P + +z /2z, P + if p 1 z. (11) Proof. Because our dyamic program recursio is almost idetical to that for the stochastic cash balace problem with symmetric orderig ad returig costs, we oly sketch the proof of the result here; for a detailed proof, iterested readers are referred to the correspodig recursio i Che ad Simchi-Levi (2009). The theorem is proved by iductio o. By assumig v +1 p is symmetric K-cocave, we ca show that L p ad v p are symmetric K-cocave by Lemma 5 part (b) ad Lemma 6 i Appedix B, respectively. The structure of the optimal policy at period comes from the properties of the symmetric K-cocavity of L p ad the cocavity of G p ad R p by usig Lemma 5 part (d) for the markup directio ad a symmetric argumet for the markdow directio. The optimal pricig policy is visualized i Figure 3, i which the dashed lies specify differet regios. Whe there is o ivetory carryover, if p 1 is i regio I, the the price should always be marked up to P ; if p 1 is i regio II ad it is optimal to chage the price, the it should be marked up to P ; otherwise, it should be kept uchaged; if p 1 is i regio III, the the price should ot be chaged; if p 1 is i regio IV ad it is optimal to chage the price, the it should be marked dow to P + ; otherwise, it should ot be chaged; ad fially, if p 1 is i regio V, the price should be marked dow to P +. Note that the pricig policy is more complicated tha the two-sided s S policy defied i Defiitio 1 ad ca be regarded as a two-sided s S A policy (see more discussio o s S A policy i Che ad Simchi-Levi 2004), which icludes regios s s + P /2 ad P + + z /2 z i which the optimal policy is ambiguous. Sheshiski ad Weiss (1977) also show the s S-type pricig policy to be optimal for a ifiite-horizo, determiistic, cotiuous-review problem with iflatio ad a

10 Che, Zhou, ad Che: Itegratio of Ivetory ad Pricig Decisios Operatios Research 59(5), pp , 2011 INFORMS 1153 Figure 3. Optimal pricig policy with o ivetory carryover. I II III IV V p s s + P 2 P z + P P z p p 1 fixed price-adjustmet cost. Nevertheless, their model settig is differet from ours, ad the mai driver of the price chage is iflatio, i.e., iflatio causes the sellig price to drift cotiuously from the iitial price level S to the level s at which poit the price is marked up to S ad this cycle the repeats itself. Nevertheless, it ca be see that the key driver of price chages i our settig is the ostatioarity of the system parameters ad the market demad. Remark. It is oteworthy that the assumptio of statioary fixed costs ca be relaxed. I particular, the precedig results hold eve for time-depedet fixed costs, deoted by K, as log as 1 K is oicreasig i. 5. Lost Sales Model I this sectio, we cosider a lost sales model, i which usatisfied demad i each period is lost. Differet from the backlog case, we do ot assume ay specific form of demad fuctio D p here but oly require it to be decreasig cocave ad twice differetiable i p ad strictly icreasig i (ote that is a scalar i this case). For example, D p = D p + or D p = D p, with D p beig decreasig cocave. We start by discussig the case of ivetory carryover. The resultig dyamic program is as follows. v x p 1 = max U p 1 p + L y p (12) yx pp p with v N +1 = 0, ad L yp =p c +1 +h EmiD p y+c +1 h c y +Ev +1 y D p + p (13) As i the backloggig model, we cosider oly variable price-adjustmet costs here, ad we do ot cosider ay additioal pealty costs of lost sales because it would become very difficult to fid the coditios uder which the value fuctios are cocave. This settig is also adopted i other ivetory-pricig models with lost sales (e.g., Sog et al. 2008). We also require that p c +1 h, which is ot too restrictive, because otherwise for those p such that p < c +1 h, the first term i J y p (the expected et reveue) is egative. It should also be oted that, differet