This paper concerns the evaluation and analysis of order

Transcription

1 ORDER-FULFILLMENT PERFORMANCE MEASURES IN AN ASSEMBLE- TO-ORDER SYSTEM WITH STOCHASTIC LEADTIMES JING-SHENG SONG Unversty of Calforna, Irvne, Calforna SUSAN H. XU Penn State Unversty, Unversty Park, Pennsylvana BIN LIU Chnese Academy of Scences, Bejng, Chna (Receved February 1997; revsed Aprl 1997; accepted May 1997) We study a multcomponent, multproduct producton and nventory system n whch ndvdual components are made to stock but fnal products are assembled to customer orders. Each component s produced by an ndependent producton faclty wth fnte capacty, and the component nventory s controlled by an ndependent base-stock polcy. For any gven base-stock polcy, we derve the key performance measures, ncludng the probablty of fulfllng a customer order wthn any specfed tme wndow. Computatonal procedures and numercal examples are also presented. A smlar approach apples to the generc mult-tem make-to-stock nventory systems n whch a typcal customer order conssts of a kt of tems. Ths paper concerns the evaluaton and analyss of order fulfllment performance measures for a multtem, assemble-to-order producton/nventory system wth stochastc leadtmes. Order fulfllment performance has become ncreasngly mportant as companes that must adapt quckly to market and technology changes move toward assemble-to-order, as opposed to the tradtonal make-to-stock, nventory plannng systems. In an assemble-to-order system, products are desgned around nterchangeable modules, and the company makes and stocks only the modules and other major components. When a customer order arrves requestng a specfc kt of modules and components, the company quckly assembles these tems and delvers the end product to the customer. Snce each customer order requres the smultaneous avalablty of several tems, the questons nventory managers ask most frequently are the followng: For any gven safety-stock level of each tem, what s the probablty a demand can be satsfed mmedately (order fll rate)? What s the probablty a customer order can be met wthn a tme wndow (customer watng tme dstrbuton; also termed order response tme relablty n ndustry)? The same ssue exsts for many mult-tem dstrbuton systems as well, ncludng the emergng mal-order busnesses (see, for example, Cohen and Lee 1990, Lee and Bllngton 1992, Song 1998). In partcular, a typcal customer order to a manufacturer conssts of a combnaton of several fnshed goods. Relable and speedy delvery of orders s one of the most crucal factors for customer satsfacton. So, the order-based performance measures, such as the order fll rate and the customer watng tme dstrbuton, are the mportant ones. Standard nventory models assume that demands are ndependent across tems, whch s a vald assumpton for some stuatons n the make-to-stock envronment but not for assemble-to-order systems, n whch one must jontly manage nventores and producton capactes across varous tems. Evdently, new models and methods are n demand to address mportant ssues n assemble-to-order systems. The purpose of ths paper s to conduct an exact analyss on a wde range of performance measures n the assembleto-order systems wth sequental, capactated stochastc producton processes. In partcular, we model the demand process as a multvarate Posson process. That s, the overall demand arrves accordng to a Posson process, but there s a fxed probablty that a demand requests a partcular kt of dfferent tems. Each tem s nventory s controlled by a separate base-stock polcy. Demands are flled on an FCFS bass. Demands for an tem that cannot be flled mmedately are queued n a backlog queue wth a certan capacty, whose value may range from zero to nfnty. Infnte capacty corresponds to the complete backlog case, whereas zero capacty corresponds to the lost-sale case. The supply system of each tem s an ndependent, sngle-machne producton faclty wth..d. exponentally dstrbuted processng tmes. For any gven base-stock polcy and backlog queue lmts, we present a procedure to evaluate the tem-based, order-based, and system-based performance measures, such as fll rate, servce level (the probablty that an order Subject classfcatons: Inventory/producton: mult-tem, operatng characterstcs, stochastc. Probablty: applcatons, Markov processes, stochastc model applcatons. Area of revew: MANUFACTURING OPERATIONS. Operatons Research X/99/ $05.00 Vol. 47, No. 1, January February INFORMS

