Outsourcing inventory management decisions in healthcare: Models and application

Transcription

1 European Journal of Operatonal Research 154 (24) O.R. Applcatons Outsourcng nventory management decsons n healthcare: Models and applcaton Lawrence Ncholson a, Asoo J. Vakhara b, *, S. Selcuk Erenguc b a Department of Management Studes, Unversty of West Indes, Mona Campus, Kngston, WI, USA b Department of Decson and Informaton Scences, Warrngton College of Busness Admnstraton, Unversty of Florda, 343 BUS, P.O. Box , Ganesvlle, FL , USA Receved 22 February 22; accepted 21 August 22 Abstract Tradtonally healthcare systems have pad lttle attenton to the management of nventores. However, wth the mplementaton of dagnostc related groups by the Unted States government (whch resulted n a pre-fxed level of compensaton for specfc medcal servces), these systems have turned ther attenton to cost contanment as a means of ncreased proftablty. Ths research addresses the ssue of managng nventory costs n a healthcare settng. The specfc problem addressed n ths paper s a comparson of nventory costs and servce levels of an n-house three-echelon dstrbuton network vs. an outsourced two-echelon dstrbuton network. In comparng nventory polces n both networks, we focus on non-crtcal nventory tems. Based on our analyss, we fnd that the recent trend of outsourcng to dstrbute non-crtcal medcal supples drectly to the hosptal departments usng them (.e., the twoechelon network) results not only n nventory cost savngs but also does not compromse the qualty of care as reflected n servce levels. Ó 23 Elsever B.V. All rghts reserved. Keywords: Health servces; Inventory; Supply chan management 1. Introducton Consder the followng scenaro: A major healthcare provder operated 7 hosptals wthn the state of Florda n the Unted States wth approxmately 2 patent departments wthn each hosptal. Non-crtcal nventory tems * Correspondng author. Tel.: ; fax: E-mal addresses: lnchols@uwmona.edu.jm (L. Ncholson), asoov@ufl.edu (A.J. Vakhara), selcuk.erenguc@cba.ufl.edu (S. Selcuk Erenguc) /$ - see front matter Ó 23 Elsever B.V. All rghts reserved. do:1.116/s (2)7-2

2 272 L. Ncholson et al. / European Journal of Operatonal Research 154(24) were receved by the central warehouse owned and operated by the provder and were then dstrbuted to each hosptal warehouse. In turn, these tems were suppled by the hosptal warehouses to departments n each hosptal. Each hosptal n the network dstrbuted stock only to departments wthn the partcular hosptal and thus, there was no stock transshpment between hosptals. A partnershp wth a major dstrbutor of medcal supples led to a restructurng of the dstrbuton network whch consoldated the central and hosptal warehousng functons nto a sngle Servce Center. The center would be operated by the dstrbutor and would drectly supply tems to ndvdual departments wthn each hosptal. Although ths restructurng effort could n the long-run reduce the suppler base, a key ssue of mmedate concern n ths converson was whether the healthcare provder realzed cost savngs and f so, dd these savngs come at the expense of lower customer servce? Ths scenaro s reflectve of a general trend of outsourcng specfc organzatonal actvtes to expert thrd-party provders. In general, the decson to outsource any organzatonal actvty s justfed by ensurng that ether: (a) the outsourcng agency can provde the product or servce more effcently than an nternal agency/department whle mantanng the same base level of the qualty of the product or servce; (b) the qualty of the product or servce provded by the outsourcer wll be hgher than the qualty of the same product or servce delvered by the nternal agency/department whle mantanng the same level of effcency n provdng the product or servce (Lunn, 2); and/or (c) the suppler base would n the longrun be reduced leadng to a reducton n procurement costs whch could be passed on downstream to the end users. Obvously, f both effcency and qualty ncrease (n the short-run) and procurement costs are lower (n the long-run) due to outsourcng, then the organzaton tends to gan substantally by choosng to outsource mmedately. In healthcare, outsourcng of nventory decsons s ndcatve of current practces (Veral and Rosen, 21) and s prmarly drven by three factors. Frst, nventory nvestments n ths ndustry are substantal and are estmated to be between 1% and 18% of net revenues (Holmgren and Wentz, 1982; Jarett, 1998). For example, for the quarter endng March 31, 21, IASIS Healthcare generated net revenues of $232,619, and ts nventory nvestment was $23,354, (1.4% of net revenues) whle Unversty Health Servces generated net revenues of $561,79, and ts nventory nvestment was $89,79, (15.97% of net revenues). Thus, any cost savngs whch can be generated through a more effcent management of nventores can lead to drect ncreases n proftablty. Second, as reported by L and Benton (1996), healthcare provders focus on qualty of servce both from an nternal and external perspectve. It has been argued that the move towards outsourcng of nventory can lead to mproved nternal performance as assessed though servce levels (Jarett, 1998). Further, mprovng nternal servce levels also postvely mpacts patent care and ths, n turn, ths should be lead to ncreases n the external measures of customer satsfacton and customer perceptons of servce qualty. Thrd and fnally, there has been an ncrease n the expertze and n the number of thrd-party provders whch offer nventory management servces n healthcare. These provders (whch nclude large management consultng frms wth healthcare specfc servces) have over a short-perod of tme accumulated substantve anectodal evdence of ther successes n stockless nventory management through outsourcng (see, for example, Rvard-Royer et al., 22). These success stores are also another key drver of the current trend of outsourcng nventory management n healthcare settngs. One mssng aspect from the accumulated anecdotal evdence n healthcare s whether the converson of an n-house three-echelon dstrbuton network to an outsourced two-echelon dstrbuton network actually results n cost savngs and the possble mpact on qualty (assessed n terms of servce levels for departments wthn hosptals). Ths forms the bass of our nvestgaton n comparng nventory costs and servce levels

