ORDER FILL RATE, LEADTIME VARIABILITY, AND ADVANCE DEMAND INFORMATION IN AN ASSEMBLE-TO-ORDER SYSTEM

ORDER FILL RATE, LEADTIME VARIABILITY, AND ADVANCE DEMAND INFORMATION IN AN ASSEMBLE-TO-ORDER SYSTEM YINGDONG LU IBM Research Dvson, T. J. Watson Research Center, Yorktown Heghts, New York 1598, yngdong@us.bm.com JING-SHENG SONG Graduate School of Management, Unversty of Calforna, Irvne, Calforna 92697, ssong@uc.edu DAVID D. YAO Department of Industral Engneerng and Operatons Research, Columba Unversty, New York, New York 127, yao@eor.columba.edu We study an assemble-to-order system wth stochastc leadtmes for component replenshment. There are multple product types, of whch orders arrve at the system followng batch Posson processes. Base-stock polces are used to control component nventores. We analyze the system as a set of queues drven by a common, multclass batch Posson nput, and derve the ont queue-length dstrbuton. The result leads to smple, closed-form expressons of the frst two moments, n partcular the covarances, whch capture the dependence structure of the system. Based on the ont dstrbuton and the moments, we derve easy-to-compute approxmatons and bounds for the order fulfllment performance measures. We also examne the mpact of demand and leadtme varablty, and nvestgate the value of advance demand nformaton. Receved Aprl 21; revson receved December 21; accepted March 22. Subect classfcatons: Inventory/producton: assemble-to-order, mult-tem, operatng characterstcs. Queues: smultaneous arrvals, nfnte server. Probablty: generatng functons, moments, stochastc comparson. Area of revew: Edtor-n-Chef. 1. INTRODUCTION In the movement toward mproved supply-chan management, more and more enterprses have adopted the assemble-to-order (ATO) system, whch s a hybrd of make-to-stock at the component (subassembly) level and assemble-to-order at the end-product (fnal assembly) level. Wth today s advanced nformaton technology to easly obtan and exchange nformaton, the ATO system appears to be an deal busness process that enables mass customzaton and quck response. The usual nventory-servce trade-off, a key factor n supply-chan management, becomes even more promnent n the ATO system because each customer order typcally nvolves a large number of components, and the stockout of any component wll cause a delay n supplyng the order. Hence, t s crtcally mportant to characterze the order fll rate and the necessary nventory nvestment. The analyss, however, s dffcult because the producton-nventory dynamcs among the components are hghly correlated, drven by a common demand stream. It s the obectve of ths paper to develop models and tools that facltate ths analyss and generate nsghts to mportant system desgn ssues. We consder an ATO system supportng multple types of demand, whch arrve at the system followng batch Posson processes. The nventory of each component s controlled by a base-stock polcy, and the replenshment leadtmes are..d. (ndependent and dentcally dstrbuted) random varables. Ths results n a set of M X /G/ queues, drven by a common multclass arrval stream. Ths model extends the sngle-product ATO system studed n Song and Yao (22). In the multproduct settng, the key challenge s that n addton to the assembly feature, whch requres the smultaneous avalablty of several components to fulfll one demand, there s the dstrbutonal characterstc of each component appearng on the bll-of-materal (BOM) of several product types the so-called component commonalty. Hence, not surprsngly, the approaches used here are mostly dfferent from those n Song and Yao (22). In addton, we study a few new topcs, such as the response-tme-based order fll rate and related nsghts. Specfcally, we derve the ont queue-length dstrbuton n terms of ts generatng functon. The exact analyss sheds lght on the dependence structure of the system, whch facltates the understandng of several manageral ssues such as the effect of product structures, the effect of demand and leadtme varablty, and the value of advance demand nformaton. It also allows us to obtan varous moments mean, varance, and covarance n smple, explct formulas, whch n turn lead to several approxmaton schemes for key performance measures that are otherwse computatonally ntractable. A bref overvew of the related lterature s n order. Studes of ATO systems dffer qute substantally n the detaled modelng assumptons and approaches. For exam- Operatons Research 23 INFORMS Vol. 51, No. 2, March Aprl 23, pp. 292 38 292 3-364X/3/512-292 $5. 1526-5463 electronc ISSN