2 132 / SONG, XU, AND LIU wll be backlogged and eventually served), watng tme dstrbuton, etc. In terms of computatonal complexty, ths procedure s very effcent to obtan the exact performance of small to medum-szed systems wth fnte backlog queues. Under certan condtons (e.g., moderate machne utlzatons), ths procedure s also capable of effcently conductng an approxmate analyss on the system wth nfnte backlog queues. Addtonally, the advantage of conductng exact analyss s to enhance our understandng of the nner workng of such systems, thereby provdng useful nsghts for effectve system desgn and control. It also provdes a benchmark result and nsghts for developng stochastc bounds, heurstc solutons for systems that are more complex n nature and thus less tractable analytcally. In our analyss, t s essental to obtan the jont equlbrum dstrbuton of the occupancy n the supply system, whch subsequently determnes other order-based performance measures. The major dffculty n evaluatng the jont dstrbuton of the occupancy s a result of correlaton of the producton facltes, caused by smultaneous arrvals. See, for example, Flatto and Hahn (1984), Shwartz and Wess (1993), and Wrght (1992) for transformaton and asymptotc results for the 2-queue system wth nfnte buffers. The key dea n our soluton procedure s that, because of the specal structure of the demand process, the occupancy n the supply system can be modeled as a quas brth-death process. Ths allows us to develop a matrxgeometrc soluton for ts jont dstrbuton. Another contrbuton of ths paper that dstngushes t from others n the lterature s that we are able to derve the exact watng tme dstrbuton for accepted orders. Such nformaton s ncreasngly mportant n the assemble-to-order envronment. To be compettve, companes often need to guarantee delvery of the product to customers n a specfed tme wndow. The watng tme dstrbuton provdes the managers the precse lkelhood such guarantees can be met (Ths s termed order response tme relablty n Hausman et al. 1998). Snce sometmes the performance measure of prmary nterest s the expected watng tme of an accepted order (as opposed to the dstrbutonal probablty), we also develop a smpler recursve evaluaton procedure for such purpose. There have been several recent research efforts n studyng mult-tem nventory systems wth correlated demands across tems. See, for example, Agrawal and Cohen (1996), Hausman et al., Schraner (1996), Song (1998), and Srnvasan et al. (1992). These studes assume a base-stock polcy and a determnstc supply system. For stochastc supply systems, Anupd and Tayur (1998) develop a smulaton model to study both tem-based and order-based performance measures n a multproduct cyclc producton system. Cheung and Hausman (1995) consder..d. replenshment leadtmes and a multvarate Posson demand model. Complete cannbalzaton s assumed, however, n order to derve the average customer backorders. In addton, two studes have been conducted ndependently and concurrently wth our research, but wth dfferent focuses. Zhang (1995) assumes a multvarate Posson demand process, and each tem s suppled by a dedcated faclty wth general..d. processng tmes. She uses the expected system-based watng tme (.e., the expected delay of an arbtrary order) as the sole performance measure. (Our model ncludes the complete backlog system as a specal case, and our prmary nterest s to obtan the dstrbutons of order-fulfllment performance measures. As a byproduct, we also obtan the dstrbuton and the expected value of the delay of an arbtrary order.) Glasserman and Wang (1998) study a model wth the same knd of supply system as n Zhang but a more general demand structure. They study the leadtme-nventory trade-off and show that the relaton s lnear, n a lmtng sense, at hgh level of servce. In Secton 1, we descrbe the model n detal and ntroduce the basc notaton. In Secton 2, we focus our analyss on the assemble-to-order system wth total order servce (TOS), where an order must be ether accepted or rejected as a whole. It s worth mentonng that, n addton to ts applcablty to some real systems, the fxed backlog buffer szes can be vewed as a measure of customer mpatence. Here, f even one tem s backlog queue s full, t sgnals the customer that there exsts a prospect of a long wat, the customer then decdes to leave, wthout watng for any tems. (L (1992) makes a smlar observaton n a sngletem system.) In Secton 3, we concentrate on the assemble-to-order envronment wth partal order servce (POS), where a customer request may be only partally accepted (guaranteed to be flled eventually), wth the rest rejected because the correspondng backlog queues are full. Ths model s mostly vald n mult-tem, make-to-stock systems n whch a typcal customer order conssts of a kt of tems n stock. One can also use ths model to measure customer mpatence, but here hs/her mpatence s assocated wth ndvdual tems as opposed to the entre order. Evdently, the TOS and POS models are dentcal n the complete backlog case. Ths rases the possblty of usng a POS model to approxmate ts TOS counterpart. An attractve feature of the POS model, as compared wth the TOS model, s that ts tem-based performance measures can be readly obtaned only through the margnal dstrbuton of the occupancy n each ndvdual producton faclty. Thus, the tem-based performance n the POS model can be convenently employed to approxmate and bound the order-based performances n both POS and TOS models. An nterestng queston s, of course, when and to what extend the former can provde useful nformaton for the latter (we address ths ssue further n Secton 4.2). To gan more nsght nto the model, Secton 4 s devoted to the smple two-tem system. In partcular, we provde detals of the general results presented n Sectons 2 and 3. Numercal experments are conducted to llustrate the results and to derve nsghts. For example, we fnd that, n general, the result from the POS model provdes a

3 relable estmate for ts counterpart n the TOS model. We also fnd that wth moderate traffc the fnte-buffer model provdes an accurate approxmaton for the nfnte-buffer model n a broad range of envronmental settngs. Fnally, n Secton 5, we make some concludng remarks and dscuss future research drectons. 1. THE MODEL DESCRIPTION Now we descrbe the specfc model assumptons and ntroduce the basc notaton. We consder an nventory system of J dfferent tems. Let {1, 2,..., J} be the set of all tem ndexes. For any subset K of, denote by K the number of elements n K. We consder an nfnte plannng horzon and assume that the assembly tmes are neglgble (compared to the producton tmes). The Demand Process The overall demand process s statonary n tme and forms a Posson process. Each customer requests at most one unt of each tem but may requre several tems smultaneously. In partcular, for any subset of tems K, we say a demand s of type K f t requres one and only one unt of each tem n K and 0 unts n K. We assume that there s a fxed probablty that a demand s of type K. Each demand s type s ndependent of the other demands types and of all other events. When K contans a sngle tem, say, we abbrevate type K by type. Smlarly, we say an order s of type j f K {, j}. Obvously, the demand process for each tem s also a Posson process. Throughout the paper, we use subscrpts to ndcate tem type and superscrpts to denote order type. For any and K, let overall demand rate, q K probablty a demand s of type K K demand rate of demand type K q K, q probablty a demand requres tem aggregate demand rate of tem q K 1, K K:K q K, K:K K q. Shpment and Backloggng Demands are flled on a Frst-Come-Frst-Serve (FCFS) bass. When an order arrves and we have some, but not all, of ts tems n stock we wll ether shp the n-stock tems f partal shpment s allowed or put asde these tems as commtted nventory f partal shpment s not allowed. However, a customer request s consdered backlogged unless t can be satsfed completely. A demand for tem that cannot be flled mmedately s queued n the backlog queue, whch has a capacty b 0, and wll be shpped out (or put asde) as soon as a unt of tem becomes avalable to fll t. (When b for all, all unflled demands are backlogged. When b 0 for all, unflled demands are lost.) We consder two knds of lost sales when an ncomng order that requests more than one SONG, XU, AND LIU / 133 tem fnds the backlog queue for at least one of ts tems s full. Suppose K 1. a. Total order servce (TOS): If a type K order sees at least one of ts tem s backlog queue s full, then the order s lost entrely. In other words, a type K order must be accepted as a whole. Ths model s vald for the assembleto-order envronment at the manufacturng level and also for some mult-tem make-to-stock systems. b. Partal order servce (POS): When a type K order arrves and the backlog queue s full for K K, then the order for tems n K s lost, whereas the order for tems n K K s satsfed, ether mmedately or n the future. Ths model fts most assemble-to-order envronments at the dstrbuton level, where customers often accept partal shpments of fnshed goods. Thus, n the POS model, customer mpatence s assocated wth ndvdual tems, whle n the TOS model t s assocated wth the whole order. The two models are dentcal n the complete backloggng case. Replenshment Polcy We assume that there s no economy of scale n replenshment. Each tem s controlled by an ndependent basestock polcy. Let s the base-stock level for tem. That s, at each demand epoch, f the nventory poston (.e., the nventory on hand plus nventory on order mnus backorders) of tem s less than s, then order up to s. Snce there s no economy of scale n replenshment, a base-stock polcy would be optmal for each tem f we were to manage the system on an tem bass and complete backloggng was assumed. Because of ts smplcty, we employ ths type of polcy as a reasonable heurstc. Wthout loss of generalty, we assume that at tme 0 tem s stocked at level s for all. Then, each demand for tem trggers an order for that tem untl a demand fnds the backlog queue s full. Hence, there can be, at most, s b outstandng orders of tem. The Supply System Replenshment orders for tem are sent to a snglemachne producton faclty, say faclty, n whch they are processed on a FCFS bass. The processng tmes at faclty are..d. exponentally dstrbuted random varables wth rate, 1, 2,..., J. Thus our supply system can be vewed as J parallel stochastc producton facltes, where faclty accepts Posson nputs wth rate and contans at most s b outstandng orders at any tme. Performance Measures We are nterested n three levels of performance measures: tem-based, order-based, and system-based performance measures. A. Performance Measures of Item,. We frst defne the tem-based performance measures that are common n