3 L. Ncholson et al. / European Journal of Operatonal Research 154(24) across two dstrbuton networks managng non-crtcal supples. 1 The specfc focus of ths research s twofold. To develop normatve tools/methodologes through whch we can examne the non-crtcal nventory tem decsons wthn each type of network. Ths would facltate a comparson of total non-crtcal tem related nventory costs across both networks under a range of scenaros. Usng these models and data for one healthcare provder, we were nterested n estmatng the potental savngs (f any) and related mpact on servce levels that would result by swtchng from the three-echelon network to a two-echelon network. The remander of ths paper s organzed as follows. In the next secton, we revew the relevant lterature on nventory management n healthcare and on mult-echelon nventory systems. The general models developed to analyze nventory polces under each scenaro are descrbed next followed by a dscusson of how each of these models was operatonalzed n ths research. Ths s followed by a comparson of the networks n terms of servce levels and total nventory costs, and fnally, we conclude the paper wth mplcatons and conclusons of our research. 2. Healthcare nventory management: A lterature revew The management and dstrbuton of nventory of all knds among and wthn hosptals s usually dscussed under the broad headng of materal management. One of the dstnct features of materal management n a hosptal s the use of a perodc revew par level (or order-up to level) servcng approach. For the threeechelon dstrbuton network, ths approach would be mplemented as follows. The nventory poston at each department s revewed at the begnnng of a revew nterval and the dfference between the departmentõs par level 2 and nventory n hand s ordered from the hosptal. Orders from each department are transmtted to the hosptal nstantaneously and they are executed as receved. After fulfllng these orders, each hosptal, n turn, revews the dfference between the par level and nventory n hand and places an order wth the central warehouse. Once agan, these orders are transmtted nstantaneously and executed as receved. Fnally, the central warehouse also orders from external supplers, the dfference between the par level and nventory n hand, and supplers replensh central warehouse stocks by the end of the revew perod. The only dfference n ths approach for the two-echelon network (Scenaro B) s that the hosptal s bypassed n the process (.e., departments communcate drectly wth the servce center). One key aspect of such a polcy s the settng of the revew nterval and practcal consderatons lead to ths nterval beng dentcal across echelons. A major ssue n settng par levels for varous tems n a healthcare settng s that these levels tend to reflect the desred nventory levels of the patent caregvers rather than the actual nventory levels needed n a department over a certan perod (see Prashant, 1991). In most cases these par levels are experence-based and poltcally drven, rather than data-drven. Ths poses a problem for warehouse managers snce the 1 In a healthcare settng, crtcal supples whch consttute a small number of tems are typcally extremely expensve (on a per unt bass), have a short shelf-lfe, and/or requre expensve storage facltes on ste (e.g., njectable medcal supples, pharmaceutcal supples, and surgcal supples). On the other hand, all the other multtude of supples are consdered as beng non-crtcal (e.g., tubng, suture sets, latex examnaton gloves, and plastc/dsposable sheetng). Whle the majorty of the total nventory nvestment s n crtcal supples (around 6%), the large number of non-crtcal supples typcally accounts for the remanng nventory nvestment for these networks. Hence, gven the nventory nvestment numbers quoted earler, non-crtcal supples account for about $9,341,6 for IASIS and $35,883,6 for Unversty Health Servces. 2 The healthcare ndustry uses par level n leu of order up to level and thus, ths term wll be used throughout the remander of ths paper.

4 274 L. Ncholson et al. / European Journal of Operatonal Research 154(24) nventory they hold s typcally based on aggregate hosptal demands whle requrements of departments when aggregated are not n lne wth such estmates. The lterature analyzng the settng of optmal par levels and revew perods for multple echelons draws upon pror work n mult-echelon nventory systems whch s dscussed next. Allen (1958) was one of the frst researchers to analyze mult-echelon dstrbuton systems. He consdered the problem of determnng an optmal redstrbuton of the stock among K locatons. There have been a number of extensons to AllenÕs model (see Smpson, 1959; Krshnan and Rao, 1965; Das, 1975; Hoadley and Heyman, 1977). Clark and ScarfÕs (1963) study represents the frst attempt to formulate and characterze an optmal polcy n a mult-perod, mult-echelon, nventory/dstrbuton model that nvolves uncertan demand. Addtonal work by Schwarz (1981) and others (e.g., Deuermeyer and Schwarz, 1981; Eppen and Schrage, 1981; Nahmas and Smth, 1994) have produced a large set of models that generally seek to dentfy optmal lot szes and safety stocks n a mult-echelon framework. A number of researchers have analyzed an arborescent dstrbuton system wth no stock at the warehouse. Thus, the warehouse acts as a break-bulk faclty by orderng goods n bulk and upon recept, breakng these quanttes nto smaller unts to shp to retalers (see Slver et al., 1998). The queston of whether or not a warehouse should keep stock extends to one of centralzaton vs. decentralzaton of stocks. More complexty s nvolved n the analyss when the warehouse can hold stock, snce the number of decson varables ncreases. These varables nclude the amount the warehouse should order from ts suppler, the amount to shp from the warehouse to the retalers each perod, and the amount to allocate to each retaler n case of shortage. The most relevant modelng studes n the context of ths research are those of Snha and Matta (1991) and Rogers and Tsubaktan (1991). Both studes analyzed a two-echelon nventory system under stochastc demand wth fxed lead tmes and a perodc revew par level servce system as descrbed above. In Rogers and Tsubaktan (1991), the focus s on fndng the optmal par levels for the lower echelons to mnmze penalty costs subject to the maxmum nventory nvestment across all lower echelons beng constraned by a budgeted value. They show that the optmal par levels are determned by a crtcal rato (for the newsboy model) adjusted by the Lagrange multpler related to the budget constrant. Snha and Matta (1991) analyze a mult-product system where they focus on mnmzng holdng costs at both echelon levels plus penalty costs at the lower echelon level. Ther results are that the par levels at the lower echelon level s determned by the crtcal rato whle the par level for the upper echelon s determned by a search of the holdng cost functon at that level. In order to adapt these modelng approaches to our settng, we needed to make several sgnfcant changes as follows: To model the three-echelon network (Scenaro A), we extend the two-echelon models of Snha and Matta (1991) and Rogers and Tsubaktan (1991). Essentally, when we model Scenaro A, our model dffers from those proposed n these studes snce t ncorporates costs and par levels for an extra echelon. In modelng Scenaro B (the two echelon network), we also needed to ncorporate the fact that over and above the backorder cost, the servce center was also gong to charge departments an addtonal penalty cost f the backorder exceeded a certan fxed amount. Ths led us to ncorporate a set of addtonal 1 decson varables n our model to capture such effects. Consequently, our modelng effort for ths scenaro whle relyng on the basc models of Rogers and Tsubaktan (1991) and Snha and Matta (1991) also dffers sgnfcantly n the context of these bnary varables to capture the realtes of the healthcare system beng modeled. Fnally, gven the par level servcng polcy descrbed earler, note that orders for each department wthn a hosptal are fulflled through hosptal stocks pror to supples beng receved by the hosptal from the central warehouse. Smlarly, the central warehouse fulflls all hosptal orders pror to supples beng receved from external dstrbutors. Ths led us to ncorporate the fact that each of the hgher echelons

