A Single-Supplier, Multiple-Retailer Model with Single-Season, Multiple-Ordering Opportunities. and Fixed Ordering Cost

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1 A Sngle-Suppler, Multple-Retaler Model wth Sngle-Season, Multple-Orderng Opportuntes and Fxed Orderng Cost Apurva Jan, Kamran Monzadeh, Yong-Pn Zhou Mchael G. Foster School of Busness Unversty of Washngton, Seattle {apurva, kamran, Abstract In ths paper, we dscuss the replenshment decson of seasonal products n a two-echelon dstrbuton system consstng of a suppler and multple retalers. Due to long manufacturng lead tme, the suppler orders ts entre stock for the season well n advance. The retalers, on the other hand, can replensh ther nventory from the suppler throughout the season as demand realzes. Demand at each retaler follows a Posson process. Each retaler order ncurs a fxed cost, and the usual understockng and overstockng costs occur. When retaler lead tme s neglgble, we show that t s optmal for the retaler to follow a tme-based, order-up-to polcy and order only when nventory s depleted. We also characterze the structure of the optmal polcy and propose a number of heurstcs for easer computaton. For the suppler, we express the dstrbuton of total demand. Ths allows the suppler to solve a classc newsvendor problem to determne the total stock for the season. We fnd that the optmal retaler polcy can sometmes cause large demand varaton for the suppler, resultng n lower suppler proft. In centralzed settngs, ths may even result n lower system proft than some naïve retaler heurstcs, creatng neffcency n the supply chan. We offer nsghts on potental causes and manageral mplcatons. B. Introducton and Lterature Revew We develop a suppler-retaler nventory model for a sngle-season, random-demand product. At the end of the season, overstockng costs are charged on leftover nventory and understockng costs are charged on lost sales. Typcally, the retaler s replenshment decson n such a settng s modeled as a newsvendor decson, wth only one orderng opportunty at the begnnng of the season. In our model, however, the retaler can place a replenshment order at any pont durng the season at a fxed orderng cost. Wth these contnuous orderng opportuntes, we determne the retaler s optmal orderng polcy and study ts mplcatons for the suppler and the system. The context of sngle-season, random-demand, and understockng and overstockng costs apples to a wde range of ndustres, such as fashon retalng, computer and consumer electroncs, and pharmaceutcal manufacturng. Typcally, the product nvolved n these stuatons has a short lfe cycle

2 (one season), and exhbts long manufacturng lead tme. The combnaton of short season and long lead tme usually leads to the assumpton of a sngle begnnng-of-season orderng opportunty and the applcaton of the newsvendor model. We observe that whle ths may be true for the manufacturer of the tem (called the suppler n our model), downstream retalers are free to replensh stocks from the suppler any tme durng the season. The retaler has to ncur a fxed orderng cost for these replenshments. For other relevant costs, we smply follow the newsvendor model conventon and charge the retaler overstockng cost on end-of-season leftover nventory and understockng cost on lost sales. Smlar costs apply at the suppler echelon. Gven that the suppler experences a long lead tme, we take the tradtonal approach of analyzng the suppler s orderng decson as a newsvendor model. The fashon apparel ndustry provdes a good example of the stuaton descrbed here. Manufacturng lead tmes are long because most manufacturers are located n Asa (see Sknner 992, Hammond 99). Demand durng the season s hghly unpredctable. Orders are receved (by the suppler n our model) at the begnnng of the season and then used to satsfy retal orders placed (by the retaler n our model) durng the season. The transportaton lead tme from suppler to retaler s short. Retalers ncur orderng costs (e.g., shppng). Overstockng and understockng costs faced by retalers n such stuatons are severe; they are estmated to be around 35% of the sales (Frazer, 986). Dstrbuton of flu vaccnes n the Unted States provdes another example. Each year, the Centers for Dsease Control and Preventon dentfes the most lkely flu stran for the upcomng flu season. Manufacturng the flu vaccne can take sx months (Russell 24), so the stock of vaccne doses s ordered well n advance of the season. Vaccnes are receved by supplers at the begnnng of the season and then are shpped durng the season to health care provders (retalers) as they place orders. The lterature on the newsvendor model s vast. Ths paper dffers from that lterature n our modelng of multple orderng opportuntes n a sngle season. We provde a quck summary of papers that model at least one more orderng opportunty n the context of a seasonal product. Lau and Lau (997) model a

3 newsvendor problem wth an opportunty to replensh n the mddle of the perod. In Gallego and Moon (993), the second replenshment opportunty occurs at the end of the season. In our model, the retaler can order at any tme. Thus, there are unlmted orderng opportuntes. Much of the lterature that allows more than one orderng opportunty n a sngle season s motvated by Quck Response (QR) n the apparel ndustry (Hammond 99). QR s a strategy under whch manufacturng lead tmes can be shortened through better use of nformaton. As a result, supply chans may beneft by more accurate forecasts. The modelng of QR and related contractng mechansms naturally calls for more than one orderng opportunty. In ther model of QR, Iyer and Bergen (997) allow a second order n the season. In the tme between the frst and second orders, however, the model only captures the sales of related tems. Fsher and Raman (996) apply the noton of QR and examne the manufacturng, replenshment, and forecastng decsons n the fashon skwear ndustry. Gurnan and Tang (999) model a retaler who has two opportuntes to order a product from the manufacturer, but the unt cost at the second opportunty s uncertan. A sub-problem n Donohue (2) also models twoopportunty orderng wth the cost at the second opportunty hgher than the cost at the frst. Tsay (999) models a quantty flexblty contract n whch the retaler s allowed to change the order sze n the second perod based on the demand sgnal receved n the frst perod. More recent lterature, such as Cachon and Swnney (2), contnues to model QR as another opportunty to place an order before the start of the sellng season. None of these papers, however, explctly model the sales of tems between the frst and second opportuntes. In our model, sales may contnuously occur between any two orders. Some other papers n ths stream of research, such as Eppen and Iyer (997) and Barnes-Schuster et al. (22), do capture sales of tems between two orderng opportuntes. The motvaton that drves these models avalablty of updated demand sgnal n the second perod s not the same as ours. At the tme retalers decde on the sze of ther frst perod order, models n these papers must take nto account the possblty of a better demand sgnal later. In our model, there s no new demand sgnal, but earler order sze