4 134 / SONG, XU, AND LIU the standard sngle-tem nventory models. Denote the followng quanttes n steady-state: I nventory on hand of tem, 0 I s, B backorders of tem, 0 B b, IO Inventory on order of tem, 0 IO s b. We shall derve the expectatons of the above measures. The fll rate of tem s denoted by F the probablty of mmedately satsfyng a demand for tem PI 0. In contrast to the fll rate, whch measures the mmedate avalablty, the servce level of an tem measures the servceablty of the tem: SL fracton of tem demands satsfed PB b. Evdently, F SL. When b 0 (the lost sale case), the two measures are dentcal. When b (the complete backloggng case), SL 1 as no demand of tem s ever lost n ths case. Now let W watng tme of an accepted request for tem. We are nterested n dervng the dstrbuton of W, whch allows one to fnd the probablty of fllng an tem n a specfed tme wndow. B. Performance Measures of a Type-K Order, K. The fll rate, servce level and watng tme of a type-k order are defned, respectvely, by F K jont probablty that all tems n a type-k order are flled mmedately PI 0 K, SL K jont probablty that all tems n a type-k order are accepted PB b K, W K watng tme to fll all tems n a type-k order max W. K Because of smultaneous demands, the components n the random vector {I, K} ({B, K}, {W, K}) are dependent. Consequently, one cannot obtan the performances of a type-k order from ts tem-based counterparts. Indeed, ths presents the major challenge of our analyss. C. Performance Measures of the System. Let F, SL and W be the fll rate, servce level, and watng tme of an arbtrary flled demand (regardless of the type). Then F q K F K, (1) K SL q K SL K, (2) K PW x q K PW K x. (3) K Because system-based performance measures are readly computable once performance measures of all type-k orders are obtaned, n the rest of the paper we are manly concerned wth the tem-based and order-based performance measures. 2. THE TOTAL ORDER SERVICE MODEL In ths secton, we focus on assembly systems. In partcular, we assume that an order s ether servced entrely or rejected entrely (total order servce). A type-k demand s lost f, upon ts arrval, at least one backlog queue s full, K. It s worth notng that ths feature causes the tembased performance measures to depend on the orderbased performance measures. Followng the standard argument n nventory models, t s easy to see that the steady-state net nventory of tem s gven by s IO. Thus, the type-k performance measures are determned by the jont dstrbuton of (IO 1, IO 2,..., IO J ). In ths secton, we frst develop a procedure to compute the jont dstrbuton of (IO 1, IO 2,..., IO J ). Then, usng ths dstrbuton, we derve the order-based performance measures for each fxed K, whch n turn s employed to obtan the tem-based performance measures The Jont Statonary Dstrbuton of Outstandng Orders Observe that IO {(IO 1 (t), IO 2 (t),..., IO J (t)), t 0}, where IO (t) s the nventory on order of tem at tme t, s a contnuous-tme Markov chan wth fnte state space {n (n 1, n 2,...,n J )0 n N,1 J} where N s b. A state transton can only occur f a demand arrves or an tem of an order s released. Specfcally, wth transton rate K, state n enters the state n (n 1,..., n J ), where n j n j 1, f j K and n N for all K, (4) n j, otherwse, and wth transton rate, state n enters the state n (n 1,...,n J ), where n j n j 1, f j and n j 0, (5) n j, otherwse. It s easy to see that ths Markov chan s rreducble, so ts statonary dstrbuton exsts unquely. Usng (4) and (5), one can establsh the balance equaton for each state J (there are j1 (N j 1) of them). The statonary jont dstrbuton s then the soluton of these balance equatons plus a normalzaton equaton, and conventonal computatonal methods for solvng lnear equatons can be employed. However, we fnd that the specal structure of the process allows us to obtan a unfed, matrx-geometrc soluton (defned n Neuts 1981) of the statonary dstrbuton of IO, as explaned below. Let us order the state space of IO n the lexcographc order (l 0, l 1,..., l N1 ), where l k s the collecton of the states when the nventory on order of tem 1 s k. That s, J l k s the j2 (N j 1) dmensonal vector