5 L. Ncholson et al. / European Journal of Operatonal Research 154(24) needed to set par levels such that the nventores were at least as large as the expected aggregate orders whch wll be placed by all the echelons drectly below. Ths s another crtcal dfference n our modelng effort as compared to pror work. In the next secton, we present the detals of the two models developed, one for each network scenaro. 3. Model development 3.1. Assumptons Our nvestgaton of the two healthcare networks assumes that each s represented as an arborescent mult-echelon dstrbuton system. Essentally, each system s represented as one where a lower echelon receves tems from only one source at a hgher echelon. In lne wth current practces, under both scenaros no transshpments among departments or among hosptals are allowed. Based on the par level orderng polcy, we also assume a perodc revew par level nventory management system for each echelon. Each model attempts to capture the key realtes underlyng the scenaros beng nvestgated. However, there are also certan assumptons we make n order to faclate the modelng effort. These assumptons and the underlyng ratonale behnd each s dscussed next. We focus on developng sngle non-crtcal tem nventory models for each network (see Footnote 2 for estmated nvestment levels and examples of non-crtcal tems). 3 Although each department wthn a hosptal does stock a large number of these types of tems, we observed that n general, demand for ndvdual tems was ndependent of the demand for other tems (dscussed below). If ths assumpton does not hold n other stuatons, the models could stll apply provded tems wthn a department could be grouped such that wthn each group tem demands are hghly correlated whle between groups, tem demand correlaton s low. In such a case, the nventory decsons for each group could be separately analyzed. Demand for the sngle tem s assumed to be ndependent across departments wthn a hosptal. The assumpton of ndependent demands at the department levels s reasonable snce a large number of dentcal non-crtcal tems are stocked wthn each department n a hosptal. The requred stocks of these tems are prmarly used to satsfy department specfc patent demands whch are usually ndependent of one another. Although n exceptonal stuatons, our assumpton may not hold (e.g., n tmes of certan epdemcs, the same tem demand may be hgher at all locatons), we feel t s reasonable n general. The relevant costs ncorporated n our models focus on the tradtonal nventory holdng and backorder costs at each echelon. Thus, we explctly assume no tem prce dfferentals across echelon scenaros. In practce, ths may not be the case, snce the outsourcer may also be n a poston to obtan smaller prces gven ts procurement volume. Further, the outsourcer could, n the long-run, reduce the total suppler base and fxed cost savngs due to such a reducton could also be passed on to the end users. However, to carry out a far and unbased comparson of the two alternatve networks, we felt that t was mportant to remove ths external effect. Gven that we focus prmarly on nventory related costs, there s another ssue that needs to be addressed. In the three-echelon network (Scenaro A), the healthcare system ncurred nventory costs at 3 The prmary dfference between management of nventores for crtcal vs. non-crtcal tems was that, as would be expected, the healthcare provders set substantally hgher servce levels for these tems (e.g., a 99% servce level would be requred for a crtcal tem as compared to a 9% servce levels for a non-crtcal tem). A secondary dfference was that transhpments of crtcal supples between departments wthn a hosptal occurred qute frequently whle for non-crtcal supples, ths was not the case.

6 276 L. Ncholson et al. / European Journal of Operatonal Research 154(24) all levels whle n the two-echelon network, the outsourcer was responsble for costs ncurred at the servce center and the nventory costs for the departments would be the responsblty of the healthcare system. Thus, when comparng the costs across networks, the healthcare system would probably ncur lower total costs. However, ths gnores the fact that the healthcare system would need to also remburse the servce provder (under Scenaro B) and such a rembursement would be based on the servce center nventory costs. Thus, when developng models for each network scenaro, we nclude all nventory related costs at all echelons. In lne wth current practce n the healthcare ndustry, the models we develop are sngle revew perod models, and we assume that all echelons have the same revew perod (Veral and Rosen, 21). We also assume that replenshment lead-tmes (ncludng transportaton tmes) are zero (or small enough n relaton to the revew perod). The routne nature of delvery, from suppler to warehouse, from warehouse to hosptals, and from hosptals to departments makes ths a vald assumpton. Consstent wth current practces n healthcare, we assume that all backorders are satsfed by emergency delveres at a cost hgher than normal delvery cost. We also assume that outsde supplers hold suffcent stock to satsfy all demand from the warehouse (servce center). In settng par levels for each department, we follow a polcy typcally establshed n the healthcare settng. Essentally, ths polcy requres each department to meet a mnmum servce level (n terms of the mnmum fracton of demand whch should be satsfed). Thus, department servce levels (and n turn par levels) are establshed such that a pre-specfed mnmum servce level s mantaned for each department. Based on these assumptons, we now proceed to descrbe each model developed to compare the two network structures. A comprehensve lst of notaton used n presentng both models s gven n Table Models Based on the assumptons stated earler, each healthcare network s analyzed usng a sngle-tem, sngle revew perod nventory model. These models are developed based on the assumpton that the underlyng revew perod demand dstrbuton for each department s known. In general, we assume that f f ðþ represents the pdf of revew perod demand then, f ðþ ¼ f <. Let l d represent the average revew perod demand wth assocated standard devaton r d for department j n hosptal. The models for each scenaro are presented next Model for Scenaro A In ths model, the decson varables are the par levels for each department (S da ), for each hosptal (S h) and the central warehouse (S w ). In determnng the department par levels, we also ensure that a mnmum pre-specfed servce level d (whch represents the proporton of demand whch s satsfed through nventory) for each department j n hosptal s mantaned. Thus, the par level S da for each department j n hosptal s determned such that tõs assocated servce level (b A ) s at least as large as the mnmum prespecfed servce level d. Based on ths, the model s presented below. Z Sw ( " Z #) Mnmze EC A ðs w ; S h ; SdA Þ¼H w ðs w x w Þf ðx w Þdx w þ XN S h H h ðs h x Þf ðx Þdx ¼1 ( " þ XN X Z #) n S da x Þf ðx Þdx ¼1 j¼1 " þ XN X n P A ¼1 j¼1 H da Z 1 ðx S da ðs da S da Þf ðx Þdx # ð1þ