ple, Hausman et al. (1998) and Zhang (1997) study perodc-revew (dscrete-tme) models wth multvarate normal demand and constant component replenshment leadtmes. Song (22) studes contnuous-revew models wth multvarate compound Posson demand and determnstc leadtmes. Song et al. (1999), Glasserman and Wang (1998), and Wang (1998) also consder multvarate (compound) Posson demand, but the supply process for each component s capactated and modeled as a sngle-server queue. In contrast, n our study the supply submodel s a parallel processng system, as reflected n the nfnte-server queues. Ths s approprate when components are procured from exogenous supplers whose capacty s deemed ample relatve to the orders placed from the ATO system n queston. Gallen and Wen (21) use the same leadtme model as ours here, but focus on a sngle demand stream and assume order synchronzaton. Cheung and Hausman (1995) also assume..d. leadtmes n the context of a repar shop. Ther multvarate Posson demand model s a specal case of ours (the unt demand case). They use a combnaton of order synchronzaton and dsaggregaton n ther analyss. The rest of the paper s organzed as follows. The formal model descrpton s presented n 2, followed by the dervaton n 3 of the generatng functon for the ont queue-length dstrbuton, the moments, and the analyss on leadtme and batch-sze varablty. Several approxmaton schemes and bounds are developed n 4. Responsetme-based order fll rates are studed 5, and the value of advance demand nformaton dscussed n 6. Numercal results are presented n 7, and fnally, bref concludng remarks are summarzed n 8. 2. MODEL DESCRIPTION We consder an nventory system of m dfferent components from whch multple types of products can be assembled upon customers requests. See Fgure 1. Let I = 1 2 m denote the set of all component ndces. Customer orders arrve at the system followng a statonary Posson process, denoted At t, wth rate. Each order may requre several components n dfferent amounts smultaneously. For any subset of components Fgure 1. The assemble-to-order system. Supplers Components Products Backorders (Items) (Customer demands) L 1 L 2 L m 1 2 m Q K 2 Q 12 1 Q 12 2 Q K m Q K 1 λ 12 λ Κ Lu, Song, and Yao / 293 K I, we say an order s of type K f t contans postve unts of component n K and unts n I \ K. We assume that there s a fxed probablty q K that an order s of type K K q K = 1. Thus, the type-k order stream forms a compound Posson process wth rate K = q K. A type-k order requests Q K unts of component K, and Q K = Q K K follows a known dscrete probablty dstrbuton. We assume that each order s type s ndependent of the other orders types and of all other events. We denote to be the set of all demand types, that s, = K I q K >. Note that s not necessarly the set of all possble subsets of I. Ths demand model s flexble n modelng a varety of stuatons. One specal case s a system wth only one demand type : All demands request m components n stock, however, the unts of each component requested can be dfferent, wth Q 1 beng the batch sze for, = 1 m. Here, f Q are constant, the system s equvalent to an assembly system wth a sngle fnal product, where Q represent the bll of materals (BOM). If Q are random varables, the model represents a system wth a varety of products, n whch the varety s characterzed by the number of unts of each component contaned n the product. Each combnaton of Q s corresponds to the BOM of a specfc product. Another specal case s a system n whch each customer demand requres one and only one component. In ths case, the system reduces to m ndependent sngle-component systems. Below we shall use the results from ths smple system to bound the performance of the orgnal, general system. For each component, let denote the famly of subsets of that contan. It s clear that the demand process for component forms a compound Posson process wth rate = K K = q and batch sze Q, the mxture of Q K for all K. (Throughout the paper we use subscrpts to ndcate the component types and superscrpts the demand types.) Demands are flled on a frst-come-frst-served (FCFS) bass. If there s enough on-hand nventory for all the components requred by a demand upon ts arrval, the demand s flled mmedately. In other words, we assume that the tme to assemble the components nto the end-product s neglgble. We also assume complete backloggng for demands that cannot be flled mmedately. When a demand arrves and some of ts requred components are n stock but others are not, we can ether shp the n-stock components or put them asde as commtted nventory. However, a demand s consdered backlogged unless t can be satsfed completely. When there are backorders, they are also flled on a FCFS bass. The nventory of each component s controlled by an ndependent base-stock polcy, wth s = the base-stock level for component That s, upon each demand arrval, f the nventory poston (.e., the on-hand nventory plus on-order poston mnus

294 / Lu, Song, and Yao backorders) of component s less than s, then order up to s ; otherwse, do not order. Ths type of polcy s known to be optmal when there are no economes of scale n replenshment and when the component demands are ndependent. When component demands are correlated as n our case, the optmal polcy should be a coordnated polcy. Unfortunately, the precse form of the optmal polcy s stll unknown. Hence, the ndependent base-stock polces are wdely used n practce, partly because of ther smplcty, and partly because they provde a benchmark on how much nventory s needed to provde a certan servce level. Because we follow a base-stock polcy and the demand arrves n batches, each component replenshment order comprses several unts. We assume that for each unt of component, the replenshment leadtmes are..d. random varables wth a common cumulatve dstrbuton G. Let L denote the generc random varable wth dstrbuton G and mean EL = l. Denote G c = 1 G. Assume the leadtmes are ndependent among the components; that s, L s ndependent of L for any. (Alternatvely, one can assume that each order