Retailers must constantly strive for excellence in operations; extremely narrow profit margins

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1 Managng a Retaler s Shelf Space, Inventory, and Transportaton Gerard Cachon 300 SH/DH, The Wharton School, Unversty of Pennsylvana, Phladelpha, Pennsylvana 90 cachon@wharton.upenn.edu Retalers must constantly strve for excellence n operatons; extremely narrow proft margns leave lttle room for waste and neffcency. Ths artcle reports a retaler s challenge to balance transportaton, shelf space, and nventory costs. A retaler sells multple products wth stochastc demand. Trucks are dspatched from a warehouse and arrve at a store wth a constant lead tme. Each truck has a fnte capacty and ncurs a fxed shppng cost, no matter the number of unts shpped. There s a per unt shelf-space cost as well as holdng and backorder penalty costs. Three polces are consdered for dspatchng trucks: a mnmum quantty contnuous revew polcy, a full servce perodc revew polcy, and a mnmum quantty perodc revew polcy. The frst polcy shps a truck when demand snce the prevous shpment equals a fxed fracton of a truck s capacty,.e., a mnmum truck utlzaton. The exact analyss of that polcy s the same as the analyss of reorder pont polces for the multechelon problem wth one-warehouse, multple retalers, and stochastc demand. That analyss s not computatonally prohbtve, but the mnmum quantty level can be chosen wth a smple economc order quantty (EOQ) heurstc. An extensve numercal study fnds the followng: Ether of the two perodc revew polces may have substantally hgher costs than the contnuous revew polcy, n partcular when the warehouse to store lead tme s short; the EOQ heurstc performs qute well; the mnmum quantty polcy s total cost s relatvely nsenstve to the chosen transportaton utlzaton, and ts total cost s close to a lower bound developed for ths problem. (Inventory Management; Stochastc Demand; Jont Setup Cost) Do not envy retalers, for they have a very tough tme earnng profts (Guar et al. 999). Hence, excellence n operatons s crtcal for them. An mportant task for retalers s the balancng of transportaton, shelf space, and nventory costs. For nstance, a retaler could choose to ncrease transportaton utlzaton, thereby lowerng ts transportaton cost, but that also ncreases the tme nterval between delveres to a store. To account for less frequent delveres, the store wll ether need to expand shelf space and nventory or sacrfce customer servce. Ths research studes the challenge of managng these nteractons. The settng consdered here s a sngle retal store that sells multple products wth stochastc demand. There are lnear nventory holdng and backorder costs as well as a lnear shelf-space cost. The latter reflects the expense of acqurng and mantanng a larger store as total shelf space s expanded. The store s replenshed from a warehouse va trucks. Each truck has a fnte capacty, C, and ncurs a fxed delvery charge ndependent of the number of unts t actually delvers. The warehouse always has enough trucks and nventory to fll the store s replenshment requests, but any shpment requres a constant transportaton tme from the warehouse to the store. The retaler must assgn a shelf-space quantty for each of 523-6/0/0303/02$ electronc ISS 200 IFORMS Vol. 3, o. 3, Summer 200, pp

2 Managng a Retaler s Shelf Space, Inventory, and Transportaton ts products, choose a replenshment polcy, and schedule truck dspatches to mnmze total expected costs per unt tme. Let S be the amount of shelf space assgned to product, and let S {S,...,S }. Three polces for dspatchng trucks are consdered. The full servce perodc revew polcy, or (S, T ) polcy for short, revews the nventory status of the store every T unts of tme and dspatches enough trucks to completely replensh the store s shelves; a base-stock polcy s the replenshment polcy for each product, where the shelf space s the base-stock level. Ths s a full servce polcy because each product s order (.e., ts demand snce the last revew epoch) s always shpped. The parameter T controls the transportaton cost, but t s possble that some delveres wll have a low transportaton utlzaton: Because of the full servce guarantee, a truck mght be dspatched wth only one unt. One desrable feature of ths polcy s that shelf space s mnmzed (for the gven revew nterval T ): Because there s no uncertanty n the supply process, each product s shelf space need only buffer the demand uncertanty durng the transportaton lead tme. If the retaler does not have the ablty to choose a revew nterval, T, the retaler could use a mnmum quantty perodc revew polcy, or a (Q, S T ) polcy for short: Every T unts of tme (an exogenous parameter) the retaler revews ts nventory and dspatches trucks so long as one truck has at least Q unts and the other trucks are full. (One mght be tempted to requre that at each revew epoch the average shpment s no less than Q unts, but that constrant results n a more complex analyss.) Due to the Q constrant, some porton of the products orders mght not be flled, where each product s order equals the dfference between ts shelf space and ts nventory poston (on-hand nventory mnus backorders plus on-route nventory). Hence, ths polcy requres an allocaton rule to determne what porton of each order s actually shpped. Ths allocaton rule creates a supply uncertanty that must be accounted for n the shelf-space decson, n addton to demand uncer- Ths polcy resembles Albert Hejn s polcy for dspatchng trucks from ts central warehouse to ts stores. Albert Hejn s a large grocery retaler n the etherlands. tanty. Due to the complexty of that problem, a heurstc s developed to choose polcy parameters. Wth a perodc revew polcy the store manager can plan the stock replenshment process (e.g., gettng extra labor to help wth unloadng and shelf replenshment) because the store receves shpments on a regular schedule. The key defcency of a perodc revew polcy s that t mght delay some truck shpments: Once there are enough orders to fll a truck, a truck should be shpped mmedately,.e., watng to shp a full truck only rases costs. A mnmum quantty polcy, or (Q, S) polcy for short, elmnates that problem: Inventory s revewed contnuously and a truck s dspatched when Q unts have been ordered, where each product s order equals ts demand snce the last shpment. (ote that all three truck dspatchng polces are coupled wth base-stock polces to govern the products replenshment decsons, where each product s base-stock level equals ts shelf space.) Wth ths polcy every shpment contans exactly Q unts, and so the transportaton utlzaton s constant across all shpments, Q/C. As wth the (Q, S T )polcy, ths polcy creates supply uncertanty, but t does so n a way that s analogous to the supply uncertanty generated by reorder pont polces n a sngle product two-echelon supply chan wth one warehouse and multple retalers. Axsäter (993) provdes an exact analyss of that model, and so those results are ncorporated nto ths model to provde a method to determne the optmal (Q, S) polcy. The analyss of the (Q, S) polcy s computatonally tractable, but a smple heurstc for choosng Q (whch, recall, determnes the transportaton utlzaton) s desrable. It s shown that the cost functon s the sum of a decreasng hyperbolc functon (transportaton cost) and an approxmately ncreasng lnear functon (nontransportaton costs). That s the same form as the cost functon n the well-known economc order quantty (EOQ) problem. Thus, the heurstc developed s analogous to the EOQ wth an adjusted holdng-cost rate. Because the EOQ cost functon s relatvely flat about ts optmum, t s reasonable to conjecture that the retaler s costs are nsenstve to the chosen transportaton utlzaton. The numercal study valdates that conjecture. 22 Vol. 3, o. 3, Summer 200