7 L. Ncholson et al. / European Journal of Operatonal Research 154(24) Table 1 Notaton Indces wðcþ Index for central warehouse (servce center) Index for hosptals ( ¼ 1;...; N) j Index for departments (j ¼ 1;...; n ) Parameters x x x w ðx c Þ l d l h l w l c r d r h r w r c d H da H db H h H w H c P A P B P B a Decson varables S da S db b da b db S h S w S c y Revew perod demand for department j n hosptal Revew perod demand for hosptal Revew perod demand for central warehouse (servce center) Mean revew perod demand for department j n hosptal Mean revew perod demand for hosptal Mean revew perod demand for central warehouse Mean revew perod demand for servce center Standard devaton of revew perod demand for department j n hosptal Standard devaton of revew perod demand for hosptal Standard devaton of revew perod demand for central warehouse Standard devaton of revew perod demand for servce center Requred mnmum servce level for department j n hosptal Unt holdng cost for department j n hosptal under Scenaro A Unt holdng cost for department j n hosptal under Scenaro B Unt holdng cost for hosptal under Scenaro A Unt holdng cost for central warehouse under Scenaro A Unt holdng cost for servce center under Scenaro B Unt penalty (backorder)cost for hosptal under Scenaro A Unt penalty (backorder)cost for hosptal under Scenaro B Unt emergency delvery cost for backorders under Scenaro B Maxmum backorders (unt) allowed for hosptal before ncurrng the cost P B under Scenaro B Par level for department j n hosptal under Scenaro A Par level for department j n hosptal under Scenaro B Servce level (correspondng to S da ) for department j n hosptal under Scenaro A Servce level (correspondng to S db ) for department j n hosptal under Scenaro B Par level for hosptal under Scenaro A Par level for central warehouse under Scenaro A Par level for servce center under Scenaro B 1 f backorders under Scenaro B do not exceed a ; otherwse subject to: Z 1 S da ðx S da Þf ðx Þdx ð1 b A Þld 6 8 and 8j; ð2þ b A P d 8 and 8j; ð3þ " Z # S h Xn S da S da ðs da x Þf ðx Þdx P 8; ð4þ j¼1 S w XN ¼1 " S h Z # S h ðs h x Þf ðx Þdx P ; ð5þ S w ; S h ; SdA P 8 and 8j: ð6þ

8 278 L. Ncholson et al. / European Journal of Operatonal Research 154(24) The objectve functon sums up the holdng costs based on expected nventory levels at all three echelons (central warehouse, hosptal warehouses, and departments wthn each hosptal) and also assesses a penalty cost for expected backorders for departments (see Eq. (1)). Constrant set (2) s the relatonshp between the expected demand satsfed from nventory for department j n hosptal based on servce level b A. Obvously b A s constraned to be at least as large as d whch s the pre-specfed mnmum servce level that should be mantaned for the department (see constrant set (3)). Constrants (4) and (5) capture the realtes of the three-echelon network for the healthcare system studed n ths paper. Recall that orders are communcated by the departments at the begnnng of the revew nterval and are executed as receved by the hosptal warehouse, and subsequently orders are communcated by hosptal warehouses at the begnnng of the revew nterval and executed as receved by the central warehouse. Based on ths, each hosptal warehouse needs to mantan adequate nventores (as determned by the par levels S h ) so as to satsfy the expected orders for all departments t supples and the central warehouse needs to mantan nventores (reflected n the par level S w ) n order to satsfy the expected orders for all hosptal warehouses. Ths feature s ncorporated n ths set of constrants. 4 Fnally, the non-negatvty constrants on the decson varables are ncorporated n constrant sets (6) Model for Scenaro B In ths model, the decson varables are the par levels for each department (S db ), and the servce center (S c ). An addtonal set of decson varables for handlng a context specfc ssue are also ncluded n modelng the problem under ths scenaro. The outsourcer operatng the servce center negotated wth the healthcare provder that f total emergency delveres to a hosptal (reflected n total backorders) exceeded a constant a unts t would charge a hgher penalty cost per unt of P B whle backorders wthn ths lmt would ncur a penalty cost of P B. In the model descrbed below, the decson varables y enable us to ncorporate ths ssue. As wth the prevous models, the department par levels are determned to ensure that a mnmum pre-specfed servce level d (whch represents the proporton of demand whch s satsfed through nventory) for each department j n hosptal s mantaned. Thus, the par level S db for each department j n hosptal s determned such that ts assocated servce level (b B ) s at least as large as the mnmum pre-specfed servce level d. Based on ths, the model s presented below. Z Sc ( " Mnmze EC B ðs c ; S db ; y Þ¼H c ðs c x c Þf ðx c Þdx c þ XN X Z #) n S db H db ðs db x Þf ðx Þdx subject to: Z 1 S db þ XN ¼1 þ XN ¼1 ð1 y Þ X n y P B j¼1 ( ðp B a Þ þ P B Z 1 S db " ¼1 j¼1 Z Xn 1 ðx j¼1 S db ðx S db Þf ðx Þdx S db Þf ðx Þdx a #) ðx S db Þf ðx Þdx ð1 b B Þld 6 8 and 8j; ð8þ b B P d 8 and 8j; ð9þ ð7þ 4 If we do not nclude these constrants n our model, then we can obtan an optmal soluton to our problem qute easly as follows. Use a newsboy type analyss at each department to determne the servce levels b A and assocated par level SdA. Based on these par levels, determne the approprate hosptal warehouse par levels S h and fnally use these to determne the central warehouse par level S w.

9 S c XN ( ( X n j¼1 X n j¼1 ¼1 X n j¼1 " Z 1 ðx S db Z 1 ðx S db L. Ncholson et al. / European Journal of Operatonal Research 154(24) S db Z # S db ðs db x Þf ðx Þdx P ; ð1þ S db Þf ðx Þdx a ) 6 ð1 y ÞM 8; ð11þ S db Þf ðx Þdx a )ð1 y Þ P 8; ð12þ y ¼f; 1g 8; ð13þ S c ; S db P 8 and 8j: ð14þ The objectve functon sums up the holdng costs based on expected nventory levels at two echelons (servce center and departments wthn each hosptal) and also assesses a penalty cost for expected backorders for all departments wthn each hosptal (see Eq. (7)). If y ¼, then total backorders for all departments wthn a hosptal exceed a, and hence the penalty costs P B (to the frst a unts) and P (for the excess unts) are charged whle f y ¼ 1, then only the penalty costs P B are charged per unt. As wth the earler model, constrant set (8) s the relatonshp between the expected demand satsfed from nventory for department j n hosptal based on servce level b B. Obvously bb s constraned to be at least as large as d whch s the pre-specfed mnmum servce level that should be mantaned for the department (see constrant set (9)). Constrant (1) captures the fact that servce center nventores must be adequate to satsfy the expected orders for all departments n all hosptals. Constrant sets (11) and (12) force y to take on the approprate value of or 1 and M s a suffcently large constant. Fnally, the bnary and nonnegatvty constrants on the decson varables are ncorporated n constrant sets (13) and (14). The models presented n ths secton apply to any general dstrbuton of revew perod demand. In the next secton, we descrbe: (a) how each of these models was operatonalzed based on emprcal data for the healthcare provder; and (b) procedures developed n order to obtan good solutons for each model. 4. Model operatonalzaton and soluton methods The models developed n the prevous secton are general n the sense that they do not assume any specfc underlyng demand dstrbuton. After collectng demand data for several non-crtcal routnely used tems n the healthcare network, we frst determned that the normal dstrbuton could be used as a close approxmaton to represent the underlyng demand process (done through a Q Q plot analyss). Although such a dstrbuton does have a postve probablty assocated wth negatve revew perod demand, such a possblty can be mnmzed f the observed coeffcent of demand s relatvely small. In our case, the largest observed value for ths coeffcent was.4 and ths effectvely elmnated the practcal possblty of a negatve revew perod demand (essentally ths translates to a probablty of negatve revew perod demand of.62 or.62%). Further, the normal dstrbuton has been used extensvely n pror research on nventory models to model the underlyng demand process (Slver et al., 1998). The exact specfcaton of both models assumng a normal dstrbuton of revew perod demand s descrbed next. Under the assumpton of ndependent revew perod demand for each department n a hosptal, we frst observe that the followng relatonshps hold:

10 28 L. Ncholson et al. / European Journal of Operatonal Research 154(24) Average revew perod demand at each hosptal (l h ) s the sum of total average revew perod demands wthn each department n the hosptal. Thus: l h ¼ Xn j¼1 l d : 2. The standard devaton of average revew perod demand for a hosptal (r h vffffffffffffffffffffffffff )s ux n ¼ t : r h r d j¼1 3. The average revew perod demand at the central warehouse (l w ) s the sum of total average revew perod demands for all hosptals. Thus: l w ¼ XN ¼1 l h : 4. The standard devaton of average revew perod demand for the central warehouse (r w )s vffffffffffffffffffffffff ux N r w ¼ t : ¼1 r h 5. For the two echelon scenaro, the servce center average revew perod demand (l c ) and the assocated standard devaton (r c ) s as follows: l c ¼ XN ¼1 X n j¼1 l d ; vffffffffffffffffffffffffffffffffffffffffff ux N X n r c ¼ t : ¼1 j¼1 r d Based on ths, f we assume that the revew perod demand for each department s normally dstrbuted wth parameters l d and rd, then we also know that: (a) aggregate revew perod demand for each hosptal s normally dstrbuted wth parameters l h and r h ; (b) aggregate revew perod demand for the central warehouse s normally dstrbuted wth parameters l w and r w ; and (c) aggregate revew perod demand for the servce center s normally dstrbuted wth parameters l c and r c. Then followng the analyss for the newsboy model under normally dstrbuted demand (Slver et al., 1998), we note that Z S w Z S h Z S da ðs w x w Þf ðx w Þdx w ¼ r w ½Z w þ GðZ w ÞŠ; ðs h x Þf ðx Þdx ¼ r h ½Zh þ GðZ h ÞŠ; ðs da x Þf ðx Þdx ¼ r d ½ZdA þ GðZ da ÞŠ;

11 L. Ncholson et al. / European Journal of Operatonal Research 154(24) Z 1 S da Z S c Z S db Z 1 S db ðx S da Þf ðx Þdx ¼ r d GðZdA Þ; ðs c x c Þf ðx c Þdx c ¼ r c ½Z c þ GðZ c ÞŠ; ðs db x Þf ðx Þdx ¼ r d ½ZdB ðx S db Þf ðx Þdx ¼ r d GðZdB Þ; þ GðZ db ÞŠ; b A ¼ F ðzda Þ; b B ¼ F ðzdb Þ; where F ðþ represents the cdf of revew perod demand and GðÞ s the unt normal loss functon. The two models developed n the pror secton are operatonalzed by substtutng these expressons. Obvously, the decson varables for these revsed models are the standard normal devates: (a) Z da, Z h, and Zw for Scenaro A; and (b) Z db,andzc for Scenaro B. Once estmates of these are obtaned, then the par levels can be set as follows: S da S db ¼ l d þ ZdA r d ; ¼ l d þ ZdB rd ; S h ¼ l h þ Z h rh ; S w ¼ l w þ Z w r w ; S c ¼ l c þ Z w r c : In both of these operatonalzatons, we are faced wth the challenge of mnmzng an objectve functon (convex n the decson varables) over a non-convex constrant set. Thus, we need to solve a reverse convex programmng problem whch has been shown to be NP-hard (Horst and Padalos, 1995). Further, there are no effectve procedures that have been developed for specfyng lower bounds to such programmng problems but upper boundng heurstc methods can of course, be developed. A frst approach to obtanng such an upper bound was through the use of LINGO where we solved each model separately wth the same parameter settngs. 5 The term solve s beng used loosely here snce we do not obtan optmal solutons to our models. Rather, we dentfed a local optmum by runnng LINGO wth a lmt on run tme. A second approach mplemented explots the newsboy type structure of both models. In ths case, we developed greedy heurstcs to obtan feasble solutons to each model and these are descrbed n the next secton Heurstc 1 (for Scenaro A) The motvaton for ths heurstc stems from the fact that based on a newsboy type analyss, the servce level for a department could be estmated to be the maxmum of the rato of penalty to holdng cost or the prespecfed mnmum servce level (d ). Based on these estmated departmental servce levels for each 5 The LINGO optmzaton software uses sequental lnear programmng (SLP) and generalzed reduced gradent methods as ts algorthm bass for non-lnear optmzaton, and the Branch and Bound procedure for nteger programmng. The prmary motvaton for usng ths software was that ts modelng language allows for quck, concse problem expresson and also, run-tme can be decreased f data nput utltes are maxmzed.

12 282 L. Ncholson et al. / European Journal of Operatonal Research 154(24) hosptal, we could determne the approprate servce levels for each hosptal whch would satsfy constrant (4). Fnally, movng up to the hghest echelon level (central warhouse), we could mpute the servce levels for the central warehouse based on these estmated hosptal servce levels to satsfy constrant (5). A formal statement of ths greedy procedure s outlned below and t also contans detals on how par levels are set based on estmated servce levels for each echelon. 1. For each department j n hosptal, let b A ¼ max P A ; d H da. Based on ths, determne Z A such that F ðz AÞ¼bA. Set the par level S da ¼ l d þ ZA rd. 2. Determne for each hosptal : nhp o Z h ¼ 1 n r h j¼1 ðld rd GðZd ÞÞ l h. Set the par level for hosptal as S ha ¼ l h þ Z hrh. 3. Determne for the central warehouse: nh PN Z w ¼ 1 r c ¼1 ðlh r h GðZh ÞÞ l o. w Set the par level for the central warehouse S c ¼ l w þ Z w r w. 4. Determne the expected total cost based on these par levels Heurstc 2 (for Scenaro B) The motvaton for developng ths heurstc s smlar to that descrbed above. However, note that n ths case, f we set department servce levels to be the maxmum of the rato of penalty to holdng cost or the prespecfed mnmum servce level (d ), we also need to be concerned wth the resultng expected total backorders whch may be greater than the allowable lmt a. The reason for ths s that any expected backorders greater than a ncur a hgher penalty cost P B. Thus, we modfy the departmental servce levels to ncorporate ths hgher addtonal penalty cost. Once we determne these departmental servce levels, then n lne wth the pror heurstc, we use these to mpute the servce center servce levels to satsfy constrant (1). A formal statement of ths greedy procedure s outlned below and, as wth the earler heurstc, t also contans detals on how par levels are set based on these estmated servce levels for each echelon. 1. For each department j n hosptal, let CR B ¼ P A. H db Based on ths, determne Q B such that F ðqb Þ¼CRB. Then, determne the expected backorder E B ¼ rd GðQB Þ where GðQB Þ s the unt loss normal functon. 2. For each hosptal : f P n j¼1 EB 6 a set b B ¼ maxfcrb ; d g; else determne P as P n P ¼ P B j¼1 Eb a! " P n P n þ P B j¼1 j¼1 Eb 1 Eb a!# P n j¼1 Eb and set ( ) b B ¼ max P ðp þ H db Þ ; d :