5 l k k, 0,...,0,0,...,k, 0,...,N J, k, 0,...,1,0,...k, 0,...,1,N J,..., k, 0,...,N J1,0,...,k, 0,...,N J1, N J,...,k, N 2,...,N J1,0,..., k, N 2,...,N J1, N J. Under the orderng (l 0, l 1,...,l N1 ), IO can be vewed as a two-dmensonal Markov chan, wth the szes of the frst J and second dmensons beng N 1 1 and j2 (N j 1), respectvely. Let p k be the statonary probabltes of those states n l k. Then one can convenently represent the statonary dstrbuton of IO by p p 0, p 1,...p N1. Snce IO 1 n a sngle transton can ncrease (decrease) by at most 1, IO can be regarded as a quas brth-and-death (QBD) process (Neuts 1981, Chapter 3). It s straghtforward to show that the nfntesmal generator of the chan IO s a block-trdagonal matrx A A I A 1 I A I A 1 I A Q I A 1 I A I A 1 1 I, J where A, A 0 and A 1 are square matrces of order j2 (N j 1). (The matrces A and A 0 can be constructed accordng to (4) and (5). Generally speakng, they are qute sparse, especally f there are only a few dentfable demand types,.e., q K 0 for only a few Ks. To avod heavy notaton, we omt the detals of these two matrces. We wll provde detals for the 2-tem system n Secton 4.) Therefore, we can express the steady-state balance equatons n the matrx form: p 0 A 1 p 1 0, p k1 A 0 p k A 1 I 1 p k1 0, 1 k N 1, p N1 1 A 0 p N1 A 1 1 I 0. The above balance equatons then lead to: p N1 p N1 1 A 0 A 1 1 I 1 p N1 1 R N1, p k p k1 R k, k 1, 2,..., N 1 1, where R k can be found recursvely usng R N1 A 0 A 1 1 I 1, R k A 0 A 1 I 1 R k1 1, k N 1 1,...,1; R 0 I. Now, p 0 satsfes p 0 A 1 R 1 0, and the normalzaton equaton N 1 N 1 p k e p 0 Rˆ k e 1, k0 k0 where e s column vector of ones, and k Rˆ k R k, k 0, 1,..., N 1. 0 Remark 2.1. In terms of computatonal complexty, the J drect approach solves j1 (N j 1) lnear equatons, whch by usng Gaussan elmnaton requres J j1 N j 1 3 J /3 j1 N j 1 2 operatons (multplcatons; we gnore addtons). See Isaacson and Keller (1966). The matrx-geometrc soluton approach, on the other hand, nvolves N 1 1 nversons J and 2N 1 multplcatons of matrces of order j2 (N j 1), whch results n J 3N 1 1 j2 SONG, XU, AND LIU / N j 1 3 J j2 N j operatons. Thus, the matrx-geometrc approach s faster than the drect approach by a factor of approxmately (N 1 1) 2 /9. In fact, snce any IO can serve as the frst dmenson of the QBD process, we can renumber the tem ndces f necessary so that N 1 max N and, consequently, accelerate the computaton by a factor of approxmately (max N 1) 2 /9. In addton, as a QBD process s solvable as long as the second dmenson of the process s fnte, our approach can be appled to obtan the exact analyss for the system wth one of the tem completely backlogged. Ths would correspond to the practcal stuaton n whch a company wshes to capture 100% demands of one of ts key products. Ths approach s also capable of provdng an approxmate soluton to the system wth more than one tem completely backlogged (see Secton 4.2 for a dscusson on buffer sze truncaton and for numercal results). In most practcal stuatons (e.g., wth moderate traffc ntensty), one can replace an nfnte backlog queue buffer wth a fnte one of rather moderate sze wthout sacrfcng computatonal accuracy. For large-scale systems, we propose the followng: () Although the total number of potental demand types can be large, by the Pareto phenomenon, often a large porton of the total dollar volume of sales s accounted for by a small number of demand types. When ths s true, one can concentrate only on these few demand types. Consequently, the system of balance equatons wll be relatvely easy to solve. () Sometmes one can partton {1, 2,..., J} nto several dsjont sets such that ther assocated IOs n dfferent sets are ether ndependent or weakly dependent. Ths wll be the case f the total arrval rate of mult-tem orders belongng to dfferent sets s small. Then one can use those dsjont sets to partton the chan IO nto several smallerdmensonal chans and derve ther solutons separately.

6 136 / SONG, XU, AND LIU 2.2. Order Fll Rates and Servce Levels The order fll rates and the servce levels can be calculated drectly from the statonary probablty vector p. In partcular, the type-k order fll rate s F K PI j 0j K PIO j s j j K, (6) and the servce level of type-k orders s SL K PB j b j j K PIO j s j b j j K. (7) 2.3. Type-K Watng Tme Dstrbuton Recall that W K represents the watng tme of a type-k order that s accepted as a whole. We assume that the jont dstrbuton of IO K : {IO j j K} s known va the jont dstrbuton of IO (IO j, j ). The basc dea of fndng the dstrbuton of W K s to condton on the state that a type-k demand observes upon ts arrval, IO K n K : (n j, j K). Frst note that a type-k order wll be flled only f none of the backlog queues, K, s full (.e., IO s b, K) when the order arrves, so we shall focus only on ths set of states, namely, C K n K n s b, K. Defne p n K PIO K n K n K C K. (8) That s, p (n K ) s the probablty that an accepted type-k order observes the system n state n K. For any subset of K, say L K, we defne C K L : n K C K s n s b, 0 n j s j, L, j K L}. In other words, C K (L) C K s the collecton of states such that tem j, j K L, of a type-k order wll be flled mmedately, whereas tem, L, of the order wll jon the backlog queue. In partcular, let 0 be the empty set. Then C K (0) s the set of states n whch all tems of a type-k order wll be flled mmedately. Condtonng on IO K n K, we obtan PW K x PW K xio K n K p n K n K C K 0 LK PW K xio K n K p n K. n K C K L (9) Snce W K 0 when n K C K (0), one has PW K xio K n K 1, n K C K 0. (10) On the other hand, f n K C K (L), the fll tme of the type-k order wll be the tme to fll all tems n set L: W K IO K n K max L W IO K n K, n K C K L. (11) Observe that, wth n K C K (L) and L, f a new type demand observes state IO n, s n s b, then there wll be n s orders n the backlog queue. Thus the new demand wll become the (n s 1)st backlogged order n backlog queue whose watng tme s just the sum of n s 1 exponental random varables wth rate, whch has an Erlang-(n s 1, ) dstrbuton. Moreover, the watng tmes of the tems n L are condtonally ndependent. Lettng G n (x) be the cumulatve dstrbuton functon of an Erlang-(n, ) random varable, we get PW K xio K n K Pmax W xio K n K L (12) G n s 1 x, n L K C K L, where n1 G n x 1 x k e x. (13) k0 k! Substtutng (10) and (12) nto (9) yelds PW K x p n K n K C K 0 LK G n s 1 n K C K L L x p n K. (14) Sometmes the prmary nterest s to fnd the mean watng tme of a type-k demand, E[W K ], not ts dstrbuton. Here s a smpler procedure for ths purpose: EW K LK LK EW K IO K n K p n K n K C K L n K C K L EV L n K p n K, (15) where V L (n K ) max L {V (n s 1)}, and V (n), K, are ndependent Erlang-(n, ) random varables wth V (0) 0. Notce that V L n K e V L n K e,, n K C K L, n K e C K L, (16) where e s the th unt vector. Usng (16) and the boundary condton EV n K n s 1, K, (17) one can compute the mean of V L (n K ) from the recursve equaton EV L n K 1 L L jl j EV L n K e. (18) Let B K denote the type-k backorders, that s, the number of type-k demands that have been accepted but have not been flled. Then, by Lttle s law, EB K PC K K EW K. (19) Remark 2.2. In fact, W K, K, follows a phase-type dstrbuton (see Neuts 1981). In other words, W K can be vewed as the tme untl absorpton n an absorbng, contnuous-tme Markov chan wth the ntal probablty vector (, m1 ) and the nfntesmal generator