4 decsons must take nto account the possblty that orders at any later tme are allowed. We remove the restrcton of only two orderng opportuntes, allow orders at any tme, and derve the optmal polcy. There are two other areas of nventory modelng research that may be nterpreted as offerng more than one orderng opportunty: emergency orderng and transshpment. When nventory obtaned from the regular source runs out, the retaler can order replenshment from an alternatve emergency source or request transshpment from an nternal source. In a contnuous-tme, nfnte-horzon settng, Monzadeh and Schmdt (99) and Axsater (27) present models wth emergency orders and Axsater (99) consders transshpment. It s dffcult to fnd optmal polces n ths settng, and these papers propose and analyze heurstc polces. In a sngle-season, fnte-horzon settng, Khouja (996) develops an example of an emergency orderng model, but the sze of the emergency order s lmted to a fxed percentage of unsatsfed demand. For an example of a sngle-season transshpment model, see Rud, Kapur and Pyke (2) n whch, after the ntal order, there may be one opportunty of transshpment between the two frms. Recent research n ths area, such as Shao et al. (2) has focused on analyzng prcng polces for transshpment. The addtonal orders n these types of models are drven by the cost and lead tme dfferences between dfferent sources. In our sngle-season, contnuous-tme model, we focus on multple orders from the same source. We put no constrants on the sze of the order or the number of orderng opportuntes. The models that come closest n sprt to our retaler s model are the fnte-horzon, perodc-revew models. The classc dynamc formulaton goes back to Arrow et al. (95) and Scarf (96) who proved the optmalty of (s,s) polcy for models wth fxed orderng cost. For nfnte-horzon models, these optmalty results can be translated to contnuous-revew settngs lke ours (see Stdham 986). Based on the smlarty of cost functons n the nfnte-horzon perodc and contnuous settngs, the effcent algorthms developed for the perodc settngs can be appled to contnuous models as well. See, for example, Federgruen and Zpkn (985), Zheng and Federgruen (99), and Zpkn (2). We are not

5 aware of smlar correspondence results for computaton of optmal polces n fnte-horzon contnuous models. Jont optmzaton of perod-length and order-up-to level n such settngs contnues to be of nterest; see, for example, Lu and Song (2). The fnte-horzon problem descrbed n ths paper can be vewed as a perodc nventory model wth nfnte perods, each wth an nfntesmal perod length but wth end-of-horzon costs only. It s not clear how the exstng perodc results can be translated to such a settng. In ths paper, we work drectly wth the contnuous-tme, fnte-horzon formulaton. The lterature on provng optmalty of nventory polces drectly wthn a contnuous-tme framework s scant. In our specfc contnuous-tme settng, our paper offers a method to fnd the optmal polcy. Our model frst addresses the stockng and replenshment decson of a retaler who experences Posson demand n a sngle season of fnte length. (Our model also allows the possblty of a nonhomogeneous Posson demand; that s, demand rate may change over tme.) At the end of the season, an overstockng cost s charged on each unt of leftover nventory. An understockng cost s charged on each unt of sales that s lost due to lack of nventory. The dfferentatng feature of our model s that, unlke other sngleseason models cted above, we allow the retaler to place a replenshment order at any pont durng the season. The retaler ncurs a fxed orderng cost for each order. We assume that the suppler has ample supply of the product or can nstantly make up for any shortage wth a unt penalty cost. Furthermore, lead tme for the retaler s orders s assumed to be neglgble. Wthout a fxed shppng cost, the problem would be trval snce the retaler wll employ an order-for-order polcy. However, the fxed per-order shppng cost s present n most real-world applcatons. Ths complcates the problem because the retaler s order quantty s not of unt sze anymore and can vary from order to order, dependng on when the order s placed. In addton, whle makng a decson on order sze, the retaler must ncorporate the possblty of placng future orders. The queston of when to place a replenshment order s also open: Is t ever optmal to forego placng an order and ncur lost sales? We contrbute to the lterature by characterzng the optmal orderng polcy n such stuatons, whch mnmzes the retaler s average total cost durng the season. Our method s somewhat unusual because we convert the contnuous-tme

6 problem to a dscrete framework by dentfyng dscrete ponts at whch optmal decsons change. We devse an exact procedure for computng the optmal polcy parameters and propose a number of heurstcs for computng the polcy parameters. We then evaluate the performance of such heurstcs through a numercal experment, whch shows that some of our heurstcs delver near-optmal results. Next, after solvng the retaler s problem, we consder a dstrbuton system consstng of one suppler and multple retalers. We assume that the suppler replenshes stock only once at the begnnng of the season and supples the retaler's orders durng the season. If the suppler runs out of stock durng the season, the excess demand wll be satsfed through a thrd party at some penalty to the suppler. We fnd the optmal order quantty for the suppler by expressng the exact dstrbuton of the suppler s demand over the season. Our numercal results provde nterestng manageral nsghts, such as the observaton that as the retaler changes ts polcy to take advantage of multple orderng opportuntes, the suppler proft decreases n general. The rest of the paper s organzed as follows. In the next secton, we consder the retaler's problem n solaton, derve the optmal polcy, and present heurstcs. In 3, we extend our model to a dstrbuton system wth one suppler and multple retalers. Usng the results developed n 2, we derve a method for computng the exact dstrbuton of the suppler s demand and propose an approxmaton for t. Secton 4 uses a numercal experment to evaluate the retaler's polcy and the effectveness of the heurstcs suggested n 2. We close ths secton by outlnng some of the manageral mplcatons resultng from operatng such models n a dstrbuton settng. Fnally, n 5, we summarze our fndngs and suggest a number of possble extensons for future work.

