A Single-Product Inventory Model for Multiple Demand Classes 1

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1 A Sngle-Product Inventory Model for Multple Demand Classes Hasan Arslan, 2 Stephen C. Graves, 3 and Thomas Roemer 4 March 5, 2005 Abstract We consder a sngle-product nventory system that serves multple demand classes, whch dffer n ther shortage costs or servce level requrements. We assume a crtcal-level control polcy, and show the equvalence between ths nventory system and a seral nventory system. Based on ths equvalence, we develop a model for cost evaluaton and optmzaton, under the assumptons of Posson demand, determnstc replenshment lead-tme, and a contnuous revew (Q, R) polcy wth ratonng. We propose a computatonally-effcent heurstc and develop a bound on ts performance. We provde a numercal experment to show the effectveness of the heurstc and the value from a ratonng polcy. Fnally, we descrbe how to extend the model to permt servce tmes, and to embed wthn a mult-echelon settng. Ths research has been supported n part by the MIT Leaders for Manufacturng Program, a partnershp between MIT and global manufacturng frms; and by the Sngapore-MIT Allance, an engneerng educaton and research collaboraton among the atonal Unversty of Sngapore, anyang Technologcal Unversty, and MIT. 2 Sawyer School of Management, Suffolk Unversty, Boston MA, , harslan@suffolk.edu 3 Leaders for Manufacturng Program and A. P. Sloan School of Management, MIT, Cambrdge MA , sgraves@mt.edu 4 Leaders for Manufacturng Program and A. P. Sloan School of Management, MIT, Cambrdge MA , troemer@mt.edu

2 . Introducton and Lterature Revew In many nventory settngs a supply frm wshes to provde dfferent levels of servce to dfferent customers. For nstance, n a servce parts network, a customer can choose amongst dfferent contracts, each wth a dfferent cost and level of servce. A gold contract mght provde a 99% fll rate wthn twenty-four hours, whle a bronze contract promses a 85% fll rate wthn two days. In other settngs, a suppler segments ts customers based on the delvery channel or the prce they pay; the suppler recognzes some customers as deservng hgher prorty over other customers. In other cases, a suppler provdes prce dscounts for delvery flexblty, and then allows a customer to choose the delvery tme when placng an order. A common approach to such scenaros s to categorze the customers nto a fnte number of demand classes. Customers wthn a demand class receve the same level of servce. The nventory challenge s then to determne how to meet the servce level expectatons for each demand class wth the least amount of nventory. In ths paper we consder a sngle-tem nventory system wth stochastc demand and multple demand classes. The key assumptons are Posson demand, a determnstc lead-tme, a contnuous-revew (Q, R) replenshment polcy, and demand backorderng. As s common n the lterature, we assume a crtcal-level polcy for ratonng the nventory across the demand classes. The key contrbuton of ths paper s to show how to map ths problem nto a seral nventory system. Ths mappng facltates the characterzaton of the steady-state behavor of the nventory system. We then develop an approxmate soluton procedure to the so-called Servce Level Problem; that s, we want to fnd the crtcal-level polcy that meets specfed fll-rate targets for each demand class wth the least nventory. We show wth both bounds and a numercal experment that ths heurstc s qute robust and near optmal. We also show how to extend the model to permt servce tmes, whereby dfferent demand classes have dfferent servce tmes by whch ther demand s to be met. Fnally, we descrbe how to use the sngle-tem nventory system to characterze the nventores and backorders n a mult-echelon dstrbuton system. We have organzed the paper nto seven sectons. In the remander of ths secton, we dscuss the relevant lterature. In the followng secton, we present our assumptons and a general framework to descrbe how we manage the nventory wth a 2

3 statonary crtcal-level polcy. In secton three, we show how to map ths nventory system nto a seral nventory system. In secton four, based on ths mappng, we develop a model for cost evaluaton and optmzaton, under the assumptons of Posson demand, determnstc replenshment lead-tme, and a contnuous revew (Q, R) polcy wth ratonng. In secton fve, we pose the Servce Level Problem, n whch we mnmze the expected nventory whle satsfyng a servce level requrement for each demand class. Furthermore, we provde a heurstc soluton approach for the Servce Level Problem. In secton sx, we provde a numercal experment both to compare our proposed heurstc wth the optmal soluton and to show the value from ratonng. In the fnal secton we dscuss possble extensons and drectons for future research. Klen and Dekker (998) gve an overvew of nventory systems wth multple demand classes and provde examples of managng nventory wth multple demand classes, rangng from arlne servce companes to petrochemcal companes. In Table we provde a hgh-level categorzaton of the lterature. Lke much of the stochastc-demand nventory lterature, we can dvde the research based on the assumed control polcy, perodc or contnuous revew, and on assumed treatment of shortages, lost sales or backorders. In addton, some of the key developments are restrcted to or prmarly focused on two demand classes, whereas other work s not. Two demand classes demand classes Perodc-Revew, Perodc-Revew, Lost Sales Backorders Evans (968) Kaplan (969) Frank et al. (2003) Venott (965) Topks (968) Katrcoglu and Atkns (996) Contnuous-Revew, Lost Sales Melchors et al. (2000) Melchors (200) Dekker et al. (2000) Contnuous -Revew, Backorders ahmas and Demmy (98) Moon and Kang (998) Deshpande et al. (2003) Dekker et al. (998) Table : Inventory Lterature for Sngle-Product, Multple Demand Classes Venott (965) analyzes an nventory model wth several demand classes for a sngle product. He proposes to use crtcal nventory levels to raton the on-hand nventory among demand classes. Topks (968) subsequently analyzes the proposed crtcal-level polcy for a perodc-revew sngle-product nventory model wth multple demand classes. Kaplan (969) and Evans (968) study perodc-revew models wth only two 3