from the backlog case, the expected reveue i the lost sales case equals the expectatio of the product of the sellig price ad the miimum of demad ad the ivetory level. This fuctio is, i geeral, ot joitly cocave. Moreover, the requiremet that the iitial ivetory at the begiig of ay period must be oegative also makes the aalysis more challegig. However, we are able to provide coditios uder which the results of the backlog case ca be exteded to the lost sales case. The proofs of the results i this sectio are give i Appedix D. Defie e x p = p c +1 + h PrD p > x p PrD p > x (14) This defiitio is similar to the elasticity of lost sales defied i Kocabiyikoglu ad Popescu (2011) with the differece of a additioal term c +1 + h i the coefficiet because we cosider a multi-period settig. I the ext two results, we provide coditios based o e x p, uder which the optimal policies we characterized for the backlog case cotiue to hold here. Theorem 5. If e x p 1/2 for all, the (a) L y p is joitly cocave i y ad p, v x p 1 is oicreasig i x ad joitly cocave i x ad p 1 ; ad (b) the optimal strategy y p has the same structure as those preseted i Theorem 1. For a sigle-period problem, Kocabiyikoglu ad Popescu (2011) show that, uder a similar coditio (as oted, e x p here is defied slightly differetly), the expected reveue pemix D p is joitly cocave. It happes that this coditio leads to the joit cocavity of our multi-period problem. Here, we give oe example that satisfies the coditio. If D p = a b p + where is expoetially distributed with cdf 1 e x, > 0, the e x p 1/2 holds for all feasible x ad p as log as p 1/2b + c +1 h. We would also like to poit out that, other papers, such as Che et al. (2005) ad Sog et al. (2008), provide coditios uder which the sigle period profit, after optimizig the orderig quatity, is quasi-cocave i the price. However, their approaches caot be applied i our settig. Uder a stroger coditio that e x p 1, we ca further show that L y p is submodular, ad hece the optimal policies whe U is piecewise liear have the same structure as those preseted i the backlog case.

11 Che, Zhou, ad Che: Itegratio of Ivetory ad Pricig Decisios 1154 Operatios Research 59(5), pp , 2011 INFORMS Theorem 6. If e x p 1 for all ad U = u, the (a) L y p is joitly cocave ad submodular i y ad p, ad v x p 1 is joitly cocave ad submodular i x ad p 1 ; ad (b) the optimal strategy y p (markup oly, markdow oly bi-directioal) has the same structure as those preseted i Theorems 2 ad 3. A example that satisfies e x p 1 ca also be foud similarly, as we did for the previous theorem. We ext cosider a case i which there is o ivetory carryover. Redefie G Q p = c Q + r EQ D p + where r still deotes the uit salvage value ad r p. It is clear that the results derived i 4 cotiue to hold if ḡ p = max Q pemid p Q + G Q p (15) is cocave i p. Let Q p be the optimal orderig quatity with a give price p. The ext theorem provides the coditio that (15) is cocave. Theorem 7. For the case i which o ivetory is carried over betwee periods, replace c +1 + h i (14) with r. If the resultig e Q p p 1/2 for all, the (a) ḡ p is cocave i p; ad (b) the optimal pricig policy for the lost sales case has the same structure as that preseted i Theorem 4. Part (a) i Theorem 7 follows from the coditio e Q p p 1/2, which ca be proved by similar argumets to those i the proof of Propositio 6 i Kocabiyikoglu ad Popescu (2011), ad thus we skip it here. Part (b) follows directly from part (a). Whe the demad fuctio takes the form D p = A p + B p, the coditio holds if has a distributio with a icreasig failure rate (IFR) ad p r A p ad p r B p are decreasig i p. See Kocabiyikoglu ad Popescu (2011) for further discussio. 