7 B2. The Retaler s Problem Consder a sngle-season nventory system of a product. Let the length of the fnte season be T. Snce we wll use backwards nducton n our dervatons, we follow the conventon to let denote the tme tll the end of the season. So =T ndcates the begnnng of the season whle = ndcates the end. Usng ths termnology, an nterval of refers to. In real tme, happens before. The retaler 2 2, 2 experences unt-sze demands for a product that occur randomly durng the season, followng a Posson process wth rate The retaler can order at any tme from a suppler. The fxed orderng and shppng cost s K for each order placed by the retaler. We assume that lead tme s zero, and the suppler s always able to meet the order. As a result, an order placed at tme can be used to satsfy a demand that occurred at tme. At the end of the season, a per-unt overstockng cost w s charged on leftover nventory. All demands durng the season that are not met from the retaler s nventory are lost and charged a per-unt understockng cost of As n a tradtonal newsvendor model, any end-of-season perunt holdng and salvage costs can be drectly captured n the overstockng cost (see Nahmas 29, pages 3-32). The retaler s objectve functon s to mnmze the total expected cost, whch s comprsed of the fxed orderng cost and the overstockng and understockng costs. It s a standard result (e.g., see Law and Kelton 2, pages ) that a nonhomogeneous Posson process wth rate (), T, can be generated from a homogeneous Posson process as follows: Frst, generate a homogeneous Posson process wth rate.then, for any arrval n the homogeneous Posson process at tme, nvert the expectaton functon T ( T ) ( s ) d s to fnd the correspondng arrval tme θ n the nonhomogeneous Posson process. Essentally, by scalng tme wth the expectaton functon (), we have a one-to-one mappng between the two processes. Moreover, any T T orderng acton taken n the homogeneous case at tme can be taken n the nonhomogeneous case at tme θ, and vce versa. The resultng system dynamcs are dentcal. Hence, there s also a one-to-one

8 mappng between the polces that can be employed between the two processes. Fnally, because all the costs (orderng, overstockng, and understockng) are not tme dependent, the correspondng polces wll have dentcal costs. As a result of ths one-to-one correspondence, n ths paper we wll focus only on the homogeneous Posson process. All the major results of ths paper contnue to hold f the demand follows a nonhomogeneous Posson process. To gan nsght nto the structure of an optmal orderng polcy, we frst study the newsvendor optmal order quantty whch s derved under the condton that only one order s allowed and analyze how t changes wth the duraton of tme t covers. Denote by D(θ) the random Posson demand durng a perod of length θ. Followng the notaton n Hadley and Whtn (963), we denote the PDF and the complementary CDF of D(θ) as: p( n, ) e ( ) n! n and P ( n, ) p ( m, ). Let T C ( S, ) be the expected newsvendor cost; that s, the costs ncurred at the end of the tme horzon, f nventory at, after any demand and orderng, s S and no orders are placed durng [, ) np( n, ) p( n, ), we have: m n. Snce T C ( S, ) w ( S ) ( w ) E D ( ) S w ( S ) ( w )[ P ( S, ) SP ( S, )]. () Let g ( ) m n TC ( S, ) be the optmal newsvendor cost for the perod [, ), and S ( ) be the S correspondng optmal newsvendor stockng level (the largest S that mnmzes T C ( S, ) present some prelmnares. All the proofs n ths paper are shown n the Appendx. ). Next, we Lemma There exst tme ponts... T for some l l such that l S ( ) on [, ), S ( ) n on [ n, ), n n l, and S ( ) l on [ l, ]. l

9 Lemma 2 g ( ) s ncreasng n θ. Lemmas and 2 show that n a newsvendor settng, as the end of the season draws near (.e., θ decreases), there s less future demand to cover and both the optmal newsvendor quantty and the optmal cost decrease accordngly. In partcular, Lemma shows that n the newsvendor settng, where only one order s placed, the optmal order quantty s an ncreasng (n θ) step functon. We wll show that ths pattern contnues to hold n the multple-order settng, though the parameters wll change. Returnng to the mult-order settng, we ntroduce some addtonal notaton frst (a summary s provded below). Let I ( ) represent on-hand nventory at θ before any demand occurrence or possble order, and I ( ) represent on-hand nventory after any demand and orderng at θ. Defne an orderng polcy to be a mappng from [ ] [ ) to [ ) such that, gven the nventory at tme [ ] s ( ) [ ) and whether there s a demand arrval at that tme ( ( ) ndcates an arrval at tme and ( ) otherwse), polcy determnes whether and how much to order. Ths results n ( ) [ ). For every, there s a correspondng cost functon ( ( ) ( )) that s a mappng from [ ] [ ) to [ ). It represents the sum of lost sales and fxed orderng costs under f ( ( ) ( )) occurs. We say polcy s feasble f ( ( ) ( )) s a measureable functon on [ ] [ ). Further, for any polcy let V ( I ( ), ) be the expected total cost ncurred durng the nterval [,). To derve ( () ) we note that ( ( ) ( )) s the cost of applyng polcy under one sample path, ( ) [ ). Let (, ) be the set of all possble sample paths on [, ) and P ts correspondng probablty measure, then we have ( () ) ( ( ) ( )).