7 T Te 0 0. We now specfy ts parameters and T. Let us arrange all the states n C K C K (0) n a lexcographc order as an m dmensonal vector, where m K (s b ) K s, the total number of states n C K C K (0). Let be the th elements of, whose value s p (n K ), for some n K C K C K (0). Let m1 n K C K (0) p (n K ). Then (, m1 ) wll be the ntal probablty. Here, the state m 1 s the absorbng state at whch all tems of a type-k order are flled. Let T n K, m K be an element of the matrx T. Then, f nk C K (L), L K, the Markov chan leaves n K wth rate L. The chan transts to state n K e wth probablty / jl j, L, correspondng to machne completes one unt of tem. Therefore, for any L K, T n K,n K e, n K C K L, L, T n K,n K, n K C K L, L. L Wth and T specfed as above, the dstrbuton of W K and ts expectaton E[W K ] can be expressed as PW K x 1 e Tx e, x 0, (20) where Tx e Tx. 0! The mean watng tme s EW T 1 e. (21) Although relatng W K to a phase-type dstrbuton may help the reader to gan nsght n the nature of ts dstrbuton, t seems computatonally more convenent to use (14) nstead of (20) to evaluate the dstrbuton of W K Item-Based Performance Measures For the later purpose of comparson between the total and partal order servce models, we now derve the tem-based performance measures. Let p (n) be the margnal dstrbuton of IO, 1, 2,..., J, whch can be readly derved va the jont dstrbuton of IO. Snce I s IO, B IO s, one can also fnd the dstrbuton and expectaton of I and B, but we omt the detals here. To avod confuson, we remark that the probabltes s 1 PIO s p n n0 and s b 1 PIO s b p n n0 SONG, XU, AND LIU / 137 are not the fll rate and servce level of tem demands, whch consst of demands for tem from all type-k orders such that K. In other words, F F and SL SL.In fact, t can be seen that F K PIO s, IO j s j b j, j K, j, KK SL K SL K. KK Next we derve the expressons of the dstrbuton and expectaton of W, 1, 2,..., J. To do so, we need to know the dstrbuton of W K, the watng tme of an tem demand n an accepted type-k order, K. Let p K n PIO n, IO K N K, PIO K N K 0, 1,..., N 1, where N s b and N K {N, K}. In words, p K (n ) s the probablty that the tem demand n a type-k order observes IO n, provded the type-k order s accepted. If the tem demand fnds that IO s, then ts watng tme s zero. If the tem demand fnds IO n, s n N, ts watng tme s an Erlang-(, n s 1) random varable. We thus obtan s 1 N 1 PW K x p K n G ns 1 x p K n, K. n0 ns The expected watng tme of the tem demand n an accepted type-k order satsfes p K n. N 1 EW K ns n s 1 Now, PW x K PW K x KK K KK N 1 ns s 1 p K n n0 G ns 1 x p K n, 1, 2,..., J, where G n (x) satsfes (13). Fnally EW K KK EW K. 3. THE PARTIAL ORDER SERVICE MODEL In ths secton we consder the assemble-to-order envronment at the dstrbuton level, where a customer order typcally requests several tems, but the partal fulfllment of an order s allowed. That s, f an order fnds only part of ts tems backlog queues not full, t wll stay n the system untl these tems have been suppled, and the rest

8 138 / SONG, XU, AND LIU of the order wll be left unflled. We derve both the tembased and the order-based performance measures Item-Based Performance Measures The partal order servce feature mples that a type demand wll be served as long as, upon ts arrval, the outstandng orders of tem s less than s b, ndependent of the status of other facltes. Thus the margnal dstrbuton of IO can be derved wthout the knowledge of the jont dstrbuton of IO. But IO s just the steady-state occupancy n an M/M/1/s b system wth arrval rate and servce rate. Accordng to the standard queueng theory, IO follows a truncated geometrc dstrbuton. That s, PIO n 1 n 1, n 0, 1, 2,..., s s b 1 b, (22) where /. From (22), t s straghtforward to obtan for tem the average number of backorders and the average nventory on hand: EB EIO s s 1 b 1 b 1 b b 1 s 1 1 b, (23) 1 EI Es IO s s 1 1 s 1 1 b. (24) 1 The fll rate and the servce level of tem can be expressed as F PI 0 PIO s 1 s 1 s b 1, (25) SL PB b PIO s b 1 s b 1. s b 1 (26) As one would expect, SL F, that s, the tem servce level s no smaller than the tem fll rate. Usng Lttle s law, we obtan the expected watng tme of an accepted request for tem : EW EB s 1 b 1 b 1 b b 1 SL s 1 1 b. (27) Clearly, wth SL fxed (.e., s b constant), the performance measures such as F, E[B ] and E[W ], wll mprove as the base-stock level s ncreases. However, the mproved customer servce comes at the expense of the hgher average nventory level E[I ]. Next, let us look at the dstrbuton of W,.e., the tme to fll an accepted demand for tem. In fact, ths s a specal case of Secton 2.3 by settng K {}. That s, the dstrbuton of W can be obtaned by condtonng on the number of outstandng orders n faclty upon the arrval of an accepted demand for tem. In partcular, as n (8), defne p n : PIO nio s b 1 n s 1 b, 0, 1,..., s b 1, (28) whch s the dstrbuton of the occupancy n an M/M/1/ s b 1 system. If an accepted demand for tem fnds postve nventory on hand, then the watng tme of the demand s zero. Thus PW 0 PIO s IO s b s 1 p n 1 s. (29) s n0 1 b For x 0, we condton on the number of outstandng orders of type and get PW x PW 0 s b 1 P0 W xio n p n. (30) ns As we argued before, gven IO n, the watng tme of the new type demand has an Erlang-(n s 1, ) dstrbuton. From (28) (30), we get PW x 1 s s 1 b s b 1 ns where G n (x) s gven by (13). G ns 1 x 1 n, x 0, s 1 b (31) Remark 3.1. Recall, b,, corresponds to the complete backloggng model, whle b 0,, corresponds to the lost sales model. Snce these are specal cases of our model and are of sgnfcant nterest n practce, they are worth some dscussons here. Complete Backloggng. Under the partal servce assumpton, the tem-based performance measures are well known n the standard nventory lterature (see Zpkn 1997, Secton 7.3.2). In partcular, IO now follows a geometrc dstrbuton: PIO n 1 n, n 0, 1,..., whch s just the lmt of (22) as b 3. Correspondngly, the performance measures for tem are the lmts (as b 3 ) of ther counterparts n the fnte backlog case. It s easy to show that the expressons n (23) and (24) are, respectvely, ncreasng and decreasng n b. Therefore, the complete backloggng scheme results n a greater average tem-backorders and henceforth a greater average watng tme, but a smaller average nventory on hand. In addton, snce (25) s a decreasng functon of b, the fll rate of tem n the complete backlog case s smaller than that n the fnte backlog case. But SL s ncreasng n b.