4 demand classes, smlar to Topks (968). Recently, Katrcoglu and Atkns (996) and Frank et al. (2003) analyze perodc-revew nventory systems wth multple stochastc demand classes. Katrcoglu and Atkns (996) requre an assocated servce level for each demand class, whch had not been analyzed n the prevous lterature. However, ther model allows negatve nventory allocatons that are hard to explan and mplement. Frank et al. (2003) apply ratonng to avod ncurrng hgh fxed orderng costs rather than savng nventory for hgh prorty demand. ahmas and Demmy (98) study a contnuous-revew nventory polcy wth two demand classes. They assume an ( QR, ) nventory replenshment polcy, a crtcal-level polcy, and at most one outstandng order at any tme. Ths last assumpton mples that whenever a reorder quantty s receved, the nventory level and nventory poston become dentcal. Ths allows them to calculate approxmate expressons for expected backorders for both demand classes. Moon and Kang (998) later extend ths model to account for compound Posson demand processes. Deshpande et al. (2003) analyze the same ( QR, ) nventory ratonng model wth two demand classes as n ahmas and Demmy (98), but wthout the restrcton on the number of outstandng orders. They ntroduce the threshold clearng mechansm to fll backorders, whch permts them to derve expressons for the expected number of backorders for both classes wthout restrctons on the number of orders outstandng. Based on these expressons, they develop algorthms to calculate the optmal orderng and ratonng parameters. They demonstrate numercally the effectveness of ther model, by comparson to a prorty-based backlog clearng mechansm, where hgh prorty backorders are flled before low prorty backorders. Melchors et al. (2000) also analyze a ( QR, ) nventory model wth two demand classes. Unlke ahmas and Demmy (98) and Deshpande et al. (2003), they consder a lost sales envronment so that demands from the low prorty class are reected whenever nventory level drops to the crtcal level. Melchors (200) extend the model n Melchors et al. (2000) to multple Posson demand classes wth stochastc replenshment lead-tmes. Moreover, he consders a non-statonary crtcal-level polcy that provdes a benchmark to evaluate the statonary crtcal-level polcy employed by ahmas and Demmy (98), Melchors et al. (2000), and Deshpande et al. (2003). Dekker et al. (998) study an nventory model wth two demand classes and one-for-one replenshment polcy. The model s smlar to the one n ahmas and 4

5 Demmy (98). They assume Posson demand processes, a determnstc replenshment lead-tme, backorderng of unflled demands, and a crtcal-level polcy to raton the nventory. Dekker et al. (998) explore how best to handle and allocate ncomng replenshment orders, whch remans an open queston n the lterature. Dekker et al. (2000) extends the model n Dekker et al. (998) to multple demand classes wth stochastc replenshment lead-tmes, but swtchng to a lost sales envronment rather than allowng backorders. They assume one-for-one replenshment polcy and a crtcal-level polcy to raton nventory among demand classes. In a lost sales envronment, handlng ncomng replenshment orders s not a dlemma; each ncomng replenshment order smply replenshes the nventory. They develop numercal soluton methods to effcently calculate the optmal base stock level and crtcal levels wth or wthout servce level constrants. Ha (997a) consders a make-to-order producton system wth a sngle producton faclty and multple demand classes for the end product. He assumes a lost sales envronment, exponentally dstrbuted producton tme, and Posson demand for each demand class. He shows that a statonary crtcal level polcy s optmal. Ha (997b) extends the study n Ha (997a) by allowng backorders to occur. Vercourt et al. (2000, 2002) consder the multple-demand class extenson of the two-demand class study n Ha (997a). They develop a characterzaton of the optmal polcy for the backorders case wth zero set-up costs and exponental lead-tmes. 2. General Framework Our work s most closely related to that of ahmas and Demmy (98) and Deshpande et al. (2003). However, whereas ther work consders two demand classes, we have no restrcton on the number of demand classes. We also develop the model n what we beleve s a more transparent and natural way. Indeed, as wll be seen, ths allows us to extend the model to permt servce tmes and to analyze a mult-echelon system wth multple demand classes. We consder a faclty that carres nventory for a sngle product to serve customer classes. We dfferentate customer classes based on ther relatve servce level requrements or shortage costs. For our analyss we requre the followng standard nventory assumptons: () We have a fxed replenshment lead-tme L > 0; 5

6 () The demand from class-, D, {, } follows a statonary Posson process wth rate λ that s ndependent of the demand from the other demand classes; () We replensh nventory wth a contnuous-revew (Q,R) polcy; (v) We backorder any demand that s not mmedately satsfed from on-hand nventory. In addton to these assumptons, we need to descrbe how we wll raton nventory across the demand classes. We number the demand classes accordng to ther relatve prorty, where class has the hghest prorty. As suggested by Venott (965), we use a crtcal-level polcy gven by c = c, c,, c + c Z {} 0 and c c. { 2 } We stop servng demand class- once the on-hand nventory reaches or falls below the crtcal stock level c ; by assumpton, we then backorder all demand for class- untl the on-hand nventory s rased above c. For class-, we set ts crtcal level c 0 = 0; thus, we contnue to fll class- demand untl the on-hand nventory s completely depleted, at whch pont we backorder any subsequent class- demand. We defne s = c c for =, - to be the reserve stock for class-, as t represents the quantty of stock that we protect or reserve for ths demand class. By the defnton of the crtcal-level polcy, each of these reserve stocks s non-negatve. For stage, we defne the reserve stock s = R c, for whch we requre no assumptons about ts sgn. We also need an assumpton wth regard to how we allocate an nventory replenshment at the tme t s receved. The prmary ssue s to decde how much of the replenshment we use to fll backorders versus use to re-buld the reserve stock for hgher-prorty demand classes. We defer untl the next secton the presentaton and dscusson of our allocaton assumpton, as t wll be easer to explan n the context of the problem mappng to a seral nventory system. 3. The Mappng to a Seral Inventory System The purpose of ths secton s to observe the equvalence between the sngle-product nventory system wth demand classes and a sngle-product nventory system wth seral stages. To ease the presentaton, we denote the former as the DCS (demand-class system) and the latter as the SSS (seral stage system). 6