3 Managng a Retaler s Shelf Space, Inventory, and Transportaton In addton to the senstvty of the cost functon about the optmum, a manager would also want to know how the optmal control varables, Q and S, are mpacted by changes n the model s parameters: lead tme, product-lne breadth, total-demand volume. For example, one could argue that the optmal transportaton utlzaton and the lead tme from the warehouse to the store are substtutes: As the lead tme s decreased the retaler can take advantage of the faster delveres by watng to fll ts trucks wth addtonal unts. Alternatvely, they are complements f the retaler should take advantage of the shorter lead tme by not watng to fll ts trucks wth addtonal unts so as to reduce supply uncertanty. In fact, the latter s correct. Overall, t s shown that transportaton utlzaton s a complement to lead tme and total demand volume but a substtute to the product lne breadth. Whereas t can be conjectured that the (Q, S) polcy does better at managng the costs explctly ncluded n ths model than the two perodc revew polces, the numercal study provdes an ndcaton of the magntude of the advantage. It s shown that the perodc revew polces perform almost as well as the (Q, S) polcy n some scenaros, but they may perform sgnfcantly worse n other scenaros. In partcular, the (Q, S T ) polcy performs poorly f T s too large, where a reasonable benchmark for too large s T greater than the average tme for total demand to equal the truck capacty. However, even f T s not too large the average performance of the (Q, S T ) polcy deterorates as the warehouse-to-store lead tme s reduced. The (S, T ) polcy also performs poorly when the warehouse-to-store lead tme s short but performs reasonably well when that lead tme s long. It s possble that a polcy exsts that s even better than the (Q, S) polcy. (The optmal polcy s not known.) A lower bound s developed for ths model to get a sense of how much better an optmal polcy mght be. The numercal study fnds that the (Q, S) polcy provdes a cost that s not much greater than the lower bound f there s a long lead tme or f the rato of backorder penalty cost to the shell-space cost s small,.e., f shelf space s relatvely expensve. The gap between the feasble cost and the lower bound s sgnfcant n partcular wth a low lead tme and a hgh backorder penalty cost, agan, relatve to the shelf-space cost. evertheless, the overall performance of the (Q, S) polcy s qute good. The remander of the artcle s organzed as follows: the next secton provdes a revew of the related lterature, 3 detals the exact evaluaton of reorder pont polces and also provdes the EOQ heurstc procedure, evaluates the fxed nterval polces, 5 descrbes the lower bound, 6 presents the numercal study, and 7 summarzes the conclusons.. Lterature Revew The retaler s problem s related to the jont replenshment problem (JRP). Whle there are many versons of the JRP, the key features are that each tem/product ncurs ts own fxed charge (a mnor setup cost) whenever t s ordered and the system ncurs a fxed charge (a major setup cost) whenever there s an order, no matter the number of tems n that order or whch tems are n the order. The retaler does not ncur an tem-specfc fxed charge, but the fee per truck delvery s smlar to the system fxed charge. The only dfference, albet a sgnfcant one, s that the truck delvery fee s a fxed charge for a lmted number of unts,.e., the capacty of the truck, whereas n the JRP the fxed charge s truly a fxed charge,.e., there s no capacty lmt. Furthermore, the JRP lterature does not consder a shelf-space cost. Several authors study the JRP wth stochastc demand. Balntfy (96) proposed can-order polces: A can-order, a must-order, and an order-up-to level are specfed for each product; nventory s revewed contnuously, and an order s placed whenever there s an tem wth an nventory poston at or below ts must-order level; ncluded n the order s any tem wth an nventory poston at or below ts can-order level, and for each of those tems the order rases ts nventory poston to ts order-up-to level. Slver (98) and Federgruen et al. (98) propose algorthms to choose can-order polcy parameters. A possble defcency of the can-order polcy s the poor coordnaton across tems: An tem mght trgger an order when there are very few other tems that need a replenshment. Vol. 3, o. 3, Summer

4 Managng a Retaler s Shelf Space, Inventory, and Transportaton To mprove coordnaton across tems, Atkns and Iyogun (988) study two perodc revew replenshment polces. In the frst polcy, tems are ordered up to a base-stock level at every revew epoch. The decson parameter s the length of the tme between revew epochs. Wth the second polcy, there s a set of tems that are revewed at every epoch, whereas the other tems are revewed less frequently (but stll only at revew epochs). The second polcy s desgned to account for dfferences n tem-specfc fxed charges. When there are no tem-specfc fxed charges, as n ths model, the two polces are the same. The full servce fxed nterval polcy n s the same as Atkns and Iyogun s frst polcy. However, ther cost evaluaton s approxmate, whereas ths artcle provdes an exact analyss. Pantumsncha (992) develops a heurstc to choose parameters for the QS polcy ntroduced by Renberg and Planche (967): Whenever the combned nventory poston of all tems reaches a reorder pont, an order s placed to rase the nventory poston of all tems to ther base-stock levels. That polcy s the same as the (Q, S) polcy consdered n ths artcle. However, here the optmal reorder pont polcy s found. Vswanathan (997) consders P(s, S) polces: every T unts of tme each product s ordered based on an (s, S) polcy, where T, s and S are chosen parameters. He shows n a numercal study that the P(s, S) polces generally perform better than the other polces mentoned. Atkns and Iyogun (988) develop a lower bound for the JRP, whch decomposes the problem nto ndependent problems by allocatng the system fxed charge (the major setup cost) among the products. In ths artcle, a new dea s used to develop a lower bound: Instead of allocatng the fxed cost across products, demands are allocated across products. Specfcally, each system demand s dvded among the products proportonal to ther average demand rates. Pryor et al. (999) study the sngle-tem nventory problem wth transportaton setup costs. That problem s closely related to the one consdered here wth the key dstncton beng that they concentrate on the sngle-tem problem. In fact, under certan condtons they fnd an optmal polcy. They also propose a heurstc polcy for the two-tem problem. There s an extensve lterature on the JRP wth determnstc demand: e.g., Jackson et al. (985), Anly and Federgruen (l99), Federgruen and Zheng (992), Vswanathan and Mathur (993), and Bramel and Smch-Lev (995). Wth determnstc demand the tmng and quantty of future orders can be antcpated, so t s not clear how to compare those polces wth those desgned for stochastc demand. Speranza and Ukovch (99) consder the determnstc verson of the retaler s problem: A frm manages transportaton and nventory along a sngle lnk (e.g., between a warehouse and a retal store), there are multple products, trucks have fnte capacty, there are nventory holdng costs, and there s a fxed cost per delvery. For that problem Blumenfeld et al. (985) show that the EOQ model can be used to choose the delvery frequency. As Speranza and Ukovch (99) dscover, that method does not provde a good soluton f the frm has a lmted set of feasble delvery frequences (e.g., t cannot shp every 2 unts of tme). Ths artcle demonstrates that a EOQ heurstc does provde good solutons n a stochastc demand settng, assumng no constrant s mposed on when the frm can dspatch trucks. There s a sgnfcant lterature on managng vehcle routng along wth nventory costs. Much of that lterature assumes determnstc demand. Federgruen and Zpkn (98a), McGavn et al. (993), Adelman and Kleywegt (999), and Reman et al. (999), are exceptons. Ths artcle does not consder vehcle routng. There has been some recent lterature on perodc revew polces n the multechelon nventory problem wth multple retalers and stochastc demand: Cachon (999), Chen and Samroengraja (996, 999), and Graves (996). A retaler n that work s analogous to a product n ths model. But those papers do not nclude a jont orderng/transportaton cost. Several authors study the allocaton of shelf space across multple products when customers may swtch ther demand among products f ther preferred product s unavalable (see Mahajan and van Ryzn 999 for a revew). Ths artcle does not consder that behavor,.e., the demand rate for each product s ndependent of the shelf-space allocaton. Gerchak and 2 Vol. 3, o. 3, Summer 200