experences L unts of leadtme, whch s ndependent of the order sze. We dscuss ths assumpton n our concludng remarks.) The performance measure of prmary nterest s the mmedate avalablty of all components needed for an arrvng demand, or the off-the-shelf fll rate. (We study a more general performance measure the response-tmebased order fll rate n 5.) It turns out that ths measure can be obtaned drectly from the steady-state ont dstrbuton of outstandng orders, as explaned n the followng. For any tme t, let I t be the net nventory of component at t, and X t be the number of outstandng orders of component at t. Then, by the nature of the base-stock control, we have I t = s X t = 1 m (1) In the next secton, we show that the ont dstrbuton of X 1 t X m t has a steady-state lmt X 1 X m. Therefore, we have the steady-state net nventory I = s X = 1 m From the property of Posson arrval see tme average, we have f = off-the-shelf fll rate of component = PX + Q s f K = off-the-shelf fll rate of type-k demand = P X + Q K s K ] f = average (over all demand types) off-the-shelf fll rate = q K f K K Thus, to study the performance of the ATO system, the key s to obtan the ont dstrbuton of X 1 X m. 3. PERFORMANCE ANALYSIS 3.1. Generatng Functons and Moments We now derve the ont dstrbuton of the outstandng orders vector Xt = X 1 t X m t and ts steadystate lmt. By the nature of the base-stock polcy, each type-k order trggers replenshment orders of all the components n subset K. Thus, the arrval processes to the supply subsystem are the same as the external demand processes to the ATO system. Also, for each component, the number of outstandng orders X t s exactly the number of obs n servce n an M Q /G / queue wth Posson arrval rate and batch sze Q. Because each order generates smultaneous arrvals at several component queues, the m queues are not ndependent. See Fgure 2. On the other hand, the m queues are dependent only through the common arrval streams; ther operatons are otherwse ndependent, n partcular the processng tmes are ndependent across the queues. In other words, gven the number of demand arrvals up to t, the X ts are ndependent of one another. Ths fact s crucal because t allows us to proceed, after condtonng upon the number of arrvals, as f we are analyzng ndependent systems. Ths way, we can derve (see detals n the Appendx) the ont dstrbuton of the outstandng orders, n terms of ts generatng functon, as summarzed n the followng proposton. Proposton 1. Let X = X 1 X m follow the lmtng dstrbuton of Xt = X 1 t X m t as t. Then, the generatng functon of X s z 1 z m m =E z X ] =exp K QK G 1 u+z 1 G c 1 ug mu K +z m G c m u 1du ] (2) where QK z 1 z m denotes the generatng functon of Q K. Fgure 2. The supply subsystem. Q 12 1 Q 12 2 λ 12 Supplers X 1 X2 X m Q K 2 Q K 1 Arrvals (Replenshment Orders) λ Κ Q K m

Below, we shall focus on the random vector X. In the specal case of unt arrvals,.e., Q 1, for all = 1 m, the generatng functon takes the followng form: z 1 z m ( ] =exp K G u+z G c )du u 1 (3) K K whch corresponds to a multvarate Posson dstrbuton. In partcular, ( ) 1 z 1 = exp K l z 1 K That s, for each, X s a Posson random varable wth parameter l = K K l, as expected. The multvarate Posson dstrbuton mples that each X can be expressed as a fnte sum of ndependent unvarate Posson dstrbutons; see, e.g., Johnson et al. (1996). Specfcally, lettng Na denote a Posson random varable wth parameter (mean) a, we can wrte X = K N K K (4) where all the Posson varables under the summaton are ndependent, and K = G c u G u du (5) K\ In prncple, we can compute the ont dstrbuton of X = X 1 X m through the above expressons. Ths, however, nvolves 2 m 1 (ndependent) Posson varates. Therefore, even n the case of unt arrvals, the exact evaluaton of the ont dstrbuton would stll be mpractcal even for moderately large m. Next, we derve the means, varances, and covarances, makng use of the generatng functon n Proposton 1. Let denote the famly of subsets that contan both and. Then, the frst two moments of X can be derved as follows: = EX = z 1 z m z =1z = l K EQ K (6) K ( 2 EX 2 = + ) z z 2 z 1 z m z =1 = EX + E 2 X + K K EQ K 2 EQ K ] G c u2 du (7) and for, EX X = 2 z z 1 z m z =1z = EX EX + K EQ K QK K Hence, from (7), we have 2 = VarX = EX + Lu, Song, and Yao / 295 G c ugc u du (8) K K EQ K 2 EQ K ] G c u2 du (9) Note that because EQ K 1 for all K,wehave EQ K 2 EQ K 2 EQ K and hence 2. Note that 2 = f and only f we have unt arrvals,.e., Q 1, whch s consstent wth the fact that X follows a Posson dstrbuton. From (8), we have, for, = CovX X = K K EQ K QK G c ugc u du (1) Note that = f s empty. That s, X and X are ndependent f there are no demand types that requre both and. In other words, the correlaton of the queues s solely nduced by the common arrvals. Indeed, (1) also ndcates that as long as there exsts common arrvals to both queue and queue, X and X are always postvely correlated. Wth (9) and (1), t s easy to calculate the correlaton coeffcents = It s nsghtful to examne the explct formulas n some specal cases. In partcular, assumng unt demand, we obtan K K = Gc ugc u du l K l K K K ( )( ) = Gc ugc u du (11) l l Here = K K s the aggregate demand rate for the component par, and and are the aggregate demand rate for components and, respectvely.