7 l l l 2 l Q - 2 D D D 2 D Fgure : Seral Inventory System wth Demand at Each Installaton We consder an -stage seral system (SSS), as shown n Fgure, and assume that t operates as follows: () Each stage, {, } operates wth a one-for-one contnuous revew () base-stock polcy wth non-negatve base-stock level s., and s replenshed by ts upstream stage + wth replenshment lead tme l=0. Stage s replenshed from an outsde suppler wth lead-tme L > 0. Stage uses a contnuous-revew reorder-pont, reorder-quantty polcy wth ts reorder-quantty equal to Q and ts reorder-pont equal to s = R c. () Demand at stage follows a statonary Posson process wth rate λ. Each stage, { 2, } s subect to nternal demand from stage -, as well as external demand; the external demand at stage follows a statonary Posson process wth rate λ. The external demand processes are ndependent. (v) At each stage we backorder all nternal and external demand that cannot be met from on-hand nventory. We contend that ths SSS s equvalent to the DCS, as descrbed n the pror secton. To establsh ths equvalence, we wll show the two systems behave the same () when a demand occurs; () when each system places an order on ts outsde suppler; and () when each system receves the order from the outsde suppler. () When a demand occurs. Let IOH represent the on-hand nventory n ether the SSS or DCS. If IOH=0, then n both systems we backorder the demand from any class and the on-hand nventory remans at zero. Consder the DCS and suppose that IOH > 0 and c < IOH c for some {, }. If the next demand were from class, {, }, t s served and the on-hand nventory level IOH s reduced by one; f the next demand were from class, {, } +, then t s backordered. 7

8 ow consder the SSS wth IOH > 0 and c < IOH c. The on-hand nventory at each stage, {, } {, } equals ts base stock s. For each stage, +, there s no on-hand nventory, whle stage has on-hand nventory equal to IOH c. If there were a demand for stage {, } demand, the on-hand nventory at each stage, {, }, the seral system flls the remans at ts base stock s, and the on-hand nventory at stage (as well as the IOH) s depleted by one. However, f there were a demand for stage, {, } +, ths demand s not flled but s backordered by stage. The on-hand nventory IOH does not change. Thus, the behavor s the same. () When each system places an order on ts outsde suppler. In the DCS, we place an order of sze Q when the nventory poston reaches the reorder pont R, where the nventory poston s the on-hand nventory, plus the on-order nventory, mnus any backorders. In the SSS, stage orders from an external suppler when ts nventory poston reaches a reorder pont equal to s = R c. We note that the nventory poston for each downstream stage {, } s always s = c c, due to the one-for-one replenshment polcy. Thus, the SSS orders from ts external suppler when the system nventory poston s: = s = R. Thus, the two systems behave the same. () When each system receves the order from the outsde suppler. Fnally, we need to establsh that both systems clear the backorders n dentcal fashon when a replenshment arrves. We wll do ths by frst descrbng our assumptons for the SSS and then nterpretng how these assumptons apply to the DCS. Consder the SSS wth on-hand nventory IOH n the system at tme t. Suppose < for some {, } that c IOH c for stage {, }. The on-hand nventory equals ts base stock s, s equal to IOH c for stage, and s zero for stage { +, }. There are no backorders at each stage, {, } {, } +, has backorders gven by:. Each stage, 8

9 () ( τ, ) and () ( τ, ) B t D t B t D t = =, k k = where D ( s, t) s the external demand durng tme nterval (s, t] at stage ; B ( ) number of backorders at tme t at stage, due ts external demand; B ( t), t s the s the number of backorders at tme t at stage, due to nternal demand from stage -; and τ s the most recent tme at whch stage stocks out. Stage stocks out once the on-hand nventory reaches c - ; thus, we determneτ to be the most recent tme epoch at whch IOH =. c Suppose stage of the SSS receves a replenshment of Q at tme t. There are two cases to consder. ( ) ( ) Q B t + B t : The replenshment quantty s suffcent to fll all backorders, at stage, as well as all downstream stages. Thus, ths replenshment returns the on-hand nventory at each stage, {, } on-hand nventory s held at stage. ( ) ( ), to ts base stock s ; any remanng Q< B t + B t : The replenshment quantty s not suffcent to fll all backorders at stage, and we need to decde how to allocate the nventory between the two types of backorders. We assume that we fll these backorders n the order of occurrence, wth no dfferentaton between external and nternal backorders. In partcular, we fnd the earlest tme s : s t Dk s = Q. k = τ < < such that ( τ, ) We set the replenshment quanttes to be Q D ( τ, s) and Q D ( τ, s) Thus, after ths allocaton the remanng backorders at stage + + are B ( t ) = D ( s, t) and B ( t ) = D ( s, t)., k k = = =., k k = We assume ths process repeats at each downstream stage. For nstance, stage - receves the replenshment of Q,-. If t s suffcent to cover the backorders at stage -, then we fll these backorders and hold the remander on hand. If the replenshment s not suffcent to cover the backorders, then we allocate Q,- to fll these backorders n ther order of occurrence, as descrbed above. We repeat ths allocaton process at each downstream stage, untl we reach a stage at whch the stage s 9

10 replenshment covers the backorders at the stage or we reach stage. We assume the same allocaton process for the DCS. amely, we assume that we fll backorders n the order of occurrence, wth no dfferentaton between external and nternal backorders. We start wth demand class, and need to decde how to splt the replenshment quantty between class- backorders and the outstandng replenshment requests from stages, {, }. These outstandng replenshment requests can ental both customer backorders and replenshments to re-buld the reserve stock at the hgher-prorty classes. If the replenshment quantty s not suffcent, we wll allocate t n the order of the demand occurrences that created the backorders or replenshment requests. Ths process repeats wth each demand class untl we reach a class for whch the replenshment covers the backorders at the class or we reach demand class. Ths allocaton scheme s not optmal. However, t seems to be reasonable gven that t allocates the nventory at each stage (or demand class) to the nternal and external backorders at the stage n the order of occurrence. Thus, at each stage t tres to balance servcng external versus nternal backorders, where the nternal backorders nclude the re-buldng of a reserve stock at downstream stages (hgher prorty demand classes). Ths allocaton process s effectvely the same as the vrtual allocaton mechansm ntroduced n Graves (996) for the analyss of a mult-echelon arborescent nventory system. As n Graves (996), ths scheme permts sgnfcant tractablty n the analyss of the nventory system, as wll be seen. Ths completes the dscusson equatng the DCS to an SSS. We fnd ths equvalence to be helpful n vsualzng the operaton of the -demand-class nventory system, and n developng an evaluaton model of ts performance, as descrbed next. 4. Model for -Demand-Class Inventory System In ths secton we develop a model for evaluatng the performance of an -demand-class nventory system, based on the mappng to a seral system from the pror secton. We buld ths model usng the termnology of the SSS, and draw upon the framework n Graves (985). We defne addtonal notaton to analyze the nventory dynamcs n the seral nventory system, where {, } ( ) : IL t = nventory level at tme t at stage ; 0