5 Managng a Retaler s Shelf Space, Inventory, and Transportaton Wang (99) consder a model n whch the mean demand rate for a product s ncreasng n the product s shelf space, but n ths model each product s demand rate s exogenous. 2. Model A sngle retaler manages a warehouse and one retal store. Trucks are used to transport nventory from the warehouse to the retal store. Each truck has a capacty of C unts and costs K per delvery, ndependent of the amount transported. Once a truck s dspatched from the warehouse, t arrves at the retal store n exactly L unts of tme. The tme to load and unload a truck s gnored. Trucks may be dspatched at any tme, and there s no lmt on the number of trucks avalable. The retaler sells products. Demands are observed contnuously. Let Dt be stochastc demand (n unts) for product over any nterval of tme of length t, and let D t be total demand over the same nterval (also n unts). Let be the mean demand rate, t E[D t], and let be the total demand rate,. Let f (x, t) andf (x, t) be the densty and dstrbuton functons of D t. Dt has a Posson dstrbuton. In some retal settngs the Posson dstrbuton s not the best representaton of the demand process (see Agrawal and Smth 99), but t does provde analytcal tractablty. Product s charged h per unt of nventory at the retal store per unt tme. A warehouse nventory holdng cost s not charged. (The warehouse probably serves multple retal stores, but ths model focuses on the cost to operate a sngle store.) ether s there a holdng cost for ppelne nventory, because the retaler cannot nfluence that cost. Product s also charged p per unt backordered per unt tme. It s assumed that all demands are backordered, whch s doubtful for most retalers. However, ntroducng lost sales would render the problem computatonally ntractable. Further, for large p values the retaler wll choose polces that lead to hgh fll rates, and thus, the behavor of ths model s an approxmaton of the behavor n a model wth lost sales. Let S 0 be the amount of shelf space the retaler allocates to product ; the retaler cannot hold any more than S unts of product at ts store, nor can product be stored n another product s shelf space. (For example, a grocery retaler does not want to stock cans of soup n the shelf space desgnated for dapers. Further, t s too costly to contnuously change the products shelf-space allocatons.) One consequence of the shelf-space constrant s that the retaler cannot load onto a truck more unts than can ft on the store s shelves f the truck were to arrve at the store mmedately. 2 The retaler ncurs a charge of a per unt of shelf space allocated to any product. There s no constrant mposed on the total shelf-space allocaton; the model s best appled before the retaler has constructed ts store. (A shelf-space constrant can be accommodated, as descrbed n 3.) ote that the shelf-space cost cannot be ncorporated nto the product s holdng cost because, whereas the average holdng cost for a product s based on that product s average nventory, the shelf-space cost s based on the product s maxmum nventory poston. The retaler s objectve s to choose a truck dspatchng polcy and a shelf-space allocaton and an nventory polcy to mnmze total expected cost per unt tme. Some math notaton: x s the greatest nteger less than or equal to x; x s the smallest nteger greater than or equal to x; [x] max{0, x}; and [x] max{0, x}. 3. The (Q, S) Polcy An ntutve polcy for dspatchng trucks s to shp a truck whenever the cumulatve orders across the products equals a constant threshold, where an order for one unt of product s generated wth each demand for product ;.e., products are ordered usng a base-stock polcy and each product s base-stock level equals ts shelf space, S. Let Q be the truck thresh- 2 Some stores have backrooms where any product may be stored, thereby allowng the retaler to load more unts onto a truck than can be placed on the shelves. However, unts n the backroom are not mmedately avalable to customers. Hence, a backroom acts lke a warehouse. Vol. 3, o. 3, Summer

6 Managng a Retaler s Shelf Space, Inventory, and Transportaton old level, where Q [, C]: It s never optmal to shp empty trucks, whch rules out Q ; nor s t optmal to delay shppng a full truck, whch rules out Q C. When Q C, the (Q, S) polcy s a full truck polcy,.e., only full trucks are dspatched. The (Q, S) polcy s smple to descrbe, but n practce t may be dffcult to mplement. The retaler would requre an nformaton system that contnuously and accurately montors nventory at the store and communcates that nformaton to the warehouse. In addton, the warehouse must have the capablty to respond to orders wthout the beneft of a perodc shpment schedule. Although t may not be ntally obvous, the (Q, S) polcy s a subset of the reorder pont polces Axsäter (993) consders n a two-echelon nventory system. Axsäter (993) provdes a recursve algorthm to exactly evaluate expected nventory and backorder costs and to choose optmal reorder pont polces. Ths secton next lnks ths model to Axsäter s model. For clarty, only the necessary results to evaluate costs and to fnd the optmal polcy are explaned. 3 The secton concludes wth a smple heurstc for choosng each product s shelf space and a heurstc for choosng Q. In Axsäter (993) there s one warehouse and retalers. There s a constant lead tme from the nventory source to the warehouse, L w, and a constant lead tme from the warehouse to each retaler, L r.demand at each retaler s Posson. Axsäter assumes an dentcal demand rate across the retalers, but that assumpton s not needed when the retalers mplement one-for-one orderng. (Axsäter 990 derves an exact analyss wth non-dentcal retalers and one-for-one orderng at both echelons.) Retalers use (R, Q ) reorder pont polces to manage ther nventory, and the warehouse nventory s managed wth an (R w, Q w ) reorder pont polcy. 3 Axsäter s results do requre several straghtforward modfcatons. He assumes dentcal retalers (n ths settng each retaler corresponds to a product), because he consders the possblty of batch orderng by the retalers. The dentcal retaler assumpton s not necessary when the retalers use one-for-one orderng, even f the warehouse uses batch orderng. Also, Axsäter does not consder an orderng/transportaton cost nor a shelf-space cost, so those costs must be ncluded n the analyss. To connect the models, let retaler correspond to product. Because a base-stock model manages each product s nventory, let Q r andr S.In Axsäter, each warehouse order contans Q w unts. In ths model, each warehouse order s a truck that contans Q unts. So, a truck n ths model corresponds to a warehouse order n Axsäter,.e., set Q w Q. In Axsäter, the warehouse orders a batch from ts source, whch shps all orders mmedately, whenever ts nventory poston s R w. In ths model, nventory s mmedately avalable to the warehouse, so the warehouse has no on-order nventory, L w 0. Further, the warehouse nventory does not ncur holdng costs, so the warehouse n ths model has no on-hand nventory ether. Hence, n ths model, the warehouse s nventory poston equals the absolute value of the number of unts backordered. Thus, a warehouse order (a truck) s placed (dspatched) when there are Q backorders at the warehouse, whch corresponds to R w Q n Axsäter. Trucks requre tme to travel to the retal store, so L L r. ote that n Axsäter, R w s a choce parameter and Q w s exogenous, whereas n ths model both are choce parameters, but Q R w Q w. ow consder the evaluaton of expected costs for an (Q, S) polcy. (The followng notaton s dfferent from that n Axsäter to streamlne the presentaton and to provde consstency wth the rest of the artcle.) Ths s done by relatng a unt s arrval tme at the retaler wth the tme the unt s demanded: If the unt arrves before ts demand, holdng cost s ncurred; whereas f the unt arrves after ts demand, backorder cost s ncurred. Averagng over all unts yelds the average cost per unt tme. A unt of product ordered and shpped at tme arrves at tme L. That unt satsfes the S th demand to occur after tme. Let ĝ (S, L) be the expected holdng and backorder costs ncurred by a unt of product f the unt s shpped once t s ordered, ĝ (y, t) p t 0 (y, x)(t x) dx h (y, x)(x t) dx, () t 26 Vol. 3, o. 3, Summer 200