9 Lost Sales. Agan, the tem-based performance measures are well known n the nventory lterature (Zpkn 1999, Secton 7.3.4). In partcular, the servce level and the fll rate n ths case are dentcal. In addton, the tem fll rate and average nventory on hand are greater than ther counterparts n the (fnte and nfnte) backlog case, as F and E[I ] are decreasng functons of b Order-Based Performance Measures Obvously, for orders that contan only one tem, the order-based performances are just the correspondng tembased performances. Thus, we need to focus only on the order types that contan more than one tem. Agan, these performance measures depend on the jont dstrbuton of SONG, XU, AND LIU / 139 tem 1 only, a type-2 customer requres only one unt of tem 2, and a type-12 customer asks for one unt of each tem. The smplcty of ths system allows us to better llustrate the results n Sectons 2 and 3, and the numercal examples help us gan more nsghts from the model The Performance Measures Clearly, for each model t suffces to get the jont dstrbuton p of (IO 1, IO 2 ). Ths, n turn, mples that we only need to specfy the square matrces A and A 0 (both wth dmenson N 2 1), n the nfntesmal generator of IO. The Total Order Servce Model. Ths s the system studed n Secton 2. In ths case, A , (IO 1, IO 2,..., IO J ). In fact, the basc structure of the nfntesmal generator of the chan IO here s exactly the same as that n Secton 2. The dfferences occur only n certan elements n the matrces A, A 0 and A 1. In partcular, the transton (5) remans the same, but the transton (4) now becomes n j n j 1, f j K and n j N j, (32) n j, otherwse. As a result, A 1 A A 0, and all the procedures and the expressons of the order-k performance measures reman vald here. Wth slght modfcaton, the approach of subsecton 2.3 can also be used to derve the watng tme dstrbuton of a partally accepted order or a subset of an accepted order. Ths wll be of nterest f a company needs to consder ts delvery schedule for some key products n an order. For example, suppose for L K, one wshes to know the dstrbuton of W K(L), the watng tme to delver all tems n L, n a type K order, gven that all tems n L are accepted. Then, followng the smlar arguments that lead to (14), t can be shown that PW KL x n L C L 0 p n L LL n L C L L L G n s 1 x p n L, L K. (33) 4. THE TWO-ITEM SYSTEM In ths secton we study the two-tem system n detal. Here, there are only three possble demand types, S {{1}, {2}, {1, 2}}. A type-1 customer requres one unt of and A Let I (N2 1) be an order (N 2 1) matrx of zeros except the last dagonal entry s 1. Then, A 1 A 12 I (N2 1) 1 I. Wth these matrces specfed, one can then follow the procedure descrbed n subsecton 2.1 to compute p, whch nvolves only matrx products and nverses. The Partal Order Servce Model. Ths s the system dscussed n Secton 3. Clearly, the type- performance measures are dentcal to the tem-based measures gven n Secton 3.1. To compute the statonary dstrbuton p, we need only to replace the value 1 by 1 n the last dagonal entres of the matrces A and A 0, and set A 1 A A Numercal Examples In ths secton we present the results of the numercal experments and dscuss our key observatons. Our goals are three-fold: 1. To nvestgate the effects of varous system parameters on order-based performances. These parameters nclude the polcy parameters (s (s 1, s 2 ) and N (N 1, N 2 )) and the envronmental parameters ( ( 1, 2 ), q (q 1, q 2, q 12 )). Ths knd of nformaton wll help us gan nsght nto how the system works. For example, how does the backlog queue capacty, traffc ntensty, and demand

10 140 / SONG, XU, AND LIU Fgure 1a. Fll rates of TOS and POS models (symmetrc cases).

11 SONG, XU, AND LIU / 141 Fgure 1b. Fll rates of TOS and POS models (asymmetrc cases). correlaton affect the system performance? Do the orderbased performances respond to the changes n the parameters smlarly to the tem-based performances? 2. To compare the outputs of the TOS model and the POS model under the same set of parameters. The purpose here s to learn under what condtons the two models behave smlarly, and whether one model can be used to approxmate or bound the other. 3. To understand the effect of the occupancy capacty N on servce levels. Ths wll shed lght on the effectveness of usng a fnte-buffer model to approxmate ts nfntebuffer counterpart. The fndngs, obvously, wll have mportant computatonal mplcatons. For all the experments, we fxed 1 2 (1 q 12 ) 9. There are another four parameter vectors that determne the system performance (ths explans the system complexty): Y Demand correlaton vector q: We chose three confguratons for ths vector: (0.4, 0.4, 0.2), (0.33, 0.17, 0.5), and (0.1, 0.1, 0.8), correspondng to the systems wth

12 142 / SONG, XU, AND LIU Fgure 2a. Servce levels of TOS and POS models (symmetrc cases).

13 SONG, XU, AND LIU / 143 Fgure 2b. Servce levels of TOS and POS models (asymmetrc cases). symmetrc and less correlated demands, asymmetrc and moderately correlated demands, and symmetrc and hghly correlated demands, respectvely. Y Traffc ntensty vector : We chose the producton rates such that takes values (0.5, 0.5), (0.9, 0.5) and (0.9, 0.9), correspondng to the systems wth symmetrc and moderate workload, asymmetrc workload, and symmetrc and heavy workload, respectvely. Y Base-stock vector s and producton buffer capacty vector N: For the symmetrc cases, we selected the followng confguratons of (s, N): (4, 4, 4, 4) (the lost sale case), (4, 4, 7, 7) and (4, 4, 8, 8). For the asymmetrc cases, we let (s, N) be (3, 5, 3, 5) (the lost sale case), (3, 5, 5, 9), and (3, 5, 6, 10). In the graphs, q1 and q12 represents q 1 and q 12, respectvely, whle F_1 means F 1, and all other notatons are smlarly defned. Fgures 1 through 4 depct the results of fll rates, servce levels, expected watng tmes and watng tme dstrbutons, respectvely, of TOS and POS models under varous parameter confguratons. Fgure 5 presents the system-based performance comparsons of TOS and POS models.