6. The Geeral Model: Heuristic ad Numerical Study Based o the aalysis ad results preseted i the previous sectios, we ca aticipate that the optimal policy of the geeral problem with both ivetory carryover ad a fixed price-adjustmet cost will be state-depedet ad might ot have a simple structure. Recall that eve for the model without fixed price-adjustmet costs, the optimal policy is state-depedet. Therefore, the optimal policy might be too complicated to implemet. To provide a cotrol policy that is relatively easy to implemet for a geeral system, i this sectio we propose a ituitive heuristic ispired by the aalysis i the two precedig special cases. We focus oly o U = u ad the backlog case, although the heuristic is also applicable to the lost sales case. Due to the existece of fixed price-adjustmet costs ad based o the aalysis i 3 ad 4, it is atural to costruct a heuristic policy from, i.e., the heuristic follows a base-stock ivetory ad two-sided s S pricig policy. To make the policy easier to implemet, we set sx = s h h, Sx = P, zx = zh h+, Zx = P with stateidepedet parameters s h P h zh h+, ad P, which we later show how to compute. The parameter for the basestock policy is ȳp 1 = y hph p 1, where p hp 1 is the sellig price after the price-adjustmet decisio. We search y hph p 1 of the value fuctio i a compact domai. We ame this policy Heuristic CZC. Because we are dealig with a fiite-horizo, ostatioary problem, the policy cotrol parameters described above eed to be computed for each period recursively, startig from period N. Specifically, after the cotrol parameters of period + 1 are computed, they are applied to derive the correspodig value fuctio of period + 1 ad, i tur, the cotrol parameters of period. Because the value fuctio resultig from the heuristic policy holds o ice structural properties, the computatio of the cotrol parameters relies o complete eumeratios over the feasible sets. For the price p i period, there is a give feasible set p p, whereas for the ivetory order-up-to level, we eed to impose a upper boud based o the rage of possible realizatios of demad. The detailed steps of the heuristic are preseted i the followig algorithm. Heuristic CZC for the Geeral System Step 1. Set = N ad defie vn h +1x p = v N +1 x p = c N +1 x. Let l N y p = R N p + G N y p + EvN h +1 y D N p N p. Step 2. Defie Y h+ P h arg max u p + l y p y p p p with s h h = mip p P u p + l Y h+ p K u P h + l Y h+ P h (if empty, sh = p) ad Y h P h+ arg maxu p + l y p with z h y p p p h+ = maxp P p u p + l Y h p K + u P h+ + l Y h P h+ (if empty, zh = p) ad y hp 1 is the global maximizer of y h p 1 arg maxl y p y

12 Che, Zhou, ad Che: Itegratio of Ivetory ad Pricig Decisios Operatios Research 59(5), pp , 2011 INFORMS 1155 Step 3. Set the price of period accordig to a twosided s S pricig policy with parameters s h P h ad z h P h+, ad the repleish the ivetory followig a state-depedet base-stock policy. Specifically, if p = P h, the y h h+ = maxx Y ad v h x p 1 = K u P h p 1 + l maxx Y h+ or else if p = P h+, the yh v h x p 1 = K + u p 1 P h+ P h + c x h = maxx Y ad + l maxx Y h P h+ + c x otherwise, if p = p 1, the y h = maxx yh p 1 ad v h x p 1 = l maxx y h p 1 p 1 + c x Step 4. If > 1, the = 1 ad let l y p = R p + G y p + Ev+1 h y D p p; go to Step 2. Otherwise, stop. It is ot difficult to see from the defiitio that Y h+ = y hp h, Y = y hp +, ad sh zh. Because we have to derive the recursio for value fuctio v hx p 1 i the algorithm, the cotrol parameters of the heuristic might ot be easy to compute. However, we expect the computatioal effort to be less tha that for the exact optimal policy, ad the heuristic policy is also ameable to practical implemetatio oce the cotrol parameters are o had. To test the effectiveess of the heuristic compared to the optimal solutios, we coduct a extesive umerical study. We first cosider istaces with a short plaig horizo, N = 4. Assume D p = 80 2p+, i which is a egative biomial radom variable (P = i = Ci+r 1 r r 1 i ) with = 05 ad r = 8, so the expected value E = 8. The other basic parameters are, for = 1 N : b = 10, h = 2, u = 3 2, K = 30, = 09, c = 6 05, p = 3, ad p = 35. Differet istaces are geerated by alteratig oe of the basic system parameters. For each istace, we cosider the possible combiatio of the iitial ivetory level x ad price p to avoid the impact of the iitial state o the performace of the heuristic. That is to say, for each set of parameters there are a total of 2,343 istaces. We defie the relative error of the heuristic as v1 x p Error% = max 0 v1 hx p 0 100% x p 0 v 1 x p 0 where v h 1 x p 0 is the resultig total discouted profit of Heuristic CZC give state x p 0. The optimal profit v 1 x p 0 is computed recursively through a exhaustive search over the feasible domai of price ad orderig quatity (i this umerical study, because the largest expected oe-period demad is 82 ad the largest variace is 80, we set a upper boud of 200 for the orderig quatity). We geerate four groups of examples. By keepig E = 8, we geerate Group 1 istaces by cosiderig the parameter r The resultig variaces V are 80, 40, 20, 16, ad 10. Group 2 is geerated by alteratig the uit orderig cost c 2 i period 2 from 2 to 10 with a step size of 2. For Group 3, we chage the backlog cost b from 20 to 80 with a step size of 20. For the last group, Group 4, we chage the uit price-adjustmet cost u 1 of period 1 from 5 to 20 with a step size of 5. To see the impact of the fixed price-adjustmet cost o the performace of the heuristic, we further test the precedig four groups of istaces with a larger fixed cost of K = 100. The results are reported i Table 1. We ca see that the heuristic performs very well i these examples. Compared to the results of K = 30, we fid that the performace of the heuristic deteriorates slightly as K becomes larger. However, it still performs quite well. The average computatio time of the heuristic for each istace (usig Matlab) is 0.76 secods of CPU time, i cotrast to secods of CPU time of the optimal policy o a PC with a GHz CPU. To further demostrate the effectiveess of the heuristic, we cosider a very simple myopic policy whose parameters are calculated based o the sigle-period profit ad apply them to Heuristic CZC. Although computatioally easier, we ca see from Table 2 that its performace is cosiderably worse after comparig with Table 1. Fially, to determie how well the heuristic performs with a loger plaig horizo, we test the previous four groups of istaces with N = 12 ad K = 100. The average ad maximum errors for the four groups are (0.00%, 0.00%), (0.15%, 0.36%), (0.92%, 2.29%), ad (0.40%, 0.56%), respectively. These results show that the resultig profit of the heuristic is still quite close to the optimal whe the plaig horizo is log. We also coduct umerical studies to compare Heuristic CZC with several other simple oes, ad the results are reported i Appedix E. Table 1. Performace of heuristic CZC. K = 30 K = 100 Error % GP 1 (r) GP 2 (c 2 ) GP 3 (b) GP 4 (u 1 ) GP 1(r) GP 2(c 2 ) GP 3(b) GP 4(u 1 ) Avg Max

13 Che, Zhou, ad Che: Itegratio of Ivetory ad Pricig Decisios 1156 Operatios Research 59(5), pp , 2011 INFORMS Table 2. Performace of myopic heuristic. K = 30 K = 100 Error % GP 1 (r) GP 2 (c 2 ) GP 3 (b) GP 4 (u 1 ) GP 1(r) GP 2(c 2 ) GP 3(b) GP 4(u 1 ) Avg Max Ivetory-Depedet Price-Adjustmet Cost I the retail idustry, the physical costs of a price adjustmet might iclude ivetory ticketig/relabelig costs, ad hece they might deped o the existig ivetory level (Celik et al. 2009). Let c I be the price-adjustmet cost per uit of ivetory. It should be stressed that this cost would be icurred oly whe there are price chages betwee periods, ad it is thus of the fixed-cost type. I additio, such a ivetory depedet price-adjustmet cost should be icurred