7 Managng a Retaler s Shelf Space, Inventory, and Transportaton where (y, t) s the densty functon of the Erlang (, y) dstrbuton. ote that t t y 0 0 y (y, x)xdx (y, x) dx ( F (y, t)), snce the probablty that (y )th demand after tme occurs before tme t s the same as the probablty that (y ) or more demands occur over the nterval [, t]. So () can be wrtten as: ĝ (y, t) [y(h p )F (y, t) t(h p )F (y, t) p (t y)]. ow suppose a unt of product s ordered at tme, but t s not shpped untl tme t, t 0. If m demands occur at product over the nterval [, t], then expected costs for that unt are ĝ (S m, L): The unt shpped at tme t wll arrve at the retaler at tme t L and satsfy the (S m)th subsequent demand after tme t. Let n be the number of products ordered n the nterval (, t] (ncludng possbly product ), where n [0, Q ]. (n 0 means that the unt trggers ts own truck shpment,.e., t 0.) Gven the set of n demands, the number of product demands n that set s bnomally dstrbuted, where n s the number of draws and / s the probablty of success. Let Z (n) be that random varable, n y ny Pr(Z (n) y) ( )(). y Fnally, n s the realzaton of a unformly dstrbuted random varable on the nterval [0, Q ]: Q n s the warehouse s nventory poston, and Axsäter (993) demonstrates that the warehouse nventory poston s unformly dstrbuted on the nterval [Q w, 0]. Thus, the expected holdng and backorder cost per unt of product s Q n Q n0 m0 Pr(Z (n) m)ĝ (S m, L). The shelf-space cost for product occurs at rate as. Transportaton cost per unt s K/Q, so transportaton cost s ncurred at an average rate (K/Q). Overall, let (S) be expected cost per unt tme, Q n K (Q, S) as Pr(Z (n) m) Q Q n0 m0 ĝ (S m, L). (2) Axsäter (993) demonstrates that the latter term n (Q, S) sconvexns. Because the shelf-space term s lnear n S, (Q, S) sconvexns. Hence, for a fxed Q t s easy to evaluate the optmal shelf space for each product. (If there s a shelf-space constrant, then a greedy algorthm fnds, for a fxed Q, the optmal shelf space for each product: Start each product wth zero shelf space, allocate one unt of shelf space at a tme to the product that generates the greatest margnal cost reducton, stop when the shelf-space constrant s bndng). Let S * (Q) beproduct s optmal shelf space gven Q. It s ntutve that S * (Q) s nondecreasng n Q: As truck utlzaton s ncreased, Q/C, the retaler never reduces a product s shelf space. Fndng the optmal shpment quantty, Q*, requres a search over the feasble nterval, [, C]: (Q, S * (Q),...,S * (Q)) may not be convex n Q. Because (Q, S) s not well behaved n Q, t s not possble to defntvely characterze the behavor of the optmal polcy wth respect to the parameters (e.g., L,, ). evertheless, clues are avalable to suggest trends. Consder the relatonshp between L and the optmal Q. L has no mpact on transportaton costs, so ts nteracton wth Q occurs wth the nontransportaton costs. Focus on the behavor of the ĝ functon. ote that ĝ (y, t) ĝ (y, t) [(h p )F (y, t) p ]. Hence, y*(t) mnmzes ĝ (y, t) fy*(t) s the largest nteger such that F (y*(t), t) p (h p ). Because F (y, t) s stochastcally ncreasng n t (.e., F (y, t) F (y, t) for all t t), the followng tends to hold for L L and nteger values of x: ĝ (y*(l) x, L) ĝ (y*(l), L) ĝ (y*(l) x, L) ĝ (y*(l), L). In words, the ĝ functon becomes flatter around ts Vol. 3, o. 3, Summer

8 Managng a Retaler s Shelf Space, Inventory, and Transportaton mnmum as L ncreases. If the ĝ functon becomes flatter, then the margnal change n nontransportaton costs wth respect to Q decreases,.e., as L ncreases, ncreasng Q generates a smaller margnal ncrease n the nontransportaton costs. That suggests the optmal Q s ncreasng n L: Because an ncrease n L has no mpact on the margnal beneft of an ncrease n Q (a lower transportaton cost) and decreases the margnal cost of an ncrease n Q, the optmal Q tends to ncrease. The lesson for a manager s that the optmal transportaton utlzaton (Q/C) should decrease f faster delveres between the warehouse and the store become avalable. The same argument can be appled to the totaldemand parameter,, and the number of products parameter,. An ncrease n, holdng all else constant, tends to ncrease each product s demand rate, whch makes the ĝ functons flatter,.e., t has the same qualtatve mpact as an ncrease n the lead tme. Hence, an ncrease n should lead to a hgher optmal Q. On the other hand, an ncrease n the breadth of the product lne, agan holdng all else constant, lowers the average product s demand rate, whch makes the ĝ functons steeper. Hence, an ncrease n should lead to a lower optmal Q. To summarze, hgh volume retalers wth narrow product lnes and long lead tmes should have hgh transportaton utlzaton. That hypothess s consstent wth the recommendaton n Fsher (997) that companes wth nnovatve products (.e., ones wth hgh demand varaton) should mplement market responsve supply chans, one consequence of whch ncludes low transportaton utlzaton. 3.. Two Heurstcs Although t s not computatonally dffcult to evaluate the optmal (Q, S) polcy, t would be useful to construct smple heurstcs to choose Q and S. Those heurstcs could provde a retaler wth a quck check on the qualty of current performance, and they also provde some qualtatve nsghts. Both heurstcs are That relatonshp does not always hold. For certan crtcal fractles, p /(h p ), t s possble that the ĝ functon s flatter near ts mnmum for smaller lead tmes. derved by replacng the stochastc varables n the cost functon (2) wth ther mean. If product s demand occurred at a determnstc rate, then at tme the expected holdng and backorder costs at tme L s g d (x), where x s the n- ventory poston at tme, g d (x) h [x L] p [x L]. ote that g d(x) ĝ (x, L) for all x and g d (x) ĝ (x, L) for large x. Replacng ĝ (x, L) wth g d (x) n (2) gves Q n K d Q Q n0 m0 as Pr(Z (n) m)g (S ). To approxmate the above, replace the bnomal random varable wth ts mean value, Q K d Q Q n0 as g (S n). (3) The above s a convex functon n S, so the optmal S s not dffcult to fnd. But workng wth dscrete S values s cumbersome, so construct the contnuous approxmaton of (3), (Q, S), S Q SQ K (Q, S) as g d(y). Q (Q, S) sconvexns. Let S (Q) be an optmal shelf space for product gven the cost functon (Q, S), 0 a p S (Q) p a L Q a p. p h ote that S (Q) s lnear n Q and greater than the mean lead tme demand. (S* (Q*) can be less than mean lead tme demand, especally f a s large relatve to p.) From the envelope theorem, (Q, S (Q)) s the sum of a decreasng hyperbolc functon and ncreasng lnear functons. That s the same structure as the well-known EOQ. Hence, (Q, S (Q)) s convex n Q. Let Q mnmze (Q, S (Q)), K, C Q mn ( p mn{a, p }) 2. p 2 p h 28 Vol. 3, o. 3, Summer 200