296 / Lu, Song, and Yao Notce that Gc ugc so u du = l l Gc ugc Gc u du u du 1 Gc u du If the proporton of the demand types that requre both and s very small, so that and, then the correlaton between X and X s neglgble, and the two queues can hence be treated as ndependent. On the other hand, f the components and are always demanded together, such as n the pure assembly (.e., sngle-product) system, then = =. In ths case, = Gc ugc u du l l Thus, although the correlaton s nduced by the common arrval, the level of correlaton s ndependent of the demand rate. The correlaton s mperfect; ts level depends on the leadtme dstrbutons. Suppose, n addton, that leadtmes L and L are exponentally dstrbuted, then and = G c ugc u du = l l (12) l + l l l 1 = l + l l /l + l /l If both leadtmes have the same dstrbuton, so that l = l, then = 5 Thus, the mperfect correlaton s a consequence of the statstcal varabltes of the leadtmes. Now, let us examne what happens f there s no statstcal varablty n leadtmes. That s, suppose both L and L are determnstc. In ths case, and G c ugc u du = mnl l (13) = mnl l maxl l Thus, the dfference between the lengths of the leadtmes determnes the level of correlaton. The bgger the dfference s, the smaller the correlaton becomes. If l = l, then the correlaton s perfect, as expected, because the two queues are completely synchronzed because of the common arrval stream and the same fxed servce tmes. Fnally, we remark that the generatng functon n Proposton 1 can also be used to obtan all other hgher moments of X. In partcular, for any subset 1 k of 1 2 m, let U be any subset of 1 k wth cardnalty U and denote Ū = 1 k \ U.Wehave k EX 1 X k = K k +1 K 1 k K l h E h Ū Q K h h U U 1 k U = ] h U G c hu du (14) 3.2. The Impactof Varablty We now compare the orgnal system wth another system, where the processng tmes of queue, are..d. random varables wth mean l (same as n the orgnal system) and a dstrbuton functon G satsfyng x G c u du x G c u du = 1 m (15) for any x. Lke G c, here G c = 1 G. All other aspects of the two systems are dentcal. Let L and L be the random varables assocated wth the dstrbutons G and G, respectvely. Then, the above condton means that L s more varable than L n the sense of the convex orderng, denoted L cx L ;.e., EfL Ef L for all convex functons f (see, e.g., Shaked and Shanthkumar 1994), and hence n partcular, VarL = EL l 2 E L l 2 = Var L Smlarly, we can address the ssue of varablty assocated wth the batch-sze dstrbuton. Suppose the new system has a batch sze of Q K for component of type- K product. Assume Q K s more varable than Q K n the same sense of convex orderng as above, whereas all other aspects of the new system are the same as the orgnal system. Proposton 2. Use tlde to denote the new system that has reduced varablty n ether leadtmes or batch szes, n the sense of convex orderng as detaled above. We have () E T X E X T, for any T 1 m. () X lo X and X uo X, where lo uo denotes lower-orthant (upper-orthant) orderng. That s, for any x 1 x m, the followng hold: PX 1 x 1 X m x m P X 1 x 1 X m x m and PX 1 x 1 X m x m P X 1 x 1 X m x m respectvely, for lower-orthant and upper-orthant orderngs. () f K f K for all K and f f. The proof of Proposton 2 s presented n the Appendx. Part () of the proposton ndcates that varablty degrades order fll rates. Part () of the proposton mples, n partcular, CovX X Cov X X

That s, reducng the varablty of leadtmes or batch szes wll result n a hgher correlaton among the queue lengths of outstandng obs. Combnng the above nequalty wth the covarance expresson n (1), and lettng the leadtmes n the modfed system be (determnstc) constants, we have CovX X mnl l K K EQ K QK Next, for any par of queues and, suppose n the orgnal system the leadtmes L and L are exponentally dstrbuted, whereas the modfed system has constant leadtmes as above, wth l = EL and l = EL ; and assume, wthout loss of generalty, l l. Then, applyng (12) and (13), we have CovX X Cov X X = l l + l whch ranges from.5 to 1, dependng on how close s l to l. 4. APPROXIMATIONS In ths secton, we develop computatonal effcent approxmatons and bounds for the ont dstrbuton of X, whch n turn form easy-to-compute approxmatons for the order fll rates. 