oly whe there is positive ivetory; for backlogged uits, the customers pay the price of the last period, ad thus there is o eed to chage the price tags whe they become available. Also ote that the labelig cost for the fresh ivetory, brought by the order that has just bee placed, is accouted for i the variable orderig cost. Therefore, whe the startig ivetory level of period is x, the total price-adjustmet cost is expressed as C p x = K + ci x+ + U With this more geeral cost fuctio, we ca aticipate that the problem will be eve more complicated tha the models aalyzed i the previous sectios. To tackle this complexity, we provide the followig two heuristic policies. The first simply applies the heuristic we developed for the case with c I = 0. The secod slightly modifies the precedig heuristic to accommodate the ew cost feature. As the fixed cost ow depeds o the ivetory level, we redefie the parameters sx ad zx as sx = s hx ad zx = zh x, where s h h x=mip pp u p+l Y h+ p K c I x + ad z h u P h +l Y h+ P h (if empty, sh x=p) h+ x = maxp P p u p + l Y h p K c I x + + u P h+ + l Y h P h+ (if empty, zh x = p) It is clear that the pricig policy of the secod heuristic becomes a state-depedet two-sided s S pricig policy. Ivetory repleishmet still follows a state-depedet base-stock policy. It becomes evidet that this policy is more difficult to compute ad implemet tha the first oe, because the thresholds that trigger price chages are ow fuctios of the startig ivetory level. We ext test the performace of these two heuristics. With c I 1 5 for all, we test the same four groups of umerical examples studied previously with N = 4. The results are tabulated i Tables 3 ad 4, i which, Avg 1 (Avg 2 ) ad Max 1 (Max 2 ) are the average ad maximum errors for the first (secod) heuristic. It ca be observed from the results that the secod heuristic outperforms the first, although the first is easier to implemet. Whe c I is small, both heuristics are quite effective. However, with a larger c I, the performace of the first heuristic deteriorates quite sigificatly. This is ituitive, because it igores the ivetory-depedet price-adjustmet costs. We therefore should ot igore such costs whe desigig the heuristic. 7. Coclusio We ivestigate a multi-period ivetory model with costly price adjustmets i this paper. A firm eeds to make its pricig ad ivetory orderig decisios simultaeously i each period to maximize total profits. Due to the complexity of a geeral system with both a fixed cost ad ivetory carryover, we characterize the optimal ivetory repleishmet ad pricig strategy for two special cases of the geeral model: oe with ivetory carryover betwee periods, but o fixed costs for price adjustmets; ad the other with fixed costs for price adjustmets, but o ivetory carryover. For the geeral problem, we provide a ituitive heuristic policy ad show with umerical examples that it is quite effective. The results ca also be exteded to the case of ifiite horizo. Cosider a ifiite horizo system with Table 3. Performace compariso of two heuristics: c I = 1. K = 30 K = 100 Error % GP 1 (r) GP 2 (c 2 ) GP 3 (b) GP 4 (u 1 ) GP 1(r) GP 2(c 2 ) GP 3(b) GP 4(u 1 ) Avg Max Avg Max

14 Che, Zhou, ad Che: Itegratio of Ivetory ad Pricig Decisios Operatios Research 59(5), pp , 2011 INFORMS 1157 Table 4. Performace compariso of two heuristics: c I = 5. K = 30 K = 100 Error % GP 1 (r) GP 2 (c 2 ) GP 3 (b) GP 4 (u 1 ) GP 1(r) GP 2(c 2 ) GP 3(b) GP 4(u 1 ) Avg Max Avg Max statioary parameters. With several techical assumptios (see Federgrue ad Hechig 1999, Che ad Simchi-Levi 2004b), the optimal policy for the fiite horizo case ca be exteded ad is statioary. First cosider the first model with markup oly. If the startig state is i regio I of Figure 1(a), the the price will be set at P i period oe ad ever chage afterward, ad the firm