10 * Defne the optmal value functon at tme θ as V ( I ( ), ) n f V ( I ( ), ) : where s the set of all non-antcpatory polces such that V ( I ( ), ) exsts for all I ( ), and. To smplfy exposton we use, nstead of I ( ), n the rest of the paper. A summary of addtonal notatons s provded below: V ( I ( ), ) Orderng polcy, [ ] [ ) [ ) Cost functon correspondng to, [ ] [ ) [ ) The expected total cost ncurred under durng [,) ( ( ) ) ( ) (, ) [ ) P probablty measure correspondng to (, ) Let N denote the set of non-negatve ntegers and I be any subset of [,T]. Defnton A polcy s called optmal on I[,T] f t satsfes * for all I V (, ) V (, ) and NA polcy s called optmal f t s optmal on [,T]. Moreover, we call any orderng acton that mnmzes * V (, ) the optmal acton at tme, and denote mnmzers exst, the largest value s chosen for S ( ). S ( ) arg m n V (, ) N *. If multple The above defnton focuses on the value functon and the unquely determned acton. In terms of the polces that may correspond to the same value functon, we note that multple optmal polces may exst. For example, changng an orderng polcy on an event wth a measure of zero (e.g. that a demand occurs at a specfc tme) does not change the system cost, but t wll result n a dfferent polcy. These polces all have the same value functon for all tme epochs and nventory levels. We say that any two polces and are equvalent f ( () ) ( () ) for all ( () ). Henceforth, the optmal polcy we dscuss below represents all the polces that are equvalent to t. We begn wth narrowng the set of polces that we must consder.

11 Proposton Suppose that under polcy there s a postve probablty that an order s placed when nventory s postve. Then, polcy s not optmal. The next step s to further narrow the set of polces to those that only order when a demand occurs. Proposton 2 Suppose that under polcy there s a postve probablty that an order s placed at a tme when no demand occurs. Then, polcy s not optmal. We note that Proposton 2 s consstent wth the result for nfnte tme horzon, where t s optmal to place orders only at demand epochs when the demand follows a Posson process and lead tme s zero (see Monzadeh and Zhou 28). Propostons and 2 mply that we need to only consder polces that order only when nventory s zero and a demand takes place. Next, we buld on Propostons and 2 to characterze the optmal acton at any tme. The expresson for value functon at tme should nclude: (a) expected orderng cost for all the orders placed n [, ), (b) expected understockng cost for all the demands lost n [, ), and (c) expected overstockng cost for leftover nventory at. We accomplsh ths n two steps. Frst, we show n Secton 2. that there exsts a pont after whch t s optmal to not place an order. Ths determnes the optmal actons for and defnes the value functon at as the newsvendor cost functon. Next, n Secton 2.2 we defne the backwards value functon nducton for and use t to show the structure of the optmal acton for the entre [,T]. In Secton 2.3, we show how to compute the optmal order quanttes and develop heurstcs.

12 2.. Exstence of θ Lemma shows that n the sngle-order, newsvendor settng, t s optmal not to order near the end of the tme horzon. The next proposton wll show that a smlar result holds for our multple-order settng. We note that when T K, t s optmal never to place any order. Thus, from now on, we wll only consder the cases where K T. We start the analyss at the end of the season ( ). If the orderng cost s reasonably hgh, then when there s relatvely lttle tme left n the season, t should be optmal not to place another order because the fxed orderng cost could outwegh the expected understockng cost. Therefore, there should exst a pont n tme, call t, after whch the optmal acton s to never place any order. On the other hand, when K Proposton 3 formalzes ths dea., t s optmal to never lose any sale. Snce we wll always be focusng on the optmal polcy, for the sake of convenence, we drop the * superscrpt from the value functon. Consequently, the value functon s expressed as V (, ) n the rest of the paper. Proposton 3 Assume K T. Let be the unque non-negatve soluton to the equaton K, and S S ( ). Then g( ) *, V (, ) TC (, ) for [,mn{,t}], and any polcy s optmal on [, ] f t never orders at tme [, ) and orders up to S when nventory s zero and a demand occurs at. For K, we have and S. Recall that the value functon V (, ) s the expected total cost n [, ) wth ( ) I ( ). It ncludes the orderng cost of all the orders placed n [, ), the understockng cost n [, ) and the overstockng cost ncurred at. Wth the beneft of Proposton 3, we know that t s optmal to not place an order at

13 , so V (, ) for s just the newsvendor cost functon. That s: V (, ) T C (, ) for Moreover, for K T, we show, n the proof of Proposton 3, that. V ( S, ) K V (, ). Ths mples that when the on-hand nventory s zero and a demand occurs at, an optmal polcy on [, ] s ndfferent between not orderng and orderng up to S. We choose to order up to S to be consstent wth the assumpton throughout the paper that n such cases the retaler wll choose the largest order-up-to level Characterzaton of Optmal Polcy We are now ready to examne the optmal orderng quanttes for. From Propostons and 2, we know that an optmal polcy orders only when nventory s zero and a demand occurs; thus the next queston s natural: when an order s placed at, how much should be ordered? To answer ths queston, we let I ( ) and defne, for any sample path ω, L (,, ) and N (,, ) to be the unts of lost sales and the number of orders placed n [, ) respectvely. Moreover, let the endng nventory at tme be j. Then, V (, ) K E N (,, ) E L (,, ) E V ( j, ). (2) (The ω subscrpt wll be suppressed later n the paper when there s no confuson.) By defnton, S ( ) arg mn V (, ) denotes the optmal order-up-to level, f nventory s zero and a demand occurs at tme. Agan, f multple optma exst, S ( ) s the largest one. In the rest of ths secton, we wll smplfy the value functon (2) and prove ts propertes. Ths wll allow us to characterze S ( ) for all.

14 Defne sup{ T, and f a demand occurs on [, ) and nventory s zero, the optmal acton s to order up to S }. The followng proposton frst shows that the set s non-empty and. Moreover, at both order-up-to levels S and S mnmze the value functon. By choosng the larger value as always, we then know the optmal order-up-to level at s S. That s, S S ( ) S. Proposton 4 () () For, the value functon can be wrtten recursvely as: () Any polcy s optmal on [, ] f t satsfes the followng: When a demand occurs at tme, f and I ( ), order up to S, f and I ( ), order up to S V (, ) K P n ( S ), p n ( S ) j, V ( j, ) p ( j, ) V ( j, ) n j n j S S, (3) otherwse, do not order. (v) V ( S, ) V ( S, ). The frst term on the rght hand sde of (3) represents the total expected orderng cost between and, and the last two terms represent all the future expected cost startng from tme (startng nventory s at tme and endng nventory s j at tme ). The dfference s that n the second term at least one order s placed between and, and n the last term no order s placed. We can now characterze the optmal order-up-to levels on the entre [,T].