9 Managng a Retaler s Shelf Space, Inventory, and Transportaton The Q-heurstc sets Q Q 0.5 and the S-heurstc sets S S (Q ) 0.5,.e., both are rounded off to the nearest nteger. Better roundng procedures could be developed, but that level of precson s unnecessary for ths exercse,.e., n practce the optmal (Q, S) polcy should be mplemented because the exact cost functon s not computatonally demandng. It s well known that the EOQ cost functon s qute flat about ts mnmum. Hence, to the extent that (Q, S) provdes a good approxmaton for the true cost functon, t s reasonable to conjecture that (Q, S * (Q)) s also relatvely flat about ts mnmum,.e., costs are relatvely nsenstve to the chosen transportaton utlzaton, Q/C. The numercal study evaluates that conjecture along wth the qualty of the two heurstcs.. Perodc Revew Polces Wth a perodc revew polcy the retaler revews ts nventory every T unts of tme. Two versons are consdered. Wth a full servce polcy, an (S, T ) polcy, the retaler dspatches a suffcent number of trucks at a revew epoch to replensh all demand snce the prevous revew epoch. Wth a mnmum quantty polcy, a (Q, S T ) polcy, the retaler requres that one of the dspatched trucks has at least Q unts and the remanng trucks are full. To control ts transportaton cost, the retaler chooses T n the (S, T ) polcy. In the mnmum quantty polcy T s exogenous, so the retaler controls the transportaton cost wth the parameter Q. (If T and Q were both choce parameters, then the retaler would surely choose T 0,.e., t would choose a (Q, S) polcy.) Wth both perodc revew polces, orders for product are generated wth a base-stock polcy, where S s the base-stock level. Expected cost wth an (Q, S T ) polcy s evaluated n two man steps. The frst evaluates the expected transportaton cost, and the second evaluates the nontransportaton costs. The expected cost of an (S, T ) polcy s the same as the expected cost of a (Q, S T ) polcy wth Q. As wth the (Q, S) polcy, a (Q, S T ) polcy s a full truck polcy when Q C. Begn wth more notaton. Let IP be product s nventory poston (on-hand plus on-route nventory mnus backorders) mmedately before a revew epoch. Let IP be product s nventory poston mmedately after a revew epoch. Defne B to be product s outstandng orders mmedately before a revew epoch, B S IP, and let B be the total number of outstandng orders at that tme, B B. ote that B s the total amount of avalable shelf space mmedately before a revew epoch. Let B be the avalable shelf space for product mmedately after a revew epoch, and let B be the total avalable shelf space; B T B D and B B D T, () where B [0, Q ]. At the revew epoch n consderaton, the retaler wll dspatch m(b ) trucks, where [ ] x Q m(x). C The probablty that at least one truck wll be dspatched at the revew epoch s Pr(B Q). The expected truck utlzaton, (Q), s therefore (Q) Pr(B x) Pr(B Q) xq C [m(x)c x] m(x). (5) C m(x) The expected transportaton cost per unt s K/(Q)C, and the expected transportaton cost per unt tme s K/(Q)C. The dstrbuton functon of B s requred to evaluate (5). From (), B s a smple convoluton of B and D T, because they are ndependent. So, t remans to evaluate the dstrbuton functon of B. There are three cases to consder: Q, Q C, and Q C. When Q, a full servce polcy s mplemented: All outstandng orders at a revew epoch are shpped. In that case x 0 Pr(B x) 0 x 0. Wth a full truck polcy, Q C, the warehouse op- Vol. 3, o. 3, Summer

10 Managng a Retaler s Shelf Space, Inventory, and Transportaton erates as f t s usng a perodc revew, reorder pont polcy wth the reorder pont equal to C: EveryC demands trggers a truck dspatch at the subsequent revew epoch. In that case, B s unformly dstrbuted on the nterval [0, Q ]. When Q C, the warehouse s outstandng orders mmedately after a revew epoch s a Markov process. Smulaton s one technque to evaluate the dstrbuton functon of B. The analytcal approxmaton that follows s an alternatve. Let B j be the number of outstandng orders mmedately after a revew epoch when the jth prevous revew epoch had zero outstandng orders and all revew epochs after that one had at least one outstandng order. For clarfcaton, the 0th prevous epoch s the epoch n consderaton; hence, Pr(B 0 0). Defne the state of the system to be the number of successve revew epochs to have occurred n whch all of the epochs had a postve number of outstandng orders; hence the system s n state 0 at tme when the last revew epoch to occur before tme had zero outstandng orders. Let j be the proporton of tme n whch the system s n state j. The system ether transtons from state j to state j orthe system transtons from state j to state 0. Let j be the probablty the system transtons from state j to state j, j. Therefore, j s the probablty the system transtons from state j to state 0. The j probabltes and the dstrbuton functons B j are evaluated wth a system of recursve equatons. Defne B j B j D T : B j s the number of outstandng orders mmedately before a revew epoch n whch there were B j outstandng orders mmedately followng the prevous revew epoch. It then follows that [Pr(B j j mc Q ) m0 Pr(B j mc)] and (6) j j j j m0 Pr(B x) [Pr(B mc x) Pr(B mc)]. Because Pr(B 0 0), the recurson begns wth B D T. ext, from (6), s evaluated and then, from (7) (7), B s evaluated. The remanng recurson s then apparent: 2, B 2, 3, B 3, etc. There are an nfnte number of states n ths Markov chan, but the probablty of reachng state j decreases n j. Therefore, as an approxmaton, suppose that the system always transtons from state M to state 0 for some large M,.e., M 0. The accuracy of the approxmaton ncreases n M, but so does the computatonal effort. It follows that and M ( ) M M. Solvng that system of equatons yelds Fnally, M j j k j j M j j k Pr(B x) k k 0 0. M Pr(B x). 0 Attenton s now turned to the nontransportaton costs. Let g (y, t) be expected holdng and backorder costs for product at tme t, t L, when the product s nventory poston s y at tme and no addtonal shpments wll be made before tme t: t t g (y, t) E{h [y D ] p [y D ] } h (y t) (h p ) xy ote that (x y) f (x, t). 220 Vol. 3, o. 3, Summer 200

11 Managng a Retaler s Shelf Space, Inventory, and Transportaton xy xy xf (x, t) tf (x, t) t[ F (y, t)], so the above can be smplfed further, g (y, t) (h p )[yf (y, t) tf (y, t)] p (y t). (8) Let G (y, T ) be expected costs over the nterval of tme [ L, L T ] when product s nventory poston s y at tme and that s a revew epoch, G (y, T) LT L g (y, t) dt. The above s easy to evaluate for y 0, snce then G (y 0, T) p LT L (y t) dt T pt L y. (9) 2 To evaluate G (y, T ) for y 0, frst dfferentate (8) wth respect to t, dg (y, t) dt [g (y, t) g (y, t)], where note that df (y, t)/dt f (y, t). Therefore, G (y, T) G (y, T) LT L [g (y, t) g (y, t)] dt LT dg (y, t) dt dt L [g (y, L T) g (y, L)]. The above mmedately provdes a recursve equaton to evaluate G (y, T ) for y 0, where (9) provdes G (0, T ), g (y, L T) g (y, L) G (y) G (y ). Thus, expected holdng and backorder costs occur at an average rate G (y)/t over the nterval. Expected holdng and backorder costs per unt tme for product occur at rate E[G (S B, T)] T Q Pr(B j)g (S j, T). T j0 The dstrbuton functon of B s requred to evaluate the above. Those dstrbutons depend on the allocaton polcy, whch s the polcy for decdng whch products wll be shpped and whch wll not be shpped. A smple allocaton polcy s frst-come-frstserve: products are loaded nto trucks n the sequence n whch they are ordered. In that case B s bnomally dstrbuted wth success probablty and B draws Q Pr(B x) Pr(B j)pr(z ( j) x). (0) jx There are probably better allocaton polces than frst-come-frst-serve. Those polces would prortze the products based on the demand and cost characterstcs so that the needest products would be assured prorty n any shpment. Unfortunately, wth those polces the analytcal evaluaton of B s very cumbersome. Thus, f a more complex allocaton algorthm were used, ether (0) can be taken as an approxmaton or B could be evaluated va smulaton. ow t s possble to express the expected average cost of a (Q, S T ) polcy, (Q, S, T ), K T (Q, S, T) as E[G (S B, T)]. (Q)C The frst term s the expected transportaton cost, the second s the shelf-space cost, and the thrd term s the expected holdng and backorder costs. For fxed Q and T, t s straght-forward to fnd the optmal S because (Q, S, T )sconvexneachs. (ote that B s ndependent of S.) The optmal (Q, S T ) polcy s found va a search over the nterval Q [, C]. The optmal (S, T ) polcy s found va a search over the parameter T. Although an upper bound on the search nterval has not been developed, t s ntutve that the search can be termnated when T s substantally Vol. 3, o. 3, Summer