4.1. Factorzed Normal Approxmaton It s qute natural to use multvarate normal dstrbuton to approxmate X 1 X m because the frst two moments, ncludng the covarances, are explctly derved n 3.1. Here, = 1 m wth gven n (6), and the covarance matrx = wth gven n (1). More specfcally, PX 1 = x 1 X m = x m 1 exp { 12 } 2 m/2 x 1/2 1 x where x = x 1 x m (a row vector), and x s the transpose of x. To compute the cumulatve dstrbuton, we have PX 1 x 1 X m x m x1 xm 1 2 m/2 1/2 exp { 12 } y 1 y dy (16) Unfortunately, however, because of the multple ntegrals nvolved, even the state-of-the-art computatonal packages of multvarate normal dstrbutons are stll lmted to rather low dmensons, e.g., m 7. (Refer to, for example, Drezner 1992.) Lu, Song, and Yao / 297 For hgher dmensons, we propose a further approxmaton on the multvarate normal dstrbuton usng the followng approach. Applyng Hölder s nequalty, we have EQ K QK ( EQ K 2 ) 1/2 ( EQ K 2 ) 1/2 and G c ugc u du 1/2 1/2 G c du] u2 G c du] u2 Hence, denote ] 1/2 1/2 = K EQ K 2 G c du] u2 (17) K and smlarly denote. Then, we can obtan a factorzed (between and ) upper bound on the covarance: = K EQ K QK G c ugc u du K ( K EQ K 2 EQ K 2 ) 1/2 K ] 1/2 G c u2 du G c u2 du The second nequalty above takes nto account. Note that we have <, whch follows from comparng (17) aganst (9), takng nto account the followng: EX = K EQ K K G c u du > K EQ K G c u2 du K Smlarly, we have <. Now, let V= V 1 V m be a multvarate normal dstrbuton wth the same mean = 1 m as before, but wth the covarance replaced by. Then, we can wrte 2 2 Z = 1 m (18) V = + Z + where Z Z 1 Z m are..d. standard normal varates. Clearly, V has the same margnal dstrbutons as. But because of ts specal form n (18), we can avod evaluatng the multple ntegrals n (16). Instead, we need only to evaluate the followng sngle ntegral: PX 1 x 1 X m x m PV 1 x 1 V m x m ( ) m x = u u du (19) =1 2 2 where and are the normal densty and dstrbuton functons, respectvely. We shall refer to the above as the factorzed normal approxmaton.

298 / Lu, Song, and Yao 4.2. Parwse Approxmaton Next, we develop a second type of approxmaton, whch we call parwse approxmaton. It s based on the followng result (proved n the Appendx). Proposton 3. For any x 1 x m, we have m PX 1 x 1 X m x m PX x (2) More generally, lettng 1 k be a partton of I = 1 m, we have k PX 1 x 1 X m x m PX x l (21) l=1 In partcular, Proposton 3 mples the followng: f K K PX + Q K s PX + Q s = f (22) K K Thus, the product of the component fll rates s a lower bound of the order fll rate. Note that n two earler works (Song 1998 and Song and Yao 22), the nequalty n (2) has been establshed for specal cases of the system consdered here (specfcally, those wth constant leadtmes and wth sngle-class demand, respectvely). Our second approxmaton scheme s smply to apply the relaton n (21) to a parwse partton of I, and use the lower bound as an approxmaton for the ont dstrbuton. (In the case of m beng odd, let the last subset k be a sngleton set.) To evaluate ths lower bound, we can frst replace each par by a bvarate normal dstrbuton, and then evaluate the product of k bvarate normal dstrbutons. The only remanng queston s how we form the k pars. In our numercal experments, we have tred to take the best (largest) lower bound among all possble ways ( m 2 /2) of formng pars. The dfference (among these dfferent ways), however, appears to be nsgnfcant. Hence, our recommendaton s to form pars n decreasng order of the covarances. That s, () Set I = I; set l = 1. () Set l = l l such that l l = arg max I () Stop, f I s ether a sngleton set or a null set. Otherwse, set I I \ l l, l l + 1; and go to (). Fnally, n the case of unt arrvals, X = X 1 X m follows a multvarate Posson dstrbuton, as dscussed n 3.1. The parwse approxmaton descrbed above apples readly here. The only modfcaton s nstead of evaluatng bvarate normal dstrbutons, here we evaluate bvarate Posson dstrbutons. In partcular, we stll form the pars n decreasng order of ther covarances. The factorzed approach could, n prncple, also be appled here. However, our numercal experments have shown that the qualty of ths approxmaton s not qute as good as n the case of normal dstrbutons. The reason s because whle we can always wrte a normal varate X as X = + Z wth Z beng the standard normal, there s no analogous expresson for a Posson varate N. (If anythng, we can wrte N = + Z; but ths goes back to the normal approxmaton.) 