15 Theorem There exst an nteger n and tme ponts T n such that the n followng polcy s optmal on [, k ], k n : ) t orders only when nventory s zero and a demand occurs, 2) t never orders at tme [, ), 3) t orders up to S S ( ) at tme [, ) for k n, and 4) t orders up to S S n at tme T n n for k n. Moreover, V (, ) s quas-convex n for all, and (, ) (, ) V S V K and V ( S, ) V ( S, ) for all k n. k k k k There may exst other polces that acheve the same value functon as the polcy n Theorem, but n the rest of the paper, we wll refer to ths polcy as the optmal polcy. As Theorem shows, both S k and S k mnmze the value functon at tme k, k n S, k to be consstent wth the conventon n the rest of the paper.. Agan, we choose to order up to the larger value, 2.3 Computatons of Retaler s Optmal Polcy and Heurstcs Smlar to (3), we have, for k n : where E N (,, ) k k s the expected number of orders (of sze S +) between and k k k, and can be calculated as: S k k k k k k k k k k k V (, ) K E N (,, ) V ( j, ) p n ( S ) j, V ( j, ) p ( j, ), j n j E [ N (,, )] P r[ N (,, ) n ] P r[ D ( ) n ( S ) ] n k k k k k k k n n P ( n )( S ),. k k k

16 These expressons allow us to recursvely calculate the value functon by gong back n tme startng wth. We now present a formal procedure for computng the optmal polcy:. If K T, optmal polcy s to never order. Go to end. 2. If K, set, S ; Else, set as the unque soluton to g( ) K. Set S S ( ) as the largest nteger such that P S, w w. Compute V (, ) TC (, ),.., S ; set k. 3. Set as the unque root n (, T ] k k of V ( S, ) V ( S, ) k k k k where we use recurson V ( S, ) K P n ( S ) S, p n ( S ) S j, V ( j, ) k k k k k k k k k k k S k n j n p ( j, ) V ( S j, ). j If no root s found, set, k T n k k k k k ; S k End. Else, set S S, k k, go to step 3. k k Compute V S k k (, ),..,. Note that unqueness of s due to Proposton 3, and unqueness of s shown n step 3 of the proof k of Proposton 4. We used the root fndng algorthm code freely avalable under GNU general publc lcense. In some cases, the algorthm may be computatonally demandng as one needs to fnd by k solvng V ( S, ) V ( S, ) k k k k. When many s are nvolved, the algorthm may run for a long tme. k For example, on a 2.33 GHz CPU Wndows XP desktop wth 4GB RAM, t took more than one hour to fnd the optmal polcy n one of the cases presented n Secton 4. Thus, t s desrable to develop effectve

17 heurstcs that yeld near-optmal results. We now outlne four heurstcs for solvng the retaler s problem. Frst, we start wth the newsvendor model as the baselne case. Other heurstcs can be compared wth ths baselne to assess the mprovement one can acheve by allowng the retaler multple order opportuntes durng the season. We denote by Heurstc H the polcy that orders only once at the begnnng of the season n the amount of S ( T ). (Recall that S ( ) s the optmal newsvendor stockng level (the largest S that mnmzes T C ( S, ) ) when tme perod [, ) s covered.) Next, we devse heurstcs whch are manly based on mprovement to the newsvendor model. One plausble heurstc, referred to as H2, can be one whch orders up to the newsvendor quantty S ( ) when I ( ) and a demand occurs ( s mplctly assumed that ths wll be the last one n the season. ). Note that H2 s stll myopc snce every tme an order s placed, t As opposed to H, where only one order s placed at the begnnng of the season, under H2 whenever nventory s zero and a demand occurs wth at least left, the optmal newsvendor quantty at that pont s ordered. The procedure s easy to employ, but s expected only to perform reasonably n stuatons where the fxed order cost s sgnfcant. In Proposton 5, we prove that the order up to level resulted from H2 serves as an upper bound to that of the optmal polcy. Proposton 5 The optmal order-up-to level for H2 s greater than or equal to that for the optmal polcy. Suppose we are to place an order at tme. To determne the order quantty, H2 uses as the margnal understockng cost n the newsvendor calculatons. Under the optmal polcy, however, f nventory runs out n the future the retaler stll has the opton to make another order. Therefore, the margnal understockng cost should be less than. The next two heurstcs am to capture ths n the calculatons.

18 That s, suppose an order s placed at tme. At a future tme x, f I ( x ) and a demand occurs, an order wll be placed to brng nventory back to S( x ) approxmate V ( S ( x ), x ) n heurstcs H3 and H4.. The expected total cost s K V ( S ( x ), x ). We wll For nstance, one may estmate V ( S ( x ), x ) by g ( ), the optmal newsvendor cost at tme. That s, the heurstc assumes that expected total future cost at x s the same as the expected total future cost at, K g ( ). We call ths Heurstc H3. Gven ths approxmaton, Heurstc H3 can be specfed as follows: (H3) If at tme ( ), I ( ) and a demand occurs, order up to S ( ), whch s the S largest nteger S satsfyng the condton [ w ( w ) P ( S j, )] p ( j, ). The dervaton of (H3) s avalable n the appendx. j Heurstc H3 can be further mproved by usng a hgher order approxmaton of V ( S ( ), ). Defne: ( ) g ( ) g( ), then V S x x g x K x ( ( ), ) ( ) ( )( ) ( ) ( )( ). We call ths Heurstc H4: (H4) If at tme ( ), I ( ) and a demand occurs, order up to S ( ), whch s the largest nteger S satsfyng the condton S ( ) P ( S, ) [ w ( w ) P ( S j, )] p ( j, ). j The dervaton of (H4) s also avalable n the appendx. Before proceedng further, some dscusson s n order. Frst, as stated n Proposton 5, the order up to level resulted by employng H2 serves as an upper bound to that of the optmal polcy. Second, our