12 Managng a Retaler s Shelf Space, Inventory, and Transportaton greater than C/: Average total demand over an nterval of length C/ equals C, so there wll be several full trucks watng to be shpped at each revew epoch when T k C/. 5. Lower Bound The optmal polcy s not known for ths problem. evertheless, the polces descrbed n the prevous sectons are ntutvely reasonable and analytcally tractable. The objectve of ths secton s to determne how much better an optmal polcy could be relatve to those feasble polces. Ths s done by evaluatng a lower bound over all feasble polces. The retaler s problem s complex because the cost of orderng one product depends on the orderng decson of the other products: Because a truck costs K per delvery no matter the number of products delvered (up to the capacty of C unts), the cost of orderng a product may be hgh f the order trggers another delvery, or t may be low f the order merely flls space n an already commtted delvery. Of course, ths complcaton dsappears f there s only one product; a sngle-product retaler would be unusual. It also dsappears f trucks carry at most one product; n that case, product s orderng costs only depend on ts order quantty and not on the order quantty of the other products. The latter nsght s the foundaton for the lower bound proposed by Atkns and Iyogun (988): Each product s delvered wth ts own truck of capacty C and ncurs a delvery charge K, where. That s, ther bound decomposes the problem nto ndependent problems, and n each of those problems the optmal polcy s known. (It s an order pont, order-up-to polcy). The bound developed n ths secton allocates demand among the products nstead of the delvery cost. The basc dea s smple. Under actual operatons each customer demands precsely one unt from one product. Under demand allocaton each customer demands / from product, so each customer s total demand s stll one unt,. Hence, under actual operatons there are two components to demand uncertanty the tmng of customer arrvals and each customer s product choce whereas wth demand allocaton there s only component to demand uncertanty the tmng of customer arrvals. The next step s to show that the mnmum cost under demand allocaton s ndeed a lower bound for cost under actual operatons. Let IP () be product s nventory poston at tme. Assume the (unknown) optmal polcy s mplemented. Under that polcy t E[g (IP () D, L)] s the expected sum of holdng and backorder costs at tme L. Defne the followng cost functon g b (y) g ( y, L) (y y )(g ( y, L) g ( y, L)), where note that g (y, L) g b (y) for nteger values of y, otherwse g b (y) s a weghted average of g ( y, L) and g ( y, L). Because g b(y) s convex, t follows that t b t E[g (IP () D, L)] E[g (IP () D )]. () The latter term s the expected sum of holdng and backorder costs under demand allocaton. So costs under demand allocaton are never greater than optmal costs under actual operatons; the optmal polcy under demand allocaton s a lower bound. It remans to evaluate the optmal polcy under demand allocaton. Begn wth the problem of mnmzng product s nventory and shelf-space costs under demand allocaton,.e., when each customer demands unts of product. Furthermore, mpose the constrant that the average shpment quantty should be no less than q unts, where q s some nteger multple of unts. When a 0, a reorder pont polcy mnmzes the nventory cost subject to the shpment constrant because gb s convex. Let r (q ) be that optmal reorder pont, q / b r q / j r (q ) mn g (r j ). The problem s more complex when a 0. Shelf space s charged n unt ncrements, but because the product s nventory poston changes only n multples of unts, the product s maxmum nventory poston under the optmal polcy may be less than 222 Vol. 3, o. 3, Summer 200

13 Managng a Retaler s Shelf Space, Inventory, and Transportaton ts shelf space. Let s be the requred shelf space f a (r (q ), q ) reorder pont polcy s mplemented, s r (q ) q, where unused shelf space s possble, s r (q ) q. It s never optmal to have more than s unts of shelf space: the (r (q ), q ) reorder pont polcy would mnmze the nventory costs, so the extra shelf space would be wasted. Further, f s unts of shelf space are assgned, then the (r (q ), q ) reorder pont polcy s optmal: Increasng or decreasng the reorder pont, whle requrng s unts of shelf space, only ncreases the nventory cost. If fewer than s unts of shelf space are assgned, then the optmal polcy s stll a reorder pont polcy, but n ths case the maxmum nventory poston must equal the shelf space. 5 Hence, let r * (q ) be the optmal reorder pont gven q, s.t. q / b r*(q ) mn a r q g (r j ) q / r j r {q, q,...,s q, s q } ote that r q s never optmal because g b (y) de- creases lnearly for y 0. The optmal shelf space for product s s * (q ) r * (q ) q. ow consder the problem of mnmzng the sum of all costs under demand allocaton. For any polcy the average transportaton cost per unt s K/Q b, where Q b s the average shpment quantty, and the average transportaton cost per unt tme s (K/Q b ). Gven Q b, the optmal polcy must mnmze the sum of the product s nventory and shelf-space costs. That s acheved wth the reorder pont r * (q ), the shelf space s *(q ), and the order quantty q Q b for each product. Wth that polcy a truck s dspatched every Q b customer arrvals wth a total of Q b unts on board, q unts for product are ncluded on each shpment, and each truck dspatch rases each product s nventory poston to r * (q ) q,.e., the nventory postons of the products are synchronzed so 5 A reorder pont polcy s optmal because () f the maxmum nventory level were less than the shelf space, then the average nventory cost could be reduced by shftng the nventory level so that the shelf-space constrant bnds, and (2) because gb s convex, a reorder pont polcy mnmzes the nventory cost subject to the constrant that the average shpment quantty s no less than q. that they all reach ther reorder pont on the same customer arrvals. Because t s never optmal to delay the shpment of a full truck, the best Q b s found va search, K Qb argmn Q Q [,C] Q b Q j as*( Q) g [r*( Q) j ]. The above polcy s optmal under demand allocaton; ts expected cost s the sought after lower bound. In the multechelon nventory lterature, several lower bounds have been developed around the dea of free-nventory rebalancng: At any moment n tme, nventory can be moved nstantly from one locaton/ product to another wthout cost (see Federgruen and Zpkn 98b, 98c, 98d; Chen and Zheng 99). In effect, for each customer arrval the retaler s able to choose whch of the products the customer demands. Gven that ablty, to mnmze costs the retaler selects the product that has the least cost mpact. That bound could be evaluated n ths system, but n some settngs that bound s not as effectve as the demand allocaton bound. To explan, consder an extreme settng n whch L 0andp p j p, where p s very large. If K were suffcently low, the retaler would assgn no shelf space across all products and shp trucks wth only one unt. Each product s nventory poston would always equal zero (due to L 0 and the sngle unt shpments), and so average cost per unt tme would just be the average transportaton cost, K. ow suppose K s suffcently large so that t may be necessary to shp more than one product per truck. Because backorders are costly, the retaler must allocate some shelf space to store product. It s expensve to allocate one unt of shelf space across all products. Furthermore, t s unnecessary. Gven the ablty to choose whch product each customer demands, the retaler need assgn only one unt of shelf space to the product wth the lowest per unt holdng cost. Then, as customer arrvals occur the retaler always selects that product for ts customers: Only one unt of shelf space s requred to accommodate the two unt shpments. But, that strategy does not work f each customer demands a lttle bt Vol. 3, o. 3, Summer