14 144 / SONG, XU, AND LIU Fgure 3a. Expected watng tmes of TOS and POS models (symmetrc cases). For each performance measure (say, fll rates), we dvde the graphs nto two subgroups: symmetrc cases (e.g., Fgure 1a) and asymmetrc cases (e.g., Fgure 1b). In each group, the horzontal pars compare the performances of the dfferent models (.e., TOS vs. POS) under the same parameter confguraton, whle the vertcal subgroups compare the performances of the same model under the dfferent parameter confguratons. The followng summarzes the key observatons and ther mplcatons. The Effects of the System Parameters. In general, we found that, n both models, the order-based performance measures respond to the parameter changes n a smlar fashon as ther tem-based counterparts Fgure 1 groups (Fgure 2 groups) show that tembased, order-based, and system-based fll rates (servce levels) are all decreasng (ncreasng) n N, for fxed s, but ncreasng (decreasng) n s, for fxed N (not reported here). These observatons are not surprsng, because larger base-stock levels mprove order-fulfllment performance Fgure 3b. Expected watng tmes of TOS and POS models (asymmetrc cases).

15 for accepted orders, whle larger buffer szes mprove the servceablty and therefore ncrease the number of accepted orders We found that the traffc ntensty has a pronounced adverse effect on each performance (see Fgures 1a-1 1a-8, 2a-1 2a-8, 3a-1 3a-4). Also, as the traffc ntensty ncreases, the tem-based performance provdes poorer nference for the order-based performance. For example, under moderate traffc, the tem-based bound mn {F 1, F 2 } provdes a reasonable estmaton for F 12 (see Fgures 1a- 5 1a-8, 1b-1 1b-6); but t s a very poor ndcaton of F 12 under heavy traffc, especally when N s large (see Fgures 1a-1 1a-4). For the asymmetrc cases, one should also observe the bottleneck effect of the heavy traffc faclty on the performance of type-12 orders (see Fgures 1b 4b). Ths ndcates the mportance of balanced workloads on the performance of assemble-to-order systems. Further, the traffc ntensty also affects the shape of the watng tme dstrbutons (see Fgure 4) As for the mpact of demand correlaton, we observed the followng. Frst, whle q 12 has no effect on the tem-based performances for the POS models, t does affect the tem-based performances of the TOS models. Fgures 1 3 show that as demands become more correlated, the tem-based performances of TOS deterorate, though mldly. Second, n both models one sees that as q 12 ncreases, F K, SL K, and P(W K x) ncrease and E[W K ] decreases (see Fgures 1 4, especally Fgure 3). In fact, Xu (1999) shows analytcally that ths property holds for mult-tem POS systems. Ths scenaro may be explaned by the fact that, as q 12 ncreases, the correlaton level of nventory (backlog queue occupancy) of dfferent tems ncreases. Ths, n turn, ncreases the chance that both tems are avalable (nether backlog queue s full) when a type-12 order arrves. Our graphs also show that whle the adverse effect of large q 12 on F, SL, and E[W ], 1, 2, s rather mld, t has a severe mpact on F 12 n general and on SL 12 and E[W 12 ] n heavy traffc. Thrd, the system-based fll rate F, servce level SL, and watng tme E[W] deterorate as q 12 ncreases, most notceably for large N. In summary, t appears that demand correlaton mproves the order-based performance, whereas t worsens the system-based and tem-based (for TOS models) performances. Also, the demand correlaton has ts greatest mpact n heavy traffc. Model Comparson. Comparng the results of the POS and the TOS models (see horzontal pars n Fgures 1 5), we made the followng observatons and nterpretatons: 2.1. Snce an order s accepted on the tem bass n POS, ts tem-based fll rates and servce levels are hgher than ther counterparts n TOS. As a result, the POS system has a hgher congeston level than that of the TOS system, wth the same parameter settng. The hgh congeston level n POS, however, has an adverse effect on ts order fll rates and servce level, as t decreases the lkelhood that an order s flled mmedately or eventually as a whole. Thus, POS has lower order-based and system-based fll rates and servce levels, as compared to TOS (e.g., see Fgures 5a-1 5a-2). Numercal results show that the dfference between the order fll rates (servce levels) of the two systems ncreases f q 12 or ncreases. However, the performance of the two systems are very close over a large set of parameter settngs we tested Because POS has a hgher congeston level than TOS, the tem-based, order-based, and system-based watng tmes n POS are stochastcally larger than ther counterparts n TOS. Ths s evdent from Fgures 5a-4 and 5b Clearly, for fxed q 12 and traffc ntensty vector, F K and SL K n POS are not affected by q 1 /q 2. However, F K and SL K n TOS vary as q 1 /q 2 vares. Numercal results show that for balanced faclty utlzatons ( 1 / 2 1), balanced arrval rates (q 1 q 2 ) yeld the hghest order fll rate and servce level; For unbalanced faclty utlzatons ( 1 / 2 1), the hgher fll rate and servce level are reached when q 1 q 2 takes a value close to 1 / In general, both models demonstrate the same qualtatve behavor, and the results from the POS model provdes rather relable estmates for the performance measures n the TOS model for a broad range of parameter settngs (see Fgure 5). The Impact of Fnte Buffer Szes. Fgures 2a-5 2a-8 show that under a moderate traffc ntensty, the system wth small backlog queue capactes (n our cases, b 1 b ) can acheve almost 100% servce levels, thus ts performance (ncludng fll rates, watng tme dstrbutons, etc.) provdes an accurate approxmaton for ts counterpart n the nfnte queue system. Numercal experments show that such results hold for a wde range of parameter confguratons, as long as the traffc ntensty s moderate. To llustrate the computatonal effcency of our procedure for large-scale systems.e., systems wth hgh traffc ntenstes and large backlog queue capactes we provde two examples. Example 1. Consder a symmetrc POS system wth (heavy traffc), q 1 q 2 0.1, q (hgh demand correlaton), s 1 s 2 8, and complete backloggng. Suppose that the desred approxmate accuracy s at least 99%. To do so, we set SL q 1 SL 1 q 2 SL 2 q 12 SL , (34) so that less than 1% of orders are lost. Usng the fact that (IO 1, IO 2 ) s postvely quadrant dependent (Xu 1999), the followng nequalty holds: SL 12 PIO 1 s 1 b 1, IO 2 s 2 b 2 PIO 1 s 1 b 1 PIO 2 s 2 b 2 SL 1 SL 2. (35) To ensure (34), t s suffcent to fnd s 1 b 1 s 2 b 2 such that SL 1 SL 2 1 s1 b1 1 s 1 1 b SONG, XU, AND LIU / 145