19 numercal tests ndcate that the order up to levels resulted by employng H3 and H4 are upper and lower bounds to those of the optmal polcy, respectvely. In closng, we would lke to stress that the latter observatons were only based on our lmted numercal expermentaton outlned n Secton The Suppler s Problem In ths secton, we consder a dstrbuton system wth one suppler and many ndependent retalers. The suppler replenshes and receves ts stock at the begnnng of the season. The suppler s stock s then used to satsfy retalers orders placed durng the season. If the suppler runs out, excess demand wll be satsfed nstantly at a penalty through a thrd party. We emphasze that whle ths assumpton may be restrctve, t represents some stuatons n practce. Clearly, stuatons n whch the suppler s back orderng wll result n delayed delvery to the retaler are nterestng and challengng, but we leave them as extensons for future work n ths area. Let w and denote the suppler s unt overstockng and understockng costs, respectvely. Snce the suppler s and the retalers problems are decoupled, one can use the results developed n the prevous secton for the retalers. The suppler, however, faces a newsvendor problem. The key here s to fnd the demand dstrbuton at the suppler, whch s comprsed of all retalers orders placed durng the season. We wll focus on one retaler for now. Consder any of the ntervals [, ) n the retaler s polcy. Gven a startng nventory of j at tme and l orders durng that nterval, the probablty that endng nventory at tme s k s: q P r( en d n g n ven to ry at k startn g n v en to ry at = j, n u m b er o f o rd ers d u rn g [, ) = l ) jkl p ( j k, ) f j k S an d l, p ( j l ( S ) k, ) f k S an d j l ( S ) k, o th erw se. The dstrbuton of retaler s total order quantty over the perod, Q, can be expressed as:

20 n n P r( Q x ) q q q. jn jn ln jn jn 2 l n j j l ln, ln,..., l ln, ln,..., l, ln ( S n ) ln ( S n 2 )... l ( S ) x jn, jn,..., j jn S n, j n, j n 2,..., j, j S n The evaluaton of the above expresson reles on consderng all possble cases that would add up to retaler orderng x unts over the season. Frst, we enumerate all possble combnatons of numbers of orders l over dfferent tme-segments [, ) such that total order quantty for all these orders adds up to x unts over the season. We can do ths because, gven retaler s polcy structure, the sze of order n each tme-segment s known. Second, for each of these combnatons, we consder all possble values of endng nventores n each tme-segment. Fnally, the probablty for each case can be specfed usng multplcaton of q values derved above. jkl As an example, consder n=2. In such a case, tme horzon [,T] s dvded nto,,,, and,t. The probablty that retaler wll places orders of sze and orders of sze 2 can be expressed as 2 q q j2 jl 2 j j l where the summaton s taken over all possble and j2, j, j j2 S, j, j S (snce the startng nventory at tme T wll be, s fxed). Therefore, for the total order quantty to be x, we smply sum over all possble and that satsfy ( ) ( ) : 2 P r( Q x ) q q. j2 j l2 j j l l2, l l2, l, l2 ( S ) l ( S ) x j2, j, j j2 S, j, j S If there are multple, but not necessarly dentcal, ndependent retalers, the total demand from all retalers s smply the convoluton of all the dstrbutons. Note that we have ntentonally let the order-upto levels, S, n be general and not requre S = S +. Ths way, the above P r( Q x) expresson can be used to evaluate any polcy of ths form.

21 Computng the exact dstrbuton of the suppler demand can be cumbersome as t requres convolutons of dstrbutons across retalers. When there are M ndependent retalers, one easy approxmaton s to assume that the total order from all the retalers to the suppler s normally dstrbuted. Such approxmatons have been employed n prevous works (See Duermeyer and Schwarz 98, Graves 985 and Zpkn and Svoronos 986). The mean and varance of ths normal dstrbuton s smply the sum of all the ndvdual retalers order mean and varance, whch can be calculated from the mass functon P r( Q x) gven above. 4. Numercal Results Our frst objectve n ths secton s to compare the performance of heurstcs H-H4 wth the optmal soluton of the retaler s problem and to understand the mpact of parameter values on the soluton. We wll then focus on developng nsghts nto the suppler s problem. 4.. The Retaler s Problem As we stated earler, the major analytcal results n Secton 2 hold for nonhomogeneous Posson retaler demand as well as homogeneous Posson retaler demand. Therefore n our dscusson below we wll focus on the homogeneous Posson demand case. To evaluate the performance of the heurstcs, we assgn three parameters,, K and, values at dfferent levels rangng from low to hgh. The other two parameters are held at constant values wthout loss of generalty: w, T. Understockng penalty cost takes values of.5,, 3, and 9, coverng a range of newsvendor fractle values:.333,.5,.75, and.9. The orderng cost K takes values, 5, and 25. Fnally, the base arrval rate s set at 5,, and 2. Ths results n a total of 36 cases. For one of the 36 cases, where the demand and under-stockng penalty are the lowest but orderng cost s the hghest

22 (=.5, K=25, =5), none of the heurstcs place any orders. We wll exclude ths case n our analyss below. Snce demand s exogenous, to solve the retaler s problem t s well known that maxmzng expected proft and mnmzng expected total cost are equvalent. Here we use the total expected cost as comparson measure for two reasons:. It follows the tradton n nventory lterature. 2. The comparson s cleaner when retaler sales prce s not nvolved. The performance of heurstcs s measured by the % devaton = x (expected cost under heurstc polcy expected optmal cost) / (expected optmal cost). Results of our numercal experment are reported n Table. max dev. mn dev. mean dev. H 46.87%.% 8.6% H %.% 7.28% H3.74%.%.93% H4 2.35%.%.3% Table : Cost Devaton from the Optmal Polcy. From Table, we can make several observatons. Frst, the mnmum devaton of all the heurstcs s at %. Ths means that each heurstc can perform as well as the optmal polcy n some scenaros. For example, ths occurs when orderng cost K s very hgh. As we can see from Proposton 3, when K goes to nfnty, approaches T. Therefore, the optmal polcy and the other heurstcs behave lke the newsvendor polcy H, orderng only once at T. Ther costs are thus the same. Under H, the retaler orders the newsvendor quantty only once, at the begnnng of the season. Its comparson wth the other polces, whch may order multple tmes, can tell us how valuable t may be to have multple order opportuntes wthn the season. Our second observaton from Table s that ths beneft s hgh, as H sgnfcantly underperforms the other heurstcs. On a practcal level, t may be