11 ( ) IP t = nventory poston (nventory level plus nventory on order) at tme t at stage ; ( ) B t = number of backorders at tme t at stage ; We can characterze how the nventory level at each stage evolves over tme usng the followng equatons for the nventory dynamcs for the SSS, where [ ] + denotes the postve part of the expresson, l s the replenshment lead-tme for stage, and {, } : IL ( t + l ) = IP () t D ( t, t + l ) B () t () t +, = () = IL () t B t + (2) ( ) ( ) ( ) B t = B t + B t (3), ( ) ( ) B t = 0; B t = 0 (4),0 +, The explanaton for () parallels that n Graves (985): at tme t the outstandng orders for stage are ether n-process to stage or are backordered at the mmedate upstream stage +. All tems that were n-process at tme t wll arrve at stage by tme t + l, by the defnton of the lead-tme. However, none of the backorders at stage + at tme t can arrve to stage by tme t + l, agan by the defnton of the lead-tme. Furthermore, stage s subect to demand from ts own external demand process, plus that for all downstream stages due to the one-for-one replenshment polcy. Any demand durng the tme nterval (t, t + l ] reduces ts nventory level and cannot be replenshed by tme t + l. Hence, the nventory level at tme t + l at stage equals ts nventory poston at tme t net of ts outstandng orders, namely the backorders at tme t and all demand durng the tme nterval (t, t + l ]. In equaton (2) we state the backorders to be the negatve part of the nventory level. In equaton (3) we decompose the backorders at stage nto backorders created by the external demand at stage and backorders from replenshment requests from the mmedate downstream stage. We stpulate boundary condtons on the model n equaton (4), namely stage serves no downstream stages and the outsde suppler for stage s relable and meets any request wthn ts lead-tme l = L. In the context of the DCS, the replenshment lead-tme l = 0 for stages, {, }. Furthermore, due to the contnuous-revew one-for-one replenshment polcy at stages, {, }, the nventory poston for each stage always equals ts

12 base-stock level s. For stage, ts lead-tme s postve, l = L. The steady-state nventory poston for stage s unformly dstrbuted on the range[ s +, s + Q], gven the assumpton of a reorder-pont, reorder-quantty replenshment system wth these parameters (Zpkn 2000, p. 93). We now use these observatons to re-wrte the steady-state form for equatons () (4): L = IL = IP D (5) IL = s B+, for =, 2, - (6) [ ] B = IL + (7), B = B + B (8) B,0 = 0 (9) where L D s the random varable for the external demand at stage over an nterval of length L; thus, t represents a Posson random varable wth mean λ L. We need to establsh one more property before we can use equatons (5) (9) to determne the steady-state dstrbuton of the nventory level at each stage. We ntend to use (5) or (6) to fnd the dstrbuton of the nventory level, and then (7) to get the dstrbuton of the total backorders B at a stage. We then need to fnd the dstrbuton of B,, the backorders at stage due to downstream demand. To do ths, we contend that the probablty dstrbuton of B,, condtoned on a realzaton for B, s a bnomal. In partcular, we have for { 0, n } that n Pr B = B = n = p p ( ) n, where p = = = λ λ. (0) As explanaton, we note that once stage stocks out, backorders occur randomly accordng to the rates for the Posson demand processes. The backorders due to external demand at stage occur at rate λ ; backorders due to nternal demand from stage - occur at rate = λ. Thus, f n backorders occur, the number of backorders due 2

13 to nternal demand s a bnomal random varable wth parameters ( n, p ). Furthermore, the allocaton scheme for fllng backorders, descrbed n the pror secton, preserves ths random dstrbuton of backorders, as t flls backorders n the order of ther occurrence. As a consequence, at any tme t, f stage has postve backorders, then B () t D ( s, t) = for some value of s<t. Due to the memory-less property and = ndependence of the Posson demand processes, the condtonal dstrbuton of B, s bnomal. We can now determne the steady-state dstrbuton of the nventory levels, gven the polcy parameters ( ) ( ) QR, and c,, c. The procedure starts from the most upstream stage and moves teratvely to each downstream stage, as follows: Step : Set =. Determne the steady-state dstrbuton of IL. We obtan the dstrbuton of IL from equaton (5) by convolvng the dstrbuton of IP wth that for L D. The former s a unform random varable on the nterval [ s, s Q] = latter s a Posson random varable wth mean L λ. = Step 2: Obtan the steady-state dstrbuton of [ ] + + ; the B = IL +, backorders at stage. Step 3: Determne the steady-state dstrbuton of B,. We use the dstrbuton for B wth (0) to get the un-condtoned dstrbuton for B,. Step 4: Set : =. Determne the steady-state dstrbuton of IL from (6). Step 5: Stop f =. Otherwse go to Step 2. Wth the steady-state dstrbuton of the nventory level at each stage, we can compute relevant performance measures, such as the expected on-hand nventory, the expected backorders, and the fll rate for each demand class. We can then pose an optmzaton problem to fnd the best choce for the control parameters, namely the reorder pont R, reorder quantty Q, and the crtcal levels{ c : =,..., } (or equvalently the reserve stocks{ s :,..., = c c = } ). In the next secton, we llustrate one such optmzaton, n whch we mnmze the expected on-hand nventory 3