16 146 / SONG, XU, AND LIU Fgure 4a. Watng tme dstrbutons of POS and TOS models (symmetrc cases). Wth and s 1 s 2 8, the above nequalty holds when b 1 b Therefore, even under heavy traffc, moderate backlog queue capactes (b 1 b 2 21) wll be capable of provdng an accurate approxmaton (wth the servce level more than 99%) to the system wth complete backloggng. For ths specfc problem, the matrx-geometrc approach s roughly (N 1 1) 2 / / tmes faster than the drect approach (see Remark 2.1). Fnally, we note that the above result s also vald for the TOS system, because our numercal examples ndcate that the servce level of a TOS system s bounded below by ts counterpart of a POS system (see Fgures 5a-2 and 5b-2). Example 2. Consder an asymmetrc POS system wth parameter settngs smlar as n Example 1, except 1 0.9, 2 0.5, and s 1 4, s 2 8. Agan, the desred approxmate accuracy s more than 99%. Followng the argument of Example 1, f we let b 1 21 (or N 1 s 1 b 1 29), b 2 3 (or N 2 s 2 b 2 7), then SL , SL , SL 12 (SL 1 )(SL 2 ) 0.991, and hence SL Agan, the matrx-geometrc soluton s roughly 100 tmes faster than the drect approach. The above examples llustrate that, n most cases, computatonal complexty s domnated by the traffc ntensty, rather than the buffer sze constrant. Therefore, t s safe to say that the fnte buffer assumpton n our soluton procedure does not lmt ts ablty to solve the complete backloggng model that has a small J. 5. CONCLUDING REMARKS We presented a model of assemble-to-order producton/ nventory systems that ncludes stochastc processng tmes. The motvaton of buldng such model was due to the prevalence of stochastc supply tmes n the computer and semconductor ndustres, where assemble-to-order manufacturng has become a common practce. The model was further talored nto two models, one wth total order servce and the other wth partal order servce. Exact procedures were developed to evaluate the order-fulfllment performance measures that are of ncreasng mportance to managers. The procedures were llustrated by a twotem system. Numercal examples were presented and ther mplcatons were dscussed. We beleve that our study s the frst exact analyss of ths knd. As such, for smplcty, we have made some restrctve assumptons, such as Posson demand and exponental processng tmes. Whle these smplfed assumptons are necessary to begn wth, t s our hope that our study wll nspre some other research efforts for alternatve models and soluton procedures. (The methodology employed n ths paper s applcable n prncple to models wth phasetype processng tme dstrbutons, but ths can only add to the computatonal and notatonal burden of the model wthout provdng addtonal nsghts.) As a result of multdmensonal Markov chans, t s not surprsng to note that the soluton procedure provded n ths paper requres consderable computatonal effort for

17 SONG, XU, AND LIU / 147 Fgure 4b. Watng tme dstrbutons of POS and TOS models (asymmetrc cases). performance evaluaton of large-scale systems (.e., wth large J and heavy traffc ntenstes). Needless to say, effcent approxmatons and heurstc procedures are n demand. (An mportant contrbuton of the current model s to provde a benchmark for testng these approxmatons.) Ths, n fact, s one of our ongong research projects: Snce tem-based performance measures are much easer to obtan, we ntend to develop certan bounds for the orderbased performance measures, and these bounds nvolve only the tem-based nformaton. To develop such bounds, structural studes wll be carred out. Another queston that nterests us s: What s the mpact of the standard ndependent-demand assumpton when the demands across tems are actually correlated? Ths lne of study wll reveal to what extent we can trust such crude assumptons and what s the value of dentfyng demand types.

18 148 / SONG, XU, AND LIU Fgure 5a. System-based performance comparsons of POS and TOS models (symmetrc cases). Fgure 5b. System-based performance comparsons of POS and TOS models (asymmetrc cases).

19 ACKNOWLEDGMENT Research of the frst author was supported n part by NSF grant DMI REFERENCES Agrawal, N., M. Cohen Optmal materal control and performance evaluaton n an assembly envronment wth component commonalty. Workng Paper. Decson and Informaton Scences Department, Santa Clara Unversty, Santa Clara, CA. Anupnd, R., S. Tayur Managng stochastc multproduct systems: model, measures, and analyss. Oper. Res. 46, Cheung, K-L., W. Hausman Multple falures n a mult-tem spare nventory model. IIE Trans. 27, Cohen, M., H. Lee Out of touch wth customer needs? Spare parts and after sales servce. Sloan Management Rev. (Wnter) Flatto, L., S. Hahn Two parallel queues created by arrvals wth two demands I. SIAM J. Appl. Math. 44, Glasserman, P., Y. Wang Leadtme-nventory trade-offs n assemble-to-order systems. Oper. Res. 46(6). Hausman, W., H. Lee, A. Zhang Jont demand fulfllment probablty n a mult-tem nventory Order response tme relablty n a mult-tem nventory system wth ndependent order-up-to polces. European J. Oper. Res. 109, SONG, XU, AND LIU / 149 Isaacson, E., H. Keller Analyss of Numercal Methods. Wley, New York. Lee, H., and C. Bllngton Materal management n decentralzed supply chans. Oper. Res. 41, L, L The role of nventory n delvery-tme competton. Management Sc. 38, Neuts, M Matrx Geometrc Solutons n Stochastc Models. Johns Hopkns Unversty Press, Baltmore, MD. Shraner, E Capacty/nventory trade-offs n assembleto-order systems. Workng Paper. IBM Watson Research Center, Yorktown Heghts, NY. Shwartz, A., A. Wess Induced rare events: analyss va large devatons and tme reversal. Adv. Appl. Prob. 25, Song, J On the order fll rate n a mult-tem, base-stock nventory system. Oper. Res. 46(6). Srnvasan, R., R. Jayaraman, R. Roundy, S. Tayur Procurement of common components n a stochastc envronment. Research Report. IBM Watson Research Center, Yorktown Heghts, NY. Wrght, P Two parallel processors wth coupled nputs. Adv. Appl. Prob. 24, Xu, S. H Structural analyss of a queueng system wth mult-classes of correlated arrvals and blockng. Oper. Res. (forthcomng). Zhang, R Tme delay n a mult-tem productonnventory system. Workng Paper. Department of Industral and Operatons Engneerng, Unversty of Mchgan, Ann Arbor, MI. Zpkn, P Foundatons of Inventory Management. Irwn, (forthcomng).