23 costly to reconfgure supply chan and mplement necessary nformaton technologes to enable retalers to order at any tme wth a neglgble lead tme. Nevertheless, by provdng mportant nformaton about the beneft of such systems, our numercal results can help n makng such manageral decsons. Once the retaler can reorder anytme durng the perod, t s possble that he or she approaches ths problem usng a heurstc smlar to H2. That s, the retaler wll order the newsvendor quantty anytme the nventory runs out wth more than left, when t becomes more economcal not to order. Note that H2 s stll a myopc polcy because the retaler doesn t take future order opportuntes nto consderaton when makng the current order. Our thrd observaton from Table s that, just by utlzng the reorder opportuntes, the retaler can sgnfcantly mprove ts performance wth H2, essentally the same myopc orderng polcy as that of H. H3 and H4 are non-myopc heurstcs. They mprove H2 by refnng the understockng cost to take nto consderaton any future order opportuntes. Our fourth observaton from Table s that on average they both perform very well aganst the optmal polcy. H4 has the better performance t s nearly optmal n many cases, and has a small maxmum devaton. Ths s consstent wth the fact that H4 employs a hgher-order approxmaton of V ( S ( ), ) than H3, and s thus more accurate. The three worstperformance cases for H4 occur when understockng penalty cost s hgh ( 9 the mddle of our range ( K 5 ) and orderng cost s n ), rrespectve of the value of arrval rate. The overall performance of H4 s so good that, beyond these three cases, the % devaton for H4 s less than % for all other nstances. The three worst-performance cases for H3 occur when K, 2, rrespectve of the value of. The next three worst performances for H3 occur at K,, rrespectve of the value of. In Tables 2 and 3 below, we further dscuss the effect of parameter values on the performance of dfferent heurstcs.

24 Fgure llustrates the order-up-to levels and the correspondng nterval break ponts for all the heurstcs and provdes further nsghts nto the performance of the varous heurstcs. For H, there s only one order-up-to level at tme θ=t. Another way to analyze polces s to focus on the average total order quantty across the horzon T. A table, avalable n the appendx, reports average order quanttes across horzon under dfferent polces for all parameter combnatons. Optmal Order up to Levels H Order-up-to level=55 Optmal H2 H3 H4 3 2 θ θ Fgure : Comparson of heurstcs wth optmal polcy: w=, =3, T=, K=5, =5. As we dscussed earler, H and H2 make myopc decsons by gnorng the possblty that the system wll be able to reorder f t runs out. As a result, H and H2 always order more than the optmal polcy. In our experment, across all cases, on average the order up to level for H and H2 at tme T s.69% more than the optmal order sze. By revsng margnal penalty cost from to somethng that acknowledges the possblty of reorderng, H3 and H4 not only mprove the performance of H and H2, they also brng the order up to levels close to near-optmal szes. For the case represented n the graph, H3 order up to levels appear to be closer to the

25 optmal than H4 order up to levels. However, takng an average across all cases, the order up to level at tme T s.54% less than the optmal for H3 and.77% less than the optmal for H4. We now dscuss the effect of parameters on the soluton. The orderng cost, K, clearly has a sgnfcant mpact on the soluton. Intutvely, when K s huge, H3, H4, and the optmal polcy wll make orderng decsons to avod future orders. In such cases, they behave lke H and H2, and the dfferences wll dmnsh. Ths s confrmed by results n Table 2. K 5 25 H 96.48% 38.46% 3.33% H2 2.3%.%.% H3 5.4%.22%.% H4.35%.55%.% Table 2: Impact of K on the Average Cost Devaton from the Optmal Polcy. The mpact of understockng and overstockng costs s smlar. The optmal polcy, as well as H3 and H4, recognze future reorderng opportuntes n makng orderng decsons. H and H2, however, tend to over-order because they don t account for future orderng opportuntes. The over-orderng becomes more severe as the newsvendor rato /(+w) becomes larger. Ths s also confrmed by the results n Table 3. / (+w) / H 3.% 48.9% 94.49% 46.36% H2.72% 2.35% 8.33% 6.98% H3.6% 2.2% 2.52% 2.7% H4.2%.9%.9%.72% Table 3: Impact of /(+w) on the Average Cost Devaton from the Optmal Polcy. The mpact of the demand rate s gven n Table 4. As ncreases over the fxed tme horzon (T=), under the optmal polcy the number of orders are expected to go up. Agan, because H and H2 do not account for future orderng opportuntes, ths drves up ther cost devaton from the optmal polcy. H3

26 and H4 mprove on H2 by ncorporatng future orders, but they only recognze the frst future order and gnore the rest (recall that we approxmate the frst future reorder pont by ). Therefore, as the number of orders goes up, ther devatons from the optmal polcy also go up. However, t should be noted that even for, H3 and H4 perform qute well. 5 2 H 56.95% 75.7%.32% H2 5.39% 6.8% 9.87% H3.3%.78% 2.88% H4.22%.3%.4% Table 4: Impact of on the Average Cost Devaton from the Optmal Polcy The Suppler s Problem Next, we study the mpact of the retaler s varous polces on the performance of the suppler. Note that all the heurstcs n Secton 2, H-H4, are developed wth only one objectve: to mnmze the retaler s cost (or, equvalently, to maxmze retaler s proft). Therefore, t s especally nterestng to see how they mpact the suppler. To underscore the fact that the optmal polcy n Secton 4. s optmal only for the retaler, we wll call t the retaler-optmal polcy n ths secton. Recall that the suppler solves a newsvendor problem. We assume that the suppler has enough data to know the dstrbuton of each retaler s total order n the perod. For example, supplers can acheve ths by knowng the end demand pattern and whch heurstc the retalers are usng. The suppler places only one order at the begnnng of the perod, and ts total demand s the aggregate order quantty across all of the retalers. However, dfferent polces used by the retalers wll result n dfferent demand dstrbutons at the suppler, so n order to compare the suppler and system performance across varous polces, we wll use proft as the measure.