14 subect to constrants on the fll rates for each demand class. 5. Servce Level Problem There are many ways to look at the tradeoff between holdng nventory and achevng a hgh level of customer servce. We consder one problem varant, n whch we mnmze the amount of nventory needed to satsfy a gven fll rate target for each demand class. In effect, we defne the demand classes by ther fll-rate targets; we would cluster customers nto the demand classes accordng to ther servce promses or expectatons, wth demand class correspondng to the hghest level of servce and so on. We formulate ths servce level problem (SLP) for a DCS as follows: SLP Mn z = = [ ] s. t. Fllrate β for =,, where s 0, nteger for =,, s nteger E IL ( IL ) ( IL ) + Pr > 0 f s > 0 Fllrate = for =,, Fllrate+ f s = 0 Fllrate = Pr > 0 The obectve s to mnmze the expected on-hand nventory, whch s the postve part of the nventory level. The reserve stocks s are the decson varables, from whch we can fnd both the crtcal levels c and the reorder pont R. To smplfy the presentaton we assume that the order quantty Q s not a decson varable, but has been pre-specfed. The constrants assure that we meet a fll-rate target β for each demand class. The computaton of the fll rate depends on the reserve stock level of the demand class. If the reserve stock for the demand class s postve, then for Posson demand the fll rate equals the probablty that the nventory level s postve. When the reserve stock for the demand class s zero, then the fll rate for demand class s the same as that for demand class +. Ths s because when s = 0, there s no dstncton n order fulfllment between a demand from class and a demand from class +. In formulatng the SLP, we expect (although don t requre) that the hgher-prorty demand classes have larger fll-rate targets; that s, we expect β β2 β. Indeed, the structure of the crtcal-level polcy guarantees 4

15 that demand class has a fll rate no worse than that for demand class +. From the model (5) (9), we see that the nventory level at demand class, IL, depends on ts reserve stock and that for lower-ranked demand classes; that s, IL s a functon of ( s,, s ) (,, ). In the followng, we wll at tmes use the notaton IL s s to make ths dependence explct. Soluton Procedure for the Servce Level Problem In ths secton we state a sequental soluton method, the Sngle-Pass-Algorthm (SPA), whch provdes us wth a good feasble soluton for the SLP. We then establsh a bound on the gap between the SPA soluton and the optmal soluton to the SLP. SPA uses the model gven by (5) - (9) to fnd the reserve stock for each demand class sequentally, startng wth stage. For each demand class, SPA fnds the mnmum value for ts reserve stock that satsfes ts fll-rate target, gven the prevously-determned reserve stocks for demand classes +,,. We state the algorthm as follows: { }. Fnd reserve stock and fll rate for stage : ˆ mn : Pr ( ( ) 0) and Fllrate ( ( ˆ Pr IL s) 0) = > ; let := Fnd reserve stock and fll rate for stage : a. If Fllrate ˆ + β : s = 0; Fllrate = Fllrate+ s = s IL s > β ; { β } b. If Fllrate ˆ ( ( ˆ ˆ + β s s IL s s + s ) ) < : = mn : Pr,,, > 0 ; and ( ( ˆ ˆ ˆ + ) ) Fllrate = Pr IL s, s,, s > 0 3. Stop f :=. Otherwse, let := - and repeat step 2. The Sngle-Pass-Algorthm yelds a feasble soluton for the SLP by constructon: at each teratve step, t sets the reserve stock for a demand class to satsfy the fll-rate constrant for ths demand class. However, there s no guarantee that the soluton s optmal; later n ths secton we provde an example that llustrates ths. We contend that the soluton for the SPA should be qute good. To develop ths argument, t wll be helpful to re-wrte the obectve functon of SLP as Q + z = E IL = s + E B + L λ 2 + [ ] ( [ ] ). () = = = 5

16 We obtan ths expresson from substtutng (5), (6) and (8) nto E[ IL ] + = E[ IL ] + E[ B ], wth the observaton that E[ IP ] Q + = s +. Thus, we 2 observe that the obectve functon conssts of the sum of the reserve stocks and the sum of the external backorders, plus a constant K: [ ]. (2) z = s + E B + K = = We wll develop a bound on z by fndng a lower bound on the sum of the reserve stocks, and then a lower bound on the sum of the external backorders. We frst show that movng one unt of reserve stock from class to class - cannot decrease the nventory level at any of the hgher-ranked classes, but can result n more backorders. Proposton : Consder two stockng polces ( s ) ( 2 2 2, s2,, s and s, s2,, s) where for some, s = s, s = s +, and s = s,,. We have that: 2 2 () ( ) ( ) IL s,, s IL s,, s for =,2,..., and () B ( 2,, 2 ) (,, s s B s s ) Proof.. = =,,,,,,, From (5) and (6) we fnd that IL( s s) = IL ( s s + s) = IL( s s),,,, Thus from (7) and (8), we obtan B( s s) B( s s) 2 2 and B, ( s s) B, ( s s) that:,,,, +. We can now use ths result n (6) to show (,, ) = (,, ) s + B, ( s,, s) = IL ( s,, s) IL s s s B s s, From IL ( 2 2 ) ( s,, s IL s,, s) and s 2 = s, =, 2, we fnd IL s,, s IL s,, s for =,2,... 2, whch proves the frst result. 2 2 that ( ) ( ) For the second result, we can make a sample path comparson. Suppose at tme B +, t < s; then there are no backorders for ether case: t, we have ( ) 2 2 ( ) ( ) B t s,, s = B t s,, s = 0. If B ( t) s = = +, then, 6

17 (,, ) = (,, ) + and ether B ( t s s ) B ( t s s ) B t s s B t s s 2 2 ( ) ( ) B t s,, s = B t s,, s +. In the former case we have,, 2 2 ( ) ( ) B t s,, s = B t s,, s + ; n the latter case we have = = ( 2,, 2 ) = (,, ) B ts s B ts s. Ths proves the result. = =,, =,, + or We now use these results to establsh bounds on the optmal soluton. We frst show that the soluton for the SPA provdes a bound on the sum of the reserve stock. Proposton 2: For all feasble solutons ( s s ),, for the SLP, we have s ˆ sfor all where ( s ˆ ˆ,, s ) s the soluton found by the SPA. = = Proof. Suppose we have a feasble soluton ( s, s2,, s ) such that s < = = sˆ. We wll teratvely construct a seres of feasble solutons, whch leads to a contradcton of the supposton. In order for ( s, s2,, s ) to be a feasble soluton, we must have s sˆ ; otherwse the fll-rate constrant for class s volated. If s ˆ s = then we must have s ˆ s by the same logc. If both s = sˆ, s ˆ = s, then we must have s ˆ 2 s 2 and so on. Iteratve Step: Let k be the largest ndex such that s > sˆ ; that s, s = sˆ,, s = sˆ, and s > sˆ. If k, we have a contradcton of the orgnal k+ k+ k k supposton that s < = = ( s, s2,, s ) ( ˆ ˆ ˆ = s, s2,, s) k sˆ. If there does not exst an ndex k, then we must have, whch s also a contradcton of the orgnal supposton. Gven k >, then we construct a new soluton: s = s, s = s +, and s = s, k, k. By applcaton of proposton (), k k k k we can show that ths new soluton s feasble. Snce k >, we have that 2 s ˆ = s < s = = =. k 7