5. RESPONSE-TIME-BASED ORDER FILL RATE A key servce measure s order response tme: the tme length between the pont when the order s placed and the pont when the product s receved by the customer. Ths response tme usually conssts of two parts: the tme needed to have all the components ready for assembly, denoted W ; and the tme for outbound logstcs, denoted W o, whch ncludes tme for processng orders, assembly tme, and transportaton tme for delverng the order. What affects servce s manly the W part when any component nventory has a stockout, the delay could be substantal, whereas W o s often nearly determnstc. Therefore, we shall focus on W and denote by f K w the probablty of havng all the components K ready wthn w unts of tme (.e., W w). (In ths sense, w s the allowed tme wndow to fll the order.) In other words, f K w s the probablty that the overall response tme to type-k orders s under w +W o unts of tme. Suppose a type-k order arrves at tme. Consder component K. Defne D t to be the cumulatve demand for component n t. For any tu, denote D t t + u= D t + u D t Then, X + D + w X + w s the total number of departures from queue n + w. Because ths s the total addton to the component s nventory between and + w, the order that arrves at can be flled by + w f and only f the net nventory at plus ths addton s nonnegatve. That s, I + X + D + w X + w Usng (1), the above nequalty smplfes to X + w D + w s K (23) Observe that X + w, the number of outstandng orders of component at tme + w, has the followng decomposton: X + w = X w QK + 1L n >w+ X + w (24) The three terms on the rght-hand sde represent the number of unts stll n queue (at + w) that correspond to the

arrvals, respectvely, n,at, and n +w. 1 denotes the ndcator functon, and L n s an ndependent copy of L representng the leadtme for the nth unt of Q K. (Note that X w s dfferent from X.) Moreover, the three terms on the rght-hand sde of (24) are ndependent of one another because of Posson arrvals and the nfnte servers n each queue. Thus, combnng (23) and (24), we know the demand at can be suppled by + w f and only f X w + X + w D + w Q K s 1L n > w K (25) When, X w has a lmt X w, whch has the followng generatng functon (see dervaton n the Appendx): { exp Q G 1 u + w+ z 1 G c 1u + w } G m u + w+ z m G c m u + w 1] du (26) Denote Y w = D + w X + w whch represents the number of obs that arrved n + w but also departed by + w. Denote Y w = Y w, whch has the followng generatng functon (derved n the Appendx): { ( ( w exp Q G c 1 u + z 1G 1 u G c m u + z mg m u 1 ] )} du (27) Moreover, X w and Y w are ndependent, because of the ndependent-ncrement property of the Posson arrval process. Denote Y = X w Y w = 1 m Then, Y = Y 1 Y m has the followng probablty generatng functon: Y z 1 z m { =exp Q G 1 u+w+z 1 G c 1 u+w G m u+w+z m G c m u+w 1du w Q G c 1 u+g 1u/z 1 G c m u+g mu/z m 1 ] } du (28) Applyng (25), we have Lu, Song, and Yao / 299 f K w = order fll rate of type-k demand wthn tme wndow w ] Q K = P Y + 1L n >ws K (29) (Kruse 198 obtaned a smlar result n a snglecomponent, unt-demand, base-stock nventory system.) The random varable Q K 1L n >ws a mxture of geometrc dstrbutons. Condtonng on Q K = k, t has the geometrc dstrbuton wth parameters k (the number of trals) and G c w (the success rate). Note that when w =, the second term n (28) vanshes, reducng the generatng functon to the one n (2). Thus, when w =, Y reduces to X, and f K concdes wth the off-the-shelf fll rate f K, as expected. Because Y s ndependent of Q K and L, as explaned earler, to derve the above fll rate, the key s the dstrbuton functon of Y, whch follows from ts generatng functon derved above. Whle ths can only be done numercally n general, we can easly derve the frst two moments n closed form from the generatng functon n (28). It s straghtforward to see that all the approxmaton schemes and bounds developed n the last secton automatcally apply here. In partcular, we have EY = z Y z 1 z m z =1 ( ) = EQ l w ( ) ( w ) VarY = EQ Gu du + G c u du w ( ) + EQ 2 EQ ( w ) G u 2 du + G c u2 du w and for, ( ) CovY Y = EQ Q ( w ) G ug u du + G c ugc u du Smlar to the propertes of X dscussed n 3, here, too, we have VarY EY for all. Also, for any par and, Y and Y are postvely correlated as long as there exsts a demand type that requres both and. Otherwse, they are ndependent. However, n the unt demand case, ( w ) EY = l w = G c u du G u du w w

3 / Lu, Song, and Yao and ( w VarY = Gu du + w G c u du ) So, for w>, VarY >EY,.e., Y does not have a Posson dstrbuton. 