13 L. Ncholson et al. / European Journal of Operatonal Research 154(24) Table 2 LINGO vs. Heurstc 1 Scenaro A Problem sze Coeffcent of varaton a % Dfference b N ¼ 1; n ¼ N ¼ 1; n ¼ N ¼ 1; n ¼ N ¼ 5; n ¼ a The coeffcent of varaton s based on the department revew perod demand parameters (.e., r d =ld ). The percentage dfference s computed as 1 ðl UÞ=L where L represents the soluton obtaned usng LINGO whle U represents the soluton obtaned usng heurstcs 1 or 2. Table 3 LINGO vs. Heurstc 2 Scenaro B Problem sze Coeffcent of varaton a % Dfference b N ¼ 1; n ¼ N ¼ 1; n ¼ N ¼ 1; n ¼ N ¼ 5; n ¼ a The coeffcent of varaton s based on the department revew perod demand parameters (.e., r d =ld ). b The percentage dfference s computed as 1 ðl UÞ=L where L represents the soluton obtaned usng LINGO whle U represents the soluton obtaned usng heurstcs 1 or Determne the par levels for each department j n hosptal as follows. Frst, determne Z B such that F ðz BÞ¼bB. Set the par level S db ¼ l d þ ZB rd.

14 284 L. Ncholson et al. / European Journal of Operatonal Research 154(24) Determne for the servce center: nhp Z c ¼ 1 N P o n r c ¼1 j¼1 ðld rd GðZd ÞÞ l c. Set the par level for the central warehouse S c ¼ l c þ Z c r c. 5. Determne the expected total cost based on these par levels. Both heurstc procedures were mplemented n Mcrosoft Excel usng the bult-n functon for generatng the CDF for the normal dstrbuton. A comparson of the solutons obtaned usng LINGO wth those obtaned usng the heurstcs are shown n Tables 2 and 3. The percentage dfferences reflect the averages from varous combnatons of nventory holdng and penalty costs. For example, a coeffcent of varaton of 4% was examned under dfferent holdng cost ratos (e.g. H h da =H ¼ :57,.63,.7 for the model for Scenaro A and H c =H db ¼ :57,.63,.7 for the model for Scenaro B). Although the percentage dfferences reflect averages, t s clear that the solutons obtaned by LINGO domnate those obtaned usng the heurstcs. Further, we also observed that n all ndvdual cases the LINGO solutons domnated those obtaned usng the heurstc methods. Thus, all the remanng dscusson n ths paper s based on the use of LINGO for obtanng solutons to both models. 5. Results 5.1. Expermental detals In order to carry out the comparson of the n-house and outsourced networks, we collected/obtaned data on revew perod (a revew perod beng defned as a week) for multple tems experenced by a sngle department of one hosptal n the health-care system beng analyzed. Based on ths nformaton and n consultaton wth the relevant system personnel, we set the revew perod demand parameters as follows. Frst, for a department n a hosptal, we set the average demand to be an average of the actual demand for all tems. Then, we adjusted ths average demand by ther estmate of the average demand at each of the other departments. In ths manner, we generated the average demand for all departments n a sngle hosptal. For departments n another hosptal n the network, we adjusted the average demand for each unt by a constant fracton (whch agan was estmated based on nput from the network personnel). An example s gven below to clarfy ths process of settng average demand for each department n each hosptal n a gven dstrbuton network. Let us arbtrarly assume that we want to specfy average sngle tem demand for two hosptals each wth 1 departments and assume that the average actual demand for the data obtaned was 5 unts. Based on ths, the average demand for department 1 n hosptal 1 s set to 5. Average demand for each of the other departments n hosptal 1, s adjusted upwards/downwards by a constant. Thus, for example, departments 2 4 wll face an average demand of 4 (.e., 8% of 5); departments 5 7 wll face an average demand of 6 (.e., 12% of 5); whle departments 8 1 wll face an average demand of 7 (.e., 14% of 5). Note that the percentage constants (.e., 8%, 12% and 14%) are based on nputs from the hosptal personnel. Next, n order to specfy the average demands for the 1 departments n hosptal 2, we smply multply the demands for each correspondng department n hosptal 1 by a constant. If we assume a constant of 9% (ndvdual constants for each hosptal are estmated through dscussons wth the health-care network personnel), average demand for each department n hosptal 2, s smply 9% of the demand of the correspondng department n hosptal 1 (.e., department 1 n hosptal 2 wll face a demand of 4.5 (9% of 5); departments 2 4 n hosptal 2 wll face a demand of 3.6 (9% of 4); and so on). The second parameter of nterest related to average demand s the standard devaton/varance of demands for ndvdual tems n each department n each hosptal. Although, we could have estmated ths