just eeds to maitai a order-up-to level Y + at the begiig of each period; if the startig state is i regio II, the after a fiite umber of periods, the state will fall ito regio I ad remai there forever; if the startig state is i regio II, IV, or V, the the firm will ever chage the price ad will repleish ivetory oly if ecessary. Similar scearios would occur i the markdow-oly ad bi-directioal price chage cases. For a model without ivetory carryover, it ca be see that the firm would chage the price at most oce. There are several other possible extesios to our models. First, a possible extesio is to study problems with capacity costraits of ivetory repleishmet. I this case, we expect that the optimal ivetory policy would become a so-called modified base-stock-type policy, i.e., if orderig, the the firm orders up to the optimal base-stock level (orders the optimal quatity) if possible; otherwise, it orders to full capacity. The optimal pricig policy is expected to be more complicated i this case as the policy parameters are more likely to deped o the ivetory level due to the capacity costrait. Secod, i this paper, we cosider maily symmetric price-adjustmet costs (see 3.2), i.e., the same variable price-adjustmet costs for markups ad markdows. However, the results i 3.2 hold eve whe markups ad markdows icur differet variable price-adjustmet costs. Whe there are asymmetric fixed price-adjustmet costs, the results i 4 ca also be exteded by employig the approach ad results for the stochastic cash balace problem with asymmetric costs i Che ad Simchi-Levi (2009). Third, we assume that demad i a period depeds oly o the price i the curret period. However, the marketig literature argues that demad might also deped o historical prices, i.e., the prices i previous periods. It would be iterestig to icorporate these more geeral demad models ito our problem ad ivestigate the correspodig optimal policies. Fially, icorporatig a fixed orderig cost ito our model would be iterestig ad importat. Aalysis of such a model is challegig ad deserves further exploratio. 8. Electroic Compaio A electroic compaio to this paper is available as part of the olie versio that ca be foud at Edotes 1. Observe that the objective fuctio U p 1 p + J y p i problem (5) is supermodular. Thus the maximum is well defied. See, for istace, Theorem of Simchi-Levi et al. (2005). 2. Because the expected reveue fuctio is strictly cocave i p, the set of optimal solutios here is a sigleto for a give x. 3. The maximum is well defied because f x p is supermodular (see Theorem of Simchi-Levi et al. 2005). Ackowledgmets The authors are grateful for the commets ad suggestios received from the associate editor ad two aoymous referees. They also thak Xiuli Chao, Jia Yag, ad Yuha Zhag for their helpful commets. The first author is partly supported by Natioal Sciece Foudatio grats CMMI , CMMI ARRA, ad CMMI ; the secod author is partly supported by Hog Kog GRF Fud CUHK #419408; ad the third author is partly supported by Hog Kog GRF Fud CUHK # Refereces Aguirregabiria, V The dyamics of markups ad ivetories i retailig firms. Rev. Ecoom. Stud. 66(2) Berge, M., S. Dutta, D. Levy, M. Ritso, M. Zbaracki Shatterig the myth of costless price chages. Eur. Maagemet J. 21(6) Capli, A., D. F. Spulber Meu costs ad the eutrality of moey. Quart. J. Ecoom. CII(4) Celik, S., A. Muharremoglu, S. Savi Reveue maagemet with costly price adjustmets. Oper. Res. 57(5) Cha, L. M. A., Z. J. Max She, D. Simchi-Levi, J. Swa Coordiatio of pricig ad ivetory. D. Simchi-Levi, S. D. Wu, Z. J. Max She, eds. Hadbook of Quatitative Supply Chai Aalysis: Modelig i the E-Busiess Era, Chapter 3. Kluwer, Bosto. Chao, X., S. X. Zhou Joit ivetory ad pricig strategy for a stochastic cotiuous-review system. IIE Tras. 38(5)

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