18 We now use the new soluton to repeat the Iteratve Step. At each step we move one unt of reserve stock from a hgher-numbered class to a lower-numbered class to create a new feasble soluton. The number of possble teratve steps s fnte as each unt of reserve stock can be moved at most - tmes. Therefore, at some step n we n have ether s sˆ for,, n n n = = or s ˆ and ( ) ( ˆ ˆ > s s+,, s = s+,, s) whch are contradctons of the orgnal supposton., both of As a specal case of ths proposton, we see that sˆ s a lower bound on the sum of = the reserve stocks n the obectve functon of the SLP. We denote sum of the reserve stocks s the reorder pont for the crtcal-level polcy. = sˆ = Rˆ, snce the Proposton 3: Consder two stockng polces ( s ) ( 2 2 2, s2,, s and s, s2,, s) wth 2 = = s s for =,, and ( ) ( 2 2,,,, ) z s s z s s. Proof. From (2) and the assumpton that 2 s = s = =. Then we have 2 s = s = =, we need to show that (, ) ( 2, 2 ) E B s s E B s s. Ths result follows drectly from = = applcaton of proposton (). Startng wth the stockng polcy ( s 2 2 2, s2,, s ) can construct a seres of new polces n whch we move one unt of reserve stock from class - to class, and eventually reach the stockng polcy ( s, s2,, s ). From proposton (), each such move reduces the external backorders n classes,2,, and has no mpact on backorders at class +,,. Thus we show that (, ) ( 2, 2 ) B s s B s s, and we get the desred result by takng = = expectatons. From ths proposton, we have a lower bound on the optmal obectve functon value of the SLP:, we 8

19 = ( 0, ˆ + 0, 0, ) Rˆ + E B s = s = s = s = R + K (3) Based on these propostons we conecture that the soluton for the SPA should be near optmal n most settngs. We see from the lower bound that any mprovement to the SPA soluton must come by means of a reducton n backorders. As most settngs have hgh servce expectatons, the fll-rate targets are such that any feasble soluton wll generate, at most, a modest amount of backorders. As a consequence, we expect there to be mnmal opportunty to mprove upon the soluton gven by the SPA. We explore ths conecture n the next secton wth a computatonal experment. Example The purpose of ths example s to provde some nsght nto why the SPA soluton need not be optmal, yet s lkely to be close to optmal. We assume three demand classes wth the Posson demand rates λ = 8 unts/year, λ 2 = 2 unts/year, and λ 3 = 6 unts/year. The replenshment lead-tme s L = 0.25 years (3 months), and the reorder quantty s Q =. The fll-rate targets are: β = 0.99, β 2 = 0.94, and β 3 = When we apply the SPA to ths problem we get: sˆ = 2, sˆ 2 =, sˆ 3 = 2 wth z = Ths translates nto the crtcal level polcy: cˆ ˆ ˆ = 2, c2 = 3 and R= 5 ; we reorder when the nventory poston reaches 5, we stop servng demand classes 3 and 2 once the on-hand nventory drops to 3 and to 2, respectvely. We can use () to break the obectve value nto ts consttuent parts: 3 3 Q + z s s s s E B L λ ( ˆ, ˆ2, ˆ3) = ˆ + [ ] + = = 2 = = = 7.09 As the expected backorders are qute small, we know from Proposton 3 that ths soluton must be very close to optmal. Indeed, the lower bound from Proposton 3 s The optmal soluton (found by exhaustve search) s s =, s2 = 0, s3 = 4 wth z = The only dfference between the optmal soluton and the SPA soluton s that one unt of reserve stock has been moved from demand class and demand class 2 to demand class 3; ths move reduces the backorders at demand class 3 wthout eopardzng the fll-rate constrant for demand class and for demand class 2. The 9

20 move does reduce the fll rate for demand class, but t stll s above To apprecate the beneft of dfferentatng the demand classes, suppose all customers were to get the hghest servce level, namely a fll rate of Then we need to set the reorder pont R = 7 and the expected on-hand nventory (z) s 9.00, whch s 27% hgher than the optmal soluton. Suppose now that the fll-rate targets are β = 0.99, β 2 = 0.93, and β 3 = The SPA fnds the soluton: sˆ = 2, sˆ 2 = 2, sˆ 3 = 0 wth z = The expected backorders for the SPA soluton s 0.24, and the lower bound on the expected backorders s The optmal soluton entals movng a unt of reserve stock from demand class to demand class 3: s =, s 2 = 2, s 3 = wth z = 6.4. In each of these two cases we see that the SPA generates a near-optmal soluton. We are able to mprove slghtly the SPA soluton by shftng a unt of reserve stock from a hgher-ranked class to a lower-ranked class. The result of ths shft s that we reduce some backorders at the lower-ranked demand class, yet stll satsfy the fll-rate target at the hgher-ranked class. 6. umercal Experment: To test the effectveness of the SPA on the SLP, we compare ts soluton to the optmal soluton on two set of test problems n two experments. For the frst experment, we examne the performance of SPA as we vary the reorder quantty, the lead tme, the fll-rate targets and the demand rates. For the second experment, we vary the number of demand classes. For each test problem n the frst experment, there are three demand classes. The reorder quantty takes on one of four values: Q =, 4, 9, or 8. The replenshment lead-tme from the outsde suppler s one of the three values: L = /24 year, ¼ year, ½ year. There are three possble values for the fll-rate target for each of the demand classes: β = 0.9, 0.95, or 0.99, β 2 = 0.8, 0.9, or 0.95, and β 3 = 0.7, 0.8, or 0.9. We only consder combnatons wth ether β > β2 β3 or β β2 > β3; thus, we have 20 combnatons of fll-rates. Fnally, we have four possble settngs for the demand rates: { λ = 8, λ2=2, λ3 = 6}, { λ = 6, λ2=2, λ3 = 8}, { λ =, λ2 = 3, λ3 = 8}, and { λ = 4, λ = 4, λ = 4} unts/year. 2 3 We specfy a test problem n the frst experment by settng the number of 20