6. CONNECTIONS TO ADVANCE DEMAND INFORMATION In recent years, there has been an ncreasng trend n ndustry to try to nduce early customer demand nformaton to streamlne operatons and mprove servce. Clearly, knowng early what customers want helps nventory plannng by orderng and stockng the rght tems n the rght amount. In ths secton, we show that the analytcal approach developed n the last secton provdes some nsght nto the mpact of advance demand nformaton on order fulfllment performance. Suppose each order arrval epoch s known w tme unts n advance, where w> s a determnstc constant. For nstance, suppose we know at t = that an order wll arrve at t = w; then, we wll send out replenshment orders for all the components nvolved, at t =. That s, we adapt the base-stock polcy to placng replenshements as soon as the order arrval nformaton becomes known (n advance). Let fa K denote the fll rate for type-k orders upon arrval (.e., the mmedately fll rate), wth the subscrpt A alludng to advance demand nformaton. Suppose a type-k order arrves at, and ths nformaton s known at w. Then, by placng component replenshment orders at w, we can fll ths order, upon ts arrval at, wth the followng probablty (cf. (25)): f K A = P {X w w+ X w D w Q K } s 1L n > w K (3) Now, compare the above wth the response-tme fll rate of the orgnal system, wthout advance demand nformaton, wthn a tme wndow of w unts, f K w n (29). From the statonary and ndependent ncrement property of the Posson arrval process, we have X w w+ X w D w d =X w + X + w D + w K d Here, = denotes equal n dstrbuton. Recall from the dscusson n the last secton that the three terms on the rghthand sde are ndependent, and so are those on the left-hand sde. (In the case of advance demand nformaton, we can shft the tme lne by w unts earler; n partcular, let the tme orgn be w nstead of.) Hence, comparng (25) and (29) wth (3), we have f K A = f K w (31) Proposton 4. In a multproduct ATO system wth stochastc leadtmes, the response-tme fll rate for any type-k product wthn a tme wndow of w unts, f K w, s equal to the mmedate fll rate wth advance demand nformaton, fa K, when order arrvals are known w unts n advance. Because, f K w f K follows trvally from (29), we can conclude fa K f K,.e., advance demand nformaton mproves the off-the-shelf fll rate. A more subtle result s to compare fa K wth the mmedate fll rate of a modfed system, n whch the leadtmes are truncated by w. Ths allows us to relate the mpact of the advance demand nformaton to the effect of leadtme reducton. The mportance of ths knd of comparson has been promnently brought out n Harharan and Zpkn (1995). To do so, the relaton n (31) enables us to replace fa K by f K w, whch relates more drectly to the modfed system, n the sense that nether the orgnal system nor the modfed system makes use of advance demand nformaton. Specfcally, the modfed system dffers from the orgnal one only n the leadtmes: For component the leadtme s reduced from L to ˆL = L w +. Note that for any u, P ˆL >u= PL w + >u= PL >u+ w = G c u + w Let ˆX = ˆX 1 ˆX m be the steady-state vector of outstandng orders n the modfed system. Then, examnng (2) and (26) we conclude that ˆX has the same dstrbuton as X w = X1 w Xw m. Thus, the mmedate order fll rate n the modfed system s { } Q K fˆ K = P X w + 1 ˆL n > s K n= Here, ˆL n s an ndependent copy of ˆL. On the other hand, for the orgnal system, from (29), we have { } Q K f K w = P Y w 1L n >w s K X w + Because L >wf and only f ˆL >, and Y w from the last two expressons,, we have, ˆ f K f K w (32) A specal case of nterest s when L w wth probablty one. In ths case, ˆL = L w, and Y w =. (Recall that Y w s the number of obs that have arrved n + w but also departed by + w; none such ob would exst f the leadtme s no less than w.) Hence, n ths case (32) holds as an equalty. Proposton 5. Consder a multproduct ATO system wth stochastc leadtmes, the orgnal system, along wth a modfed system where the leadtme for each component, L,

s reduced to ˆL = L w +, wth w> a gven constant. For any product type K, let fˆ K be the mmedate fll rate n the modfed system, and f K w be the fll rate wthn a response-tme wndow of w n the orgnal system. Then, we have fˆ K f K w. Furthermore, f L w wth probablty one (and hence ˆL = L w) for all K, then fˆ K = f K w. Note that the equalty part of the above proposton extends a smlar result n the lterature when the leadtmes are all determnstc; see e.g., Harharan and Zpkn (1995) and Song (1998). Connectng (32) wth (31), we have the followng corollary. Corollary 6. Consder the orgnal and modfed systems n Proposton 5. Suppose demand arrvals are known w unts n advance n the orgnal system, and let fa K denote the mmedate fll rate n ths case (no change n the modfed system). Then, we have fˆ K fa K. Thus, knowng demand n advance (by w tme unts) s more effectve, n terms of order fll rate, than reducng the supply leadtmes of components. In fact, advance demand nformaton can do even better. The nequalty n Corollary 1 holds when both systems mantan the same base-stock levels (for component nventory). Suppose fˆ K n (32) corresponds to the optmal base-stock levels, n terms of maxmzng fˆ K, n the modfed system, and let the orgnal system (wth advance demand