14 Managng a Retaler s Shelf Space, Inventory, and Transportaton Table K/C Average Truck Utlzaton wth Optmal (Q, S) Polcy C/ L 0 C/ L % 56.0% 9.0% 37.0% 7.% 98.% 3.0% 78.6% 99.5%.0% 3.3% 6.8% 8.2% 3.8% 69.7% 20.3% 39.2% 7.6% % 57.6% 57.6% 8.0% 5.6% 56.2% 7.2% 52.8% 5.6% Table 2 Dstrbuton of the Rato of the Heurstc Cost to the Optmal (Q, S) Polcy Cost Polcy Mnmum Medan 90th Percentle 95th Percentle Maxmum Average Q-heurstc/S-optmal Q-optmal/S-heurstc Q-heurstc/S-heurstc ( ) of every product, because then each customer demand generates backorders for the products that have not been allocated shelf space. Hence, wth the demand allocaton bound t s necessary to allocate shelf space across all products; t yelds a better bound. 6. umercal Study Ths secton detals a numercal study that evaluates the polces developed n 3 and as well as the lower bound developed n the prevous secton. From all combnatons of the followng sets, 972 scenaros were constructed: h {} {, 6, 32} C {, } p {, 6, 32} {, 6, 32} K {C/, C, C} a {, } {/} L {0,, }. In all scenaros the products are dentcal (same mean demand, holdng and backorder cost rates). Truck capacty s chosen relatve to total system demand; when C, average total demand flls a truck each unt of tme, whereas average demand takes four tmes longer to fll a truck when C. The transportaton cost s defned as the mnmum possble transportaton cost per unt: Each unt ncurs a K/C {¼,, } transportaton cost f 00% utlzaton s mantaned. For each scenaro, the optmal (Q, S) polcy was evaluated. Table presents data on truck utlzaton, Q/C, wth the optmal (Q, S) polcy. As expected, truck utlzaton ncreases sharply wth the mnmum transportaton cost per unt K/C. As conjectured, truck utlzaton ncreases wth L and, and decreases as the product lne becomes more fragmented ( ncreases). However, the mpact of or s less sgnfcant than the mpact of ether K/C or L. Table 2 presents data on the performance of three heurstc (Q, S) polces, where the polces dffer on whch parameters are chosen by heurstc. The frst polcy uses the Q-heurstc but chooses the optmal shelf space,.e., Q Q and S S * (Q). That polcy provdes excellent performance relatve to the optmal (Q, S) polcy: Average cost across the scenaros s only 3.7% hgher than the optmal (Q, S) polcy cost, and for 95% of the scenaros that polcy s cost s wthn 7.% of the optmal cost. The second polcy uses the optmal Q and uses the S-heurstc to choose shelf space. Although medan performance of ths polcy s reasonable (wthn % of the optmal), there are a number of scenaros n whch performance s poor: Ten percent of scenaros have costs that are at least 8.9% hgher than optmal. The thrd polcy uses both heurstcs. That polcy yelds the worst performance: The medan cost ncrease over the optmal (Q, S) polcy s 5.%, but for 0% of the scenaros that polcy s cost s more than 52% hgher than optmal. To sum- 22 Vol. 3, o. 3, Summer 200

15 Managng a Retaler s Shelf Space, Inventory, and Transportaton Fgure Cost Functon wth a (Q, S) Polcy Fgure 2 A Scenaro n Whch the Q-Heurstc/S-Optmal Polcy Makes a Poor Choce ( 32, L, a, p 32, L 0, C 6, K 6) marze, the Q-heurstc provdes a good choce for Q, hence, t provdes retaler wth an easy way to check ts transportaton utlzaton, Q/C. However, the S- heurstc does not provde suffcently robust performance. Fgure dsplays the cost functon for one scenaro n whch the frst heurstc polcy (Q-heurstc/S-optmal) performs well ( 6, 6, a, p 32, L 0, C 6, K 6). It s clear from the fgure why the Q-heurstc s effectve: The nontransportaton cost s an approxmately lnear ncreasng functon. Even wth the frst heurstc polcy there are a few scenaros n whch performance s sgnfcantly worse than optmal: There are sx scenaros (out of 972) n whch the polcy s cost s more than 50% hgher than the optmal cost. Table 3 ndcates when the frst polcy performs well: long lead tme and narrow product lne (low ). Fgure 2 graphs expected cost for a scenaro n whch the frst heurstc polcy performs poorly. It s clear that expected cost s not convex n Q. The heurstc makes a poor choce because t fals to recognze the beneft of operatng wth very low transportaton utlzaton. Fgures 3 and reveal the senstvty of costs to the chosen Q. In Fgure 3 two cases are consdered: Q s set 25% above the optmal, Q mn{.25q*, C}, or Q s set 50% above the optmal, Q mn{.5q*, C}. Two scenaros are also consdered n Fgure : Q s set to 75% of the optmal, Q 0.75Q*, or Q s set to half of the optmal, Q 0.5Q*. In all cases Q s rounded to the nearest nteger, and the optmal shelf space s chosen gven Q. Scenaros are placed nto 0 groups, based on ther optmal polcy transportaton utlzaton. The fgures dsplay averages and maxmums for each group of scenaros. Each modfcaton of Q generally ncreases costs by less than 0%. However, there are some scenaros n whch a sgnfcant penalty can occur by ncreasng Q f the optmal transportaton utlzaton s qute low (say 25%). There may also be sgnfcant penaltes for choosng Q too low f the optmal transportaton utlzaton s qute hgh (say 95%). evertheless, the data ndcate that costs are relatvely nsenstve around the optmal, Q*, assumng the optmal shelf space s chosen wth the mplemented Q. Table provdes data on the performance of (Q, S T ) polces for the followng values of T/(C/) Table 3 The Rato of the Q-Heurstc Polcy Cost (wth Optmal Shelf-Space Choce) to the Optmal (Q, S) Polcy Cost L Average Maxmum Vol. 3, o. 3, Summer

16 Managng a Retaler s Shelf Space, Inventory, and Transportaton Fgure 3 Senstvty of Costs to Increases n Q, Assumng Optmal Shelf-Space Assgnment Fgure Senstvty of Costs to Decreases n Q, Assumng Optmal Shelf-Space Assgnment 226 Vol. 3, o. 3, Summer 200

17 Managng a Retaler s Shelf Space, Inventory, and Transportaton Table Rato of (Q, S T) Polcy Cost to Optmal (Q, S) Polcy Cost T/(C/) Mnmum 5% Average 95% Maxmum Table 5 T(C/) Average Rato of (Q, S T) Polcy Cost to Optmal (Q, S) Polcy Cost L {0., 0.2, 0.5,, 2}. 6 The T values are chosen relatve to the rato C/ because C/ s the tme requred for average total demand to fll a truck. The table ndcates that perodc revew polces may perform reasonably well for low values of T, but generally perform qute badly for large values of T. Thus, f a retaler chooses to operate wth a perodc shppng nterval durng whch mean demand approxmately equals one truck load, then that retaler s cost probably could be reduced substantally f t were to swtch to a contnuous revew shppng polcy. Accordng to Table 5, ths s partcularly true f the retaler has a small lead tme between ts warehouse and ts store; when the warehouse to store lead tme s short, each product s nventory cost s senstve to devatons about the deal nventory poston, so an ncrease n T s partcularly costly n that case because ncreasng T reduces the retaler s ablty to keep each product s nventory poston close to ts deal. Table 6 dsplays data on the performance of full servce perodc revew polces (Q ) when the retaler can choose T. Table 7 gves the optmal perod length 6 Frst-come, frst-serve allocaton s assumed. To evaluate the B dstrbuton functon, M s chosen such that the probablty of reachng state M s less than relatve to the average tme to fll a truck. These polces do qute well (even n the worse case) when there are long lead tmes and low transportaton costs. In contrast, ther performance deterorates sharply as L decreases. Hence, f the retaler has the capablty to make quck delveres between ts warehouse and ts store, t should be wary of operatng a full servce perodc revew polcy. However, f the retaler beleves that the beneft of these polces (e.g., the operatonal smplcty of knowng that every order wll always be flled) outweghs the cost, the retaler should generally choose a perod length that s sgnfcantly smaller than the average tme to fll a truck (Table 7). Among the set of feasble polces consdered, the contnuous revew (Q, S) polcy clearly performs the best. Table 8 ndcates how that polcy performs relatve to the best lower bound. The gap between the best feasble polcy and the lower bound s qute small when the rato p/a s small. Indeed, n about 5% of the scenaros ( of them) the bound s tght, whch means that the (Q, S) polcy s n fact optmal. evertheless, the gap ncreases sgnfcantly as the p/ a rato ncreases. Backorder costs domnate when p/ a s large, and so, the optmal (Q, S) polcy wll tend to assgn a sgnfcant amount of shelf space to each product. It s possble that there exsts a better feasble polcy for managng that shelf space; however, t s also possble that the bound s poor n those scenaros. Addtonal research s needed to resolve that ssue. (Incdentally, for all of the scenaros tested, the demand allocaton bound provded a better bound than ether the setup cost allocaton bound or the nventory rebalancng bound.) 7. Concluson Ths research studed the management of transportaton, shelf space, and nventory costs for a retaler that sells multple products wth stochastc demand. Three operatng polces were compared. The contnuous revew, mnmum quantty polcy, or (Q, S) polcy, performed better than the two perodc revew polces. It even compared well aganst a lower bound developed for ths model. However, the advantage of the (Q, S) polcy over the perodc revew polces should be tempered by the addtonal mplementaton Vol. 3, o. 3, Summer