21 demand classes ( canddate), the replenshment lead-tme (3 canddates), the reorder quantty (4 canddates), the set of desred fll-rates (20 canddates), and the set of demand rates (4 canddates). Ths provdes a total of 960 test problems. For each test problem, we compute the SPA soluton ( sˆ ˆ ˆ, s2, s 3) and ts cost from equaton (2); the lower bound from equaton (3); and the optmal soluton and ts cost. We fnd the optmal soluton by a search algorthm. We frst compute zs ( = 0, s ˆ ˆ ˆ 2 = 0, s3 = s+ s2 + s3+ ), whch s a lower bound on the cost for any soluton for the SLP wth total reserve stock equal to sˆ + sˆ 2 + sˆ 3 +. In all test problems, we fnd zs ( ˆ ˆ ˆ, s2, s 3) to be less than zs ( = 0, s ˆ ˆ ˆ 2 = 0, s3 = s+ s2 + s3+ ). Ths observaton together wth the results n Proposton 2 and Proposton 3 guarantee that the total reserve stock n the optmal soluton must be sˆ + sˆ 2 + sˆ 3. ext, we fnd the optmal soluton by searchng over the nteger solutons n the space: s sˆ ; s + s sˆ + sˆ ; and s + s + s = sˆ + sˆ + sˆ The results of ths numercal experment support our ntuton that the SPA s qute effectve. The SPA fnds the optmal soluton n 274 problem nstances or 29% of the cases. The cost of the SPA soluton s on average 0.57% hgher than the cost of the optmal soluton and.28% hgher than the lower bound. The maxmum error for SPA s 3.24%. In Table 2, we examne how the relatve performance of the SPA heurstc changes as we vary the problem parameters. Each cell of the table provdes the average cost ncrease for the SPA soluton for all test problems wth the sngle parameter fxed. For nstance n the cell wth L=/24, we report the average performance of the SPA for the 320 test problems wth lead tme L=/24. For the fll-rate targets, we have dvded the 20 combnatons accordng to the spread between the fll-rate targets for class and class 3 ( β β3). There are sx combnatons and 288 test problems wth < 025., β β3 <, eght combnatons (384 problems) wth β β3 and sx combnatons (288 problems) wth 025. β β3. The performance of the SPA seems qute nsenstve to the settngs for the reorder quantty, and the replenshment lead-tme. However, the performance seems to depend on the dstrbuton of demand rates and the spread n fll-rate targets. The performance mproves slghtly when there s a hgher percentage of demand n the hgher-prorty demand class (class ). In addton, the SPA performs best when the 2

22 spread n servce levels s smallest. For each test problem we also compute the cost for the optmal nventory polcy n whch we provde the hghest fll-rate β for each demand class. Admttedly ths s a sub-optmal polcy as there s no ratonng of nventory between demand classes; nevertheless, we observe ths polcy n practce as t satsfes the servce requrements and s easy to mplement. For ths set of test problems, the cost of a no-ratonng polcy s on average 8% hgher than the optmal crtcal-level polcy. In the second experment, we specfy four test problems, one for each settng for the number of demand classes: = 2, 3, 4, or 5. We set the replenshment lead-tme L = ¼ year and the reorder quantty Q = 4 for each test problem. In Table 3, we specfy the fll-rate targets and the demand rates for each test problem. As wth the frst experment, we compute the SPA soluton and cost for each test problem, as well as the optmal soluton and ts cost. We report the results n Table 3. On ths set of test problems, the relatve performance of the SPA heurstc mproves as the number of demand classes ncreases. Ths observaton s consstent wth our ntuton that the SPA performs better when there are smaller dfferences between the fll-rate targets for consecutve demand classes. Lead Tme L = / % L = ¼ 0.66% L= ½ 0.54% Reorder Quantty Q= 0.58% Q=4 0.56% Q=9 0.58% Demand Rates { λ = 8, 2, 6} { λ = 6, 2, 8} { λ =, 3, 8} 0.64% 0.46% 0.65% Servce Target 005. β β3< β β3< β β 3 Spread 0.32% 0.56% 0.84% Table 2: Average ncrease n SPA soluton relatve to optmal soluton Q=8 0.57% { λ = 4, 4, 4} 0.53% Servce Targets Demand Rates SPA Soluton Optmal Soluton Percent Dfference λ = 8, % 2 { 0.99, 0.8} β = { } 3 { β = 0.99, 0.9, 0.8} { 8, 2, 6} 4 { β = 0.99, 0.95, 0.9, 0.8} { 4, 6, 0, 6} 5 { β = 0.99, 0.95, 0.9, 0.85, 0.8} { 4, 6, 8, 8, 0} λ = % λ = % λ = % Table 3: Test problems and results from second computatonal experment 7. Extensons In ths paper we consder a sngle-product nventory system wth multple 22

23 demand classes. We show how to map ths system nto an equvalent sngle-product seral nventory system. We then apply a modelng framework for mult-echelon dvergent systems to obtan a characterzaton of the steady-state performance of the -demand-class nventory system for a crtcal-level polcy. To fnd the best crtcal-level polcy, we pose an optmzaton problem to mnmze the on-hand nventory subect to fll-rate constrants for each demand class. We provde a computatonally-effcent approxmate procedure for solvng ths problem, and demonstrate ts effectveness on a set of test problems. In ths secton we frst show how to ncorporate servce tmes nto the model and how to use the model to characterze a mult-echelon system wth multple demand classes. We then dscuss possble extensons to ths research. Servce Tmes. In the presentaton so far, we assume that the servce tme for each demand class s zero. That s, customers n each demand class expect ther demand to be flled at the tme of ts occurrence. In many contexts, however, t s common to have non-zero servce tmes, whereby a customer expects demand to be flled wthn some specfed tme wndow. Indeed, ths can be the bass for defnng demand classes. Demand class mght be the customers, who, say, have a twenty-four-hour servce tme. The other demand classes mght have longer servce tmes, say three days for class 2, one week for class 3, and so on. For nstance, for servce parts nventory systems, these servce tmes are part of the contract between the customer and the provder of the servce parts. Another example s where customers select the tme of delvery, as s avalable from most e-talers. In ths manner the customer defnes (and pays for) a desred servce tme. We need to descrbe how the crtcal-level polcy apples when servce tmes are non-zero. Let w be the servce tme for demand class. We assume each w < L. We say that a demand from class that arrves at tme t s due at tme t + w. Let IOH represent the on-hand nventory n the system. Suppose that c < IOH c for some {, }. Then, f the next demand due were from class, {, }, t s served and the nventory level IOH s reduced by one; f the next demand due were from class, {, } +, then t s backordered. Ths polcy s not optmal as t gnores nformaton about demand that s not yet due. evertheless, the polcy would be relatvely easy to mplement and does allow for 23