27 If we denote the suppler s demand by, then the suppler maxmzes ts proft = [ ] ( ), where p s the unt wholesale prce and TC s the newsvendor functon as defned n equaton () when the demand s. The 36 problem nstances we use are the same as those used n Secton 4.. In addton, we assume there are eght dentcal retalers. We assume that the suppler smplfes the calculatons by assumng that the total demand follows a normal dstrbuton (wth eght dentcal retalers, ths s a reasonable assumpton). In each problem nstance, we use the order-up-to quanttes developed n Secton 4. for the heurstcs. The suppler then fnds the frst two moments of ts total order and uses the newsvendor model to calculate ts own order quantty. For suppler-specfc parameters, we assume the followng values for unt overstockng (w ) and understockng ( ) costs: w = w=, and /=.5,, or 2 representng cases where the suppler s understockng cost s lower than, equal to, or hgher than the retaler s understockng penalty cost per unt. Fnally, to calculate profts, we let the wholesale prce p be 25%, 5%, and 2% of w. The results across all the possble cases are presented n Table 5. For each heurstc, we compute the proft devaton from the retaler-optmal polcy, both for the suppler and for the system. Suppler Proft Total System Proft max dev. mn dev. mean dev. max dev. mn dev. mean dev. H 8.7% -6.55% 7.3% 2.4% -4.68% -.35% H2 65.%.% 5.6% 3.49%.%.92% H3 6.7% % -.85%.64% -.73%.2% H4 5.82% % -2.89%.6% -.38% -.5% Table 5: Proft Devaton from the Retaler-Optmal Polcy. It s nterestng to note the followng:

28 Whle the retaler-optmal polcy naturally domnates H-H4 for the retalers, H-H2 mostly outperform the retaler-optmal polcy for the suppler. Gong from H to H4, whle the retalers' performance mproves, the suppler's proft decreases n general n ths stuaton, what s good for the retalers s bad for the suppler. The result for total system proft s mxed. o Usng H, retalers do not take advantage of multple order opportuntes, thus resultng n much hgher retaler cost than the retaler-optmal polcy. Ths also means, however, that the retaler s order to the suppler s a fxed number wthout varance (.e., t s determned at the begnnng of the perod, not subject to the realzaton of random demand). Ths reduces the suppler s overstockng and understockng costs to zero. Overall, however, these two costs offset each other, and the total system performance of H s very close to that of the retaleroptmal polcy. For H3 and H4 the opposte s true: whle they perform well for the retalers they result n lower proft for the suppler. Ther overall total system proft s close to that of the retaler-optmal polcy. o H2 s qute dfferent. Because H2 ncreases the suppler s proft much more sgnfcantly than H3-H4, ths more than compensates for the slghtly bgger retaler cost ncrease. Consequently, H2 results n the hghest system proft overall. It s nstructve to contrast the retaler-optmal polcy and the heurstcs based on t (H3 and H4), wth the heurstcs based on the newsvendor model (H and H2). Wth the former group, the goal s to optmze retaler performance. Ths s acheved through dynamcally adjustng retaler order sze over tme. Whle ths dynamc adjustment makes the retaler better off, t creates hgher demand uncertanty at the suppler, who makes a newsvendor order decson. It s ths ncreased demand varablty that can sometmes sgnfcantly reduce the suppler s proft. H, on the other hand, performs poorly for the retaler because t foregoes the multple order opportuntes, but t works best for the suppler. We can see ths nterestng tenson between what s good for the retaler and what s good for the suppler. It seems that H2 s a good

29 polcy that strkes the balance: It s based on H but t also modfes H to account for multple order opportuntes. Among the heurstcs we study, t s ndeed the best n terms of total system proft. The overall optmal polcy for the centralzed system (.e., one that maxmzes total system proft) s unknown, and would ndeed be a good topc for future research. 5. Concluson and Future Research In ths paper we analyze a fnte-horzon dstrbuton system wth one suppler and multple retalers. Our work apples to a wde varety of stuatons n whch the suppler must get all supply before the season starts, but the retalers can place orders any tme durng the season. The retalers must carefully decde when to reorder and how much to order each tme by balancng the fxed orderng cost aganst overstockng and understockng penalty costs. We show that the optmal choce s a tme-based, order-upto polcy n whch the order-up-to level decreases over tme. Moreover, we are able to calculate the optmal order-up-to level at any tme and fnd the tme break ponts at whch order-up-to levels change. Ths completely characterzes the optmal retaler polcy. It s worth notng that even though we present our results n terms of homogeneous Posson demand, they apply to non-homogenous Posson demand as well. Ths s especally mportant for products wth a short lfe cycle, where demand often changes over tme (e.g., demand dmnshes towards the end of the season). For practcal purposes, we also propose heurstcs H3 and H4, whch are easer to calculate. The suppler faces a classc newsvendor problem. We express the probablty dstrbuton of suppler s demand. However, evaluaton of the demand dstrbuton, whch nvolves condtonng over tme ntervals and convolutons across retalers, can be complex. Our numercal experment results n nterestng fndngs. Frst, we show that by havng multple order opportuntes wthn the same season, retalers can sgnfcantly reduce ther costs (H2 versus H). If retalers use more sophstcated orderng polces (H3, H4, and optmal), they can acheve further cost