18 Managng a Retaler s Shelf Space, Inventory, and Transportaton Table 6 K/C Rato of Full Servce Polcy Mnmum Cost to Optmal (Q, S) Polcy Mnmum Cost L 0 L Mn. Average Max. Mn. Average Max. L Mn. Average Max Table 7 Rato of Full Servce Polcy T to C/ L 0 L L K/C Mn. Average Max. Mn. Average Max. Mn. Average Max Table 8 Rato of Lower Bound Cost to Optmal (Q, S) Polcy Cost p/a Mnmum Average Maxmum L challenges of operatng wth contnuous (.e., real tme) nventory revew and truck dspatchng. Furthermore, the retaler should be aware that the advantage of the (Q, S) polcy depends strongly on the warehouse to store lead tme: The advantage s small when the lead tme s long, but the advantage grows sgnfcantly as the lead tme s reduced. Hence, f a retaler s able to reengneer ts supply chan so that ts warehouse to store lead tme s decreased, the retaler wll gan addtonal operatonal benefts by swtchng from perodc truck replenshment to contnuous revew truck replenshment. Even though the model studed has stochastc demand, the behavor of the model s remarkably lke the well-known determnstc demand economc order quantty (EOQ). In partcular, the retaler s cost s relatvely nsenstve to the optmal transportaton utlzaton: For example, f a retaler operates wth a transportaton utlzaton that s one and a half tmes greater than the optmal transportaton utlzaton, then (for the scenaros tested) the retaler s total expected cost s generally no more than 0% hgher than optmal. In addton, the EOQ structure leads to a smple, but very effectve, heurstc for choosng the retaler s transportaton utlzaton,.e., the Q n the (Q, S) polcy. In addton to the connecton to the EOQ model, there s a strong relatonshp between ths model and the multechelon nventory models wth multple retalers and stochastc demand. Indeed, the analyss of the (Q, S) polcy s exactly the same as the analyss of reorder pont polces n Axsäter (993). However, although the lower bounds developed for the multechelon nventory models could be appled to ths settng, a better bound for ths model was developed. That bound relaxes the constrant that each demand occurs only for one product,.e., n the demand allocaton lower bound each system demand s propor- 228 Vol. 3, o. 3, Summer 200

19 Managng a Retaler s Shelf Space, Inventory, and Transportaton tonally dvded among all of the products. Future research wll determne f the demand allocaton lower bound can mprove upon the current bounds for multechelon nventory models. References Adelman, D., A. Kleywegt Prce drected nventory routng. Workng paper, Unversty of Chcago, Chcago, IL. Agrawal,., S. A. Smth. 99. Estmatng negatve bnomal demand for retal nventory management wth lost sales. aval Res. Logstcs Anly, S., A. Federgruen. 99. Capactated two-stage mult-tem producton/nventory model wth jont setup costs. Oper. Res. 39 (3) Atkns, D., P. Iyogun Perodc versus can-order polces for coordnated mult-tem nventory systems. Management Sc. 3 (6) Axsäter, S Smple soluton procedures for a class of two-echelon nventory problems. Oper. Res. 38 () Exact and approxmate evaluaton of batch-orderng polces for two-level nventory systems. Oper. Res. () Balntfy, J. 96. On a basc class of mult-tem nventory problems. Management Sc Blumenfeld, D., L. Burns, J. Dltz, C. Dagano Analyzng tradeoffs between transportaton, nventory and producton costs on freght networks. Transportaton Res. 9B Bramel, J., D. Smch-Lev A locaton based heurstc for general routng problems. Oper. Res Cachon, G Managng supply chan demand varablty wth scheduled orderng polces. Management Sc. 5 (6) Chen, F., R. Samroengraja A staggered orderng polcy for one-warehouse mult-retaler systems. forthcomng Oper. Res., Order volatlty and supply chan costs. Workng paper, Columba Unversty, ew York., Y-S. Zheng. 99. Lower bounds for mult-echelon stochastc nventory systems. Management Sc. 0 () Federgruen, A., H. Groenevelt, H. Tjms. 98. Coordnated replenshments n a mult-tem nventory system wth compound Posson demands and constant lead tmes. Management Sc , Y. S. Zheng The jont replenshment problem wth general jont cost structures. Oper. Res , P. Zpkn. 98a. A combned vehcle routng and nventory allocaton problem. Oper. Res. 32 (5) ,. 98b. Approxmatons of dynamc, multlocaton producton and nventory problems. Management Sc ,. 98c. Computatonal ssues n an nfnte-horzon, multechelon nventory model. Oper. Res. 32 () ,. 98d. Allocaton polces and cost approxmaton for mult-locaton nventory systems. aval Res. Logstcs Fsher, M What s the rght supply chan for your product? Harvard Busness Rev. (Mar Apr) Gerchak, Y., Y. Wang. 99. Perodc revew nventory models wth nventory-level-dependent demand. aval Res. Logstcs Graves, S A multechelon nventory model wth fxed replenshment ntervals. Management Sc. 2 () 8. Guar, V., M. Fsher, A. Raman What explans superor retal performance. Presented at the 999 IFORMS Sprng meetng n Cncnnat, OH. Jackson, P., W. Maxwell, J. Muckstadt The jont replenshment problem wth power-of-two ntervals. IIE Transactons Mahajan, S., G. van Ryzn Retal nventores and consumer choce. Quanttatve Models for Supply Chan Management. S. Tayur, R. Ganeshan and M. Magazne, eds. Boston, Kluwer. McGavn, E., L. Schwarz, J. Ward Two-nterval nventory allocaton polces n a one-warehouse -dentcal retaler dstrbuton system. Management Sc. 39 (9) Pantumsncha, P A comparson of three jont orderng nventory polces. Decson Sc Pryor, K., R. Kapuscnsk, C. Whte A sngle tem nventory problem wth multple setup costs assgned to delvery vehcles. Workng paper, Unversty of Mchgan, Ann Arbor. Renman, M., R. Rubo, L. Wen Heavy traffc analyss of the dynamc stochastc nventory-routng problem. Transportaton Sc. 33 () Renberg, B., R. Planche Un modèle pour la geston smultanèe des n artcles d un stock. Revue Francase d Informatque et de Recherche Opératonnelle Slver, E. 98. Establshng reorder ponts n the (S, c, s) coordnated control system under compound Posson demand. Internatonal J. Producton Res Speranza, M. G., W. Ukovch. 99. Mnmzng transportaton and nventory costs for several products on a sngle lnk. Oper. Res. 2 (5) Vswanathan, S Perodc revew (s,s) polces for jont replenshment nventory systems. Management Sc. 3 (0) 7 53., K. Mathur Integratng routng and nventory decsons n one-warehouse mult-retaler multproduct dstrbuton systems. Management Sc. 3 (3) The consultng Senor Edtor for ths manuscrpt was Leroy B. Schwarz. Ths manuscrpt was receved on July 7, 999, and was wth the author 36 days for 2 revsons. The average revew cycle tme was 97 days. Vol. 3, o. 3, Summer