24 stock ratonng so as to protect the servce to hgher prorty demand classes. We do assume that when a demand arrves from any demand class, the system nventory poston s reduced by one, and a reorder s placed once the system nventory poston reaches the reorder pont. Wth these assumptons, we can re-state the steady-state equatons analogous to (5) and (6): ( ) (4) IL = IP D L w = IL = s B+, for =, 2, - (5) where D ( τ ) s the random varable for the external demand at stage over an nterval of length τ ; thus, t represents a Posson random varable wth mean λτ. The equatons (7), (8), (9) are the same. As explanaton, we refer to the operaton of the -stage seral nventory system. When a demand (ether external or nternal) occurs at tme t at stage {, } wth a due date of t + w, we assume that stage does not fll ths demand untl ts due date; f the stage cannot fll the demand on the due date, then the stage backorders the demand untl t has nventory to fll t. We also assume that at tme t stage {, } ntates a one-for-one replenshment from ts upstream suppler but wth the due date of t + w. When a demand (ether external or nternal) occurs at tme t at stage wth a due date of t + w, we assume that stage reduces ts nventory poston by one; when ts nventory poston reaches a reorder-pont equal to s = R c, stage orders on an external suppler. The last step n developng the model for non-zero servce tmes s to ndcate the allocaton scheme for fllng backorders at each stage. We assume that at each stage we fll the backorders n the order of ther due dates, wth no dfferentaton between external and nternal backorders. As a consequence, f B ( t ) > 0 for some stage, then we can express the nternal and external backorders as: () = ( τ, ) B t D w t w = where τ < t s the most recent tme at whch stage stocks out. Wth ths assumpton we have that the probablty dstrbuton of nternal backorders, condtoned on the total 24

25 backorders at a stage, s bnomal, as gven by (0). We thus see that the model (5) (9) extends drectly to permt non-zero servce tmes, wth the same computatonal requrements. evertheless, there s an open queston as to how effectve s the (myopc) crtcal-level polcy for ths extenson. Mult-Echelon Systems As a second extenson, we descrbe how one mght develop a model of a mult-echelon nventory system wth multple demand classes. For nstance, consder a servce-parts dstrbuton system n whch there s a central warehouse that replenshes several local stes. We assume each of the local stes s subect to Posson demand from classes, operates wth a crtcal-level polcy, and reorders on the central warehouse wth an order quantty Q =. We assume the central warehouse replenshes ts nventory wth a one-for-one replenshment from an external suppler wth a determnstc lead-tme, and flls order from the local stes on a frst-come, frst-served bass wth a determnstc lead-tme. These assumptons are qute typcal for low-volume, hgh-value servce parts. We can use the model (5) (9) for each ste, but wth one modfcaton. When the central warehouse stocks out, replenshment requests from the local stes are delayed. Thus, we need re-state equaton (5) for the nventory at local ste k as: L k, = k, k, 0, k = IL IP D B (6) where the second subscrpt refers to the local ste, and where B0,k denotes the backorders at the central warehouse that are due to local ste k. For Posson demand we can use ether the exact or approxmate model n Graves (985) to characterze the backorders at the central warehouse as a functon of ts base stock level. Thus, we can model the performance of a mult-echelon system wth -demand classes for Posson demand, one-for-one replenshment polces, and determnstc lead-tmes. We can use ths model to optmze the nventory parameters, namely the base stock at the central warehouse and the crtcal levels and reorder pont at each of the local stes. One approach would be to do a sngle-dmenson search over possble settngs for the base stock at the central warehouse. Gven a base stock at the central warehouse, we can characterze the backorders to each of the local stes. We can then use (6) and (6) (9) to optmze the nventory parameters at each local ste, as 25

26 descrbed n ths paper. General Demand Process. We assume a Posson demand process. The model () (4) remans vald for demand processes wth ndependent ncrements, e.g., compound Posson demand. However, we would need to re-vst the next steps n the model development f demand were from a compound Posson process. The dstrbuton of the nventory poston n equaton (5), IP, s no longer unform, as t wll depend on the compoundng dstrbuton. Smlarly, the condtonal dstrbuton of backorders for demand class due to downstream demand s not bnomal, as gven by (0). The computaton of fll rate at each demand class s also more complcated. Alternatvely, one mght approxmate the demand for each demand class by an ndependent Brownan moton process,.e., D (, ) and standard devaton gven by ( ) and s t s normally dstrbuted wth mean t s µ t sσ. As ths process has ndependent ncrements, we can apply the model () (4), but some care s needed n the subsequent analyss. As wth the case of compound Posson demand, we would have to adapt the condtonal dstrbuton of B, n (0), as t s no longer bnomal. One would also need to examne how best to specfy and measure servce, when demand s approxmated by a Brownan moton process. Allocaton Process. We assume that when the replenshment quantty Q s not suffcent to cover all backorders, we fll the backorders n the order of occurrence wth no dfferentaton between external and nternal backorders. The ntent s to allocate the replenshment quantty farly between fllng the backorders at lower-ranked demand classes and restorng the reserve stock for hgher-ranked demand classes. evertheless, ths process s ndependent of any obectve functon, and s not optmal. It would be of nterest to understand better how ths allocaton scheme performs for varous problem crtera. Lost Sales. The development of the model n ths paper depends on the assumpton that demand s backordered when t cannot be met from stock. We have not found an easy way to modfy the current model to accommodate a lost sales assumpton. We leave ths for future research. 26