15 L. Ncholson et al. / European Journal of Operatonal Research 154(24) parameter for our experments n the same manner as descrbed for the average demand, we found that ths parameter was not really dfferent across departments and hosptals but t vared more across tems. Thus, we choose to vary ths parameter n an expermental manner and the range of values explored s based on the varances n multple tem demand data that was collected/obtaned from the healthcare network. Gven the average demand for an tem descrbed above, we vared the standard devaton of demand such that the coeffcent of varaton levels were set at.4,.35, and.3. Once demand parameters for departments n each hosptal was set, we now descrbe how the cost and network sze parameters were set n order to facltate an analyss and evaluaton of the network structures beng compared. One of the key ssues to note n our expermentaton s that unt holdng costs were set to be equal across all departments whch led to equal unt holdng costs for each hosptal and equal unt backorder costs for each hosptal. Table 4 below summarzes the settngs used for our experments and a dscusson and ratonale for each settng follow. As s obvous from the Table 4, all cost parameters are ether an explct or mplct functon of the holdng cost per unt for a department n a hosptal (H da and H db shown n the table above, we frst arbtrarly set the value of H da departments n all hosptals. Then: ). Hence, to obtan the parameter settngs (and equal to H db ) to be a constant for all 1. Based on the coeffcent of varaton settng to be nvestgated, fx the standard devaton of demand for each department n each hosptal (see Table 4 for dfferent settngs nvestgated for ths parameter). 2. Fx the unt holdng cost for each department n each hosptal. For: the three-echelon network, fx the unt holdng cost for each hosptal (H h) and the warehouse (H w ) dependng upon the parameter settng to be nvestgated (see Table 4); and the two-echelon network, fx the unt holdng cost for the servce center (H c ) dependng upon the parameter settng to be nvestgated (see Table 4). 3. Set the unt backorder cost for each hosptal (P A ¼ P B ) dependng upon the parameter settng to be nvestgated (see Table 4). 4. Based on dscussons wth the outsourcer, the emergency unt delvery cost for Scenaro B (P B ) s set to be 12% of P B. 5. Run the experment usng LINGO and collect nformaton on the relevant performance measures. For each experment we carred out, we collected nformaton on the servce levels for each department n each hosptal, servce levels for each hosptal (for the three-echelon dstrbuton network Scenaro A), Table 4 Expermental parameters Parameter sets Ratos Parameter levels Holdng costs H h da =H.7,.63,.5 H w =H h.7,.63,.57 H c =H db.7,.63,.57,.5,.44 Penalty costs P A =H da 15.3, 23., 46. P B =H db 15.3, 23., 46. Coeffcent of varaton r d =ld.3,.35,.4 Number of departments n 8 1, 15, 2 Number of hosptals N 3, 5, 1

16 286 L. Ncholson et al. / European Journal of Operatonal Research 154(24) servce level for the warehouse/servce center, and total costs. Before dscussng the results n detal, three nput parameters were set as follows: The mnmum requred servce level for a department set at 9% (ths was ndcatve of current practce n the healthcare settng for non-crtcal nventory tems). The unt holdng cost for each department n each hosptal set at $1 (.e., H da ¼ H db ¼ $1 8 and 8j). Although we dd nvestgate dfferent startng values for the unt holdng cost for each department n each hosptal, we found that ths dd not change the resultng servce levels wthn departments, wthn hosptals, and at the warehouse/servce center. However, total costs were obvously greater when we ncreased the unt holdng cost for each department n each hosptal. When runnng the experments for Scenaro B, we set the value of a ¼ 15 unts 8. Thus, f expected backorders for all departments n a hosptal were greater than 15 unts, the emergency delvery penalty cost P B was charged. We now proceed to separately dscuss the results n terms of servce levels, total costs and a comparson of Scenaros A and B Results for servce levels For both scenaros, the average servce levels for departments are qute comparable and changes n these levels are prmarly due to changes n the rato of penalty costs to department holdng costs. As would be expected, when the rato of penalty cost to departmental holdng costs ncreases, average department servce levels also ncrease. More specfcally, when P A =H da ¼ P B =H db ¼ 15:3, average servce levels for departments are approxmately 93.75%, when P A =H da ¼ P B =H db ¼ 23, average servce levels for departments are approxmately 95.77%, and when P A =H da ¼ P B =H db ¼ 46, average servce levels for departments are approxmately 97.86%. Average servce levels for hosptals are, of course, only relevant for Scenaro A. In ths case, these servce levels ncrease as the rato of penalty costs to holdng costs ncreases and these are n the range %. The nterestng dfferences n servce levels that are notceable across scenaros are for the warehouse as compared to the servce center. In general, the servce center servce levels are always hgher than those of the warehouse (warehouse servce levels are n the range % whle servce center servce levels are n the range %). Thus, f we compare scenaros, ths ndcates that greater nventores are held at the servce center (Scenaro B) as compared to the warehouse (Scenaro A). Further, these servce levels are mpacted by the rato of Penalty to departmental holdng costs and the sze of the network (n terms of hosptals and departments for Scenaro A and the number of departments for Scenaro B). As earler, warehouse and servce center servce levels ncrease when the rato of penalty costs to average department holdng costs ncreases. Further, n lne wth pror research on mult-echelon nventory models, hgher servce levels for the warehouse and servce center are acheved when the sze of the network decreases snce the possblty of backorders decreases upstream n a smaller network as compared to a larger network Results for total costs After the dscusson related to servce levels, two results for total costs are obvous: For both scenaros, as the rato of penalty costs to holdng costs ncreases, there s an ncrease n aggregate total costs. These results are a drect functon of the fact that ths rato tends to ncreases servce levels (and hence, nventores) whch leads to an ncrease n total costs.

17 L. Ncholson et al. / European Journal of Operatonal Research 154(24) Expected Total Costs $ 25,. $ 2,. $ 15,. $ 1,. $ 5,. $. H=1 H=5 H=3 Number of Hosptals n=2 n=15 n=1 Fg. 1. Scenaro A effects of network sze. Expected Total Costs $ 25,. $ 2,. $ 15,. $ 1,. $ 5,. $. H=1 H=5 H=3 Number of Hosptals n=2 n=15 n=1 Fg. 2. Scenaro B effects of network sze. In lne wth the tradtonal sngle tem newsboy analyss, ncreases n the coeffcent of varaton of demand leads to ncreases n nventory levels at each echelon and ths, n turn, ncreases aggregate costs. In terms of network sze, the results are graphcally represented n Fgs. 1 and 2. The prmary reason for ths ncrease n costs due to an ncrease n network sze stems from the fact that when ncreasng the number of departments n a hosptal or the number of hosptals, there s an ncreased demand that each system must accommodate. Thus, f we compare a network of three hosptals wth 1 departments vs. a network of three hosptals wth 15 departments, the latter system experences hgher demand due to a larger number of departments. Although we could have compared costs by assumng the same demand spread over dfferent sze networks, ths was not carred out for two reasons. Frst, note that our focus s prmarly on comparng the mpact of the two generc network structures (the three-echelon vs. the two-echelon) rather than the mpact of demand beng spread over fewer or greater number of departments. Second, and more mportantly, we found that n practce, demand was typcally greater n larger hosptal networks and hence, our experments reflect ths scenaro Comparng Scenaro A vs. Scenaro B Gven that the pror results are n lne wth expectatons and thus, provde face valdty for our modelng effort, we now address the ssue of comparng the two alternatve network structures. Our analyss n ths context attempts to: (1) evaluate the cost savngs assocated wth swtchng from an n-house three echelon network to an outsourced two-echelon network; and (2) compare the servce levels for each department n these two scenaros. In carryng out such a comparson, we frst had to equalze the echelon holdng costs across scenaros. Two specfc holdng cost cases where the holdng costs across echelons were approxmately equal are:

18 288 L. Ncholson et al. / European Journal of Operatonal Research 154(24) Table 5 Expected total costs Scenaro A vs. Scenaro B Case Expected total costs Mnmum Maxmum Average Scenaro A Scenaro B Dfference Scenaro Scenaro Dfference Scenaro Scenaro Dfference A B A B ($) (%) ($) (%) ($) (%) The % dfference n costs s computed as 1ðEC A EC B Þ=EC A where EC A s the expected cost under Scenaro A and EC B s the expected cost under Scenaro B. Case 1. H h Case 2. H h =H da =H da ¼ :7 and H w =H h ¼ :7 and H w =H h ¼ :7 for Scenaro A vs. H c =H db ¼ :63 for Scenaro A vs. H c =H db ¼ :5 for Scenaro B; and ¼ :44 for Scenaro B. The results descrbed below are based on these two cases. For both these scenaros, Scenaro B always domnates Scenaro A n terms of total costs for any fxed network sze, a fxed coeffcent of varaton of demand, and a fxed penalty cost to department holdng cost rato. In terms of magntude of savngs, these are obvously dependent on the specfc holdng costs, penalty costs and emergency delvery costs for the system and a summary of cost savngs across all expermental parameters are shown n Table 5. As can be observed from Table 5, average savngs across both cases are approxmately $2 wth a range of $7 5 (wth percentage savngs rangng between 2% and 3%). Gven that our analyss s for a sngle nventory tem, obvously these savngs would be sgnfcantly larger for several non-crtcal tems whch are typcally stocked. Further, note that here we are comparng cost savngs n terms of total costs for both healthcare networks. If the servce center s actually operated by an outsourcer, then ths entty would be responsble for the costs of the servce center nventores at that stage. Thus, the advantages to the current healthcare system would be even larger and could be used to estmate how much the system would be wllng to pay to contract out nventory operatons to the thrd-party and/or n negotatons wth thrd-party provders. The surprsng result that also stems from our analyss s that related to the department servce levels snce Scenaro B provdes equvalent average department servce levels for departments as compared to Scenaro A. Further, the servce center servce levels for Scenaro B are always hgher than the warehouse servce levels and lower than the hosptal servce levels for Scenaro A. Ths ponts to the fact that the servce center holds smaller nventores than the total nventores held by the central warehouse and hosptals. 6 Analytcally from a mult-echelon nventory management perspectve, the result that the nventory costs of the two echelon system are lower than that of the three echelon system s n lne wth expectatons snce the former system elmnates an echelon level (and related nventores) n the analyss. However, what s unusual s that the hghest echelon servce levels are greater for the two echelon network as compared to the three echelon network. Thus, ths ponts to the fact that although end-user (.e., department) servce levels are equvalent between the two scenaros, we are also able to provde hgher customer servce at the hghest echelon level. 6 Lower total costs due to an outsourced hybrd stockless system are also reported n a case study by Rvard-Royer et al. (22). In addton to these cost reductons, they also found that outsourcng resulted n a sgnfcant reducton n full-tme equvalent labor hours requred by the healthcare provder.

19 L. Ncholson et al. / European Journal of Operatonal Research 154(24) Ths concludes our analyss of the two alternatve networks. As we have seen, the results ndcate that our modelng approaches have strong face valdty. Further, these approaches can also be used successfully to compare alternatve networks for managng nventores. 6. Conclusons and recommendatons Outsourcng the dstrbuton of non-crtcal tems n the health ndustry s always a vable alternatve. The network structure of Scenaro B s a typcal mode of outsourcng, where an outsde agent manages the recevng, nventory holdng, and dstrbuton components of the dstrbuton system. Based on observaton and reports from admnstratve healthcare network personnel, ths arrangement resulted n freeng up the hosptalsõ facltes and staff. The mplcaton for ths on a broader scale should allow those n the health sector to concentrate more on ther core responsbltes of beng health provders. Scenaro B s more or less an mplementaton of a centralzed dstrbuton system (a pull system, snce the lowest echelon dctates dstrbuton of materals). However, n order for such a centralzed system to be fully effectve, there needs to be a well-coordnated nformaton system to support the dstrbuton of tems. The problems encountered n gettng data on the dstrbuton of basc tems for ths research suggests that there s lttle coordnaton among the varous facltes. Our models are also approprate for other settngs where par level servcng polces are used to manage nventory nvestments (e.g., retalng). However, care should be taken n translatng our cost savngs n terms of other settngs snce these are a functon of the parameter settngs used n our paper. Both models have some shortcomngs, whch should help to form a platform for future research agendas n the area of dstrbuton systems wthn the healthcare ndustry. One of these shortcomngs s the assumpton of zero lead-tmes between echelons. Whle there are many stuatons, such as the one that s the focus of ths research that can ft wthn ths assumpton; there are many other stuatons that devate somewhat from ths assumpton. Thus, one extenson of ths research could be the ncluson of postve lead-tme between echelons, especally between the outsde suppler and the warehouse/servce center. Ths s not as straghtforward as t may seem on the surface, as the structure of the model mght have to be modfed to accommodate ths. A second ssue s to extend these models to ncorporate the fact that sngle tem demand may be correlated across departments wthn a hosptal or departments across hosptals. Ths would requre the formulaton of more complcated cost functons reflectng jont probablty densty functons of tem demands as well as constrants where the servce levels are also jontly determned. A thrd extenson of our models would reflect mult-tem demands and hence, lookng at jont replenshment polces for these tems. Two other more drect extensons of the model could be () to nvestgate the trade-off of havng a predetermned mnmum system-wde servce level compared to a predetermned mnmum servce level at the department level; and () to nvestgate the tradeoffs of basng the desred target nventory for the warehouse (or servce center) on ncremental echelon cost, as aganst nstallaton cost. The frst case would be a complex undertakng snce one would have to be concerned wth the jont probabltes among the echelons. Ths complexty would be ncreased as the number of hosptals (N) and the number of departments wthn each hosptal (n ) ncrease. The second case would requre fundamental changes n the formulaton to ensure that the order-up-to-level at each upper echelon becomes a functon of the echelon stock rather that the nstallaton stock. In addton, the expressons for endng nventory are desgned for nstallaton stock (Dks et al., 1996). In examnng a 2-echelon nventory system n a capactated productonnventory system, Rappold and Muckstadt (1998) showed that the desred target nventory for each dstrbuton center based on nstallaton costs wll be less than or equal the target nventory level based on echelon costs.

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