nformaton) follow the same base-stock levels. Then, the nequalty stll holds. We can then optmze the base-stock levels n the orgnal system to further ncrease fa K. The advance demand nformaton can be mplemented n other ways,.e., other than the drect, base-stock control mechansm that leads to fa K above. For example, suppose a customer order that wll arrve at tme s known at w. We can then mmedately place a replenshment order for each component nvolved n the order, smlar to what we dd earler n obtanng fa K but wth the followng twst: We poll the suppler of each component for a quoted leadtme, and f the quoted leadtme s shorter than w, then we ask that the replenshment order be delvered exactly w unts later. (Otherwse, we do not nterfere wth the delvery, whch wll take place at the quoted leadtme.) Hence, the replenshment order wll arrve at w + maxl w= + L w + = + ˆL.e., at the same tme f the order s placed at (nstead of at w) whle the leadtme s reduced to ˆL. Effectvely, ths equates the orgnal system wth advance demand nformaton to the modfed system wth reduced leadtme n Proposton 5. And, t can be shown that the resultng offthe-shelf fll rate, denoted fˆ A K, s the same as the offthe-shelf fll rate n the modfed system of Proposton 5, fˆ K. Lu, Song, and Yao / 31 The above mechansm s remnscent of a smlar procedure n Harharan and Zpkn (1995) for sngle component systems. Connectng the equalty, fˆ A K = fˆ K, wth Proposton 5 and Corollary 1, we have fˆ A K f A K. That s, we can compare two dfferent mechansms of makng use of the advance demand nformaton, n terms of ther correspondng order fll rates. (Of course, that the second mechansm yelds a lower fll rate s hardly surprsng, because t delays the delvery of any replenshment wth a leadtme L w.) 7. NUMERICAL RESULTS In ths secton we llustrate the effectveness of the approxmatons through numercal results. We focus on an ATO system of desktop computers studed n Cheng et al. (2), usng a slghtly smplfed BOM (through combnng certan common components of dfferent products). Specfcally, a famly of product conssts of sx dfferent types of products, whch are assembled from a set of 14 dfferent components. The components are obtaned from external supplers wth dfferent supply leadtmes. Table 1 provdes the lst of all the components and ther average leadtmes, along wth the BOM of each product. The aggregate demand process for the sx products forms a Posson process wth mean = 5. The proportons of the sx products are q 1 = 1 q 2 = 15 q 3 = 4 q 4 = 2 q 5 = 1 q 6 = 5 We consder four dfferent sets of base-stock levels at the components as summarzed n Table 2. The data (basestock levels) n Table 2 are chosen such that the off-theshelf fll rates (lower bounds) are acheved at about 98%, 95%, 9%, and 8% (respectvely for the four groups) for all sx products. To demonstrate the mpact of leadtme varablty on order fll rates, we consder both exponental dstrbu- Table 1. Average component leadtmes and the BOM of each product. Descrpton l K: 1 Shell 7 1 1 1 1 1 1 2 Processor: 45 MHz 12 1 3 Processor: 5 MHz 12 1 4 Processor: 55 MHz 12 1 1 1 5 Processor: 6 MHz 12 1 6 64 MB Memory 15 1 1 1 1 1 1 7 6.8 GB Hard Drve 18 1 1 1 8 13.5 GB Hard Drve 18 1 1 1 9 Hard Drve (Common Parts) 8 1 1 1 1 1 1 1 Software 1. 4 1 1 1 1 11 Software 2. 4 1 12 CDROM 1 1 13 VGA Card 6 1 1 1 1 1 14 Ethernet Card 1 1

32 / Lu, Song, and Yao Table 2. Four dfferent groups of base-stock levels. Component 7 8 9 1 11 12 13 14 Level 1 52 13 17 67 9 96 52 87 67 3 1 36 54 36 Level 2 5 12 16 62 8 94 44 83 59 29 1 35 46 35 Level 3 48 12 16 6 7 9 42 83 59 27 1 34 42 34 Level 4 47 1 14 6 6 86 46 78 6 25 9 3 44 3 Table 3. Computaton of order fll rates smulatons and approxmatons. Lead Tme: Exponental (L), Base-Stock: Level 1 Product 1 Product 2 Product 3 Product 4 Product 5 Product 6 Smulaton.979.975.963 977.985.983 Lower Bound (Margnal).976.973.98.974.982.981 Lower Bound (Parwse).976.974.981.975.965.982 Normal Approxmaton 1.985.983.988.985.988.988 Normal Approxmaton 2.985.984.987.984.987.988 Lead Tme: Exponental (L), Base-Stock: Level 2 Product 1 Product 2 Product 3 Product 4 Product 5 Product 6 Smulaton.948.948.961.956.963.96 Lower Bound (Margnal).945.943.952.948.955.955 Lower Bound (Parwse).947.946.955.951.958.957 Normal Approxmaton 1.965.964.971.969.971.972 Normal Approxmaton 2.962.963.967.965.967.971 Lead Tme: Exponental (L), Base-Stock: Level 3 Product 1 Product 2 Product 3 Product 4 Product 5 Product 6 Smulaton.94.92.918.915.921.92 Lower Bound (Margnal).897.896.896.897.91.892 Lower Bound (Parwse).896.897.94.94.99.898 Normal Approxmaton 1.926.925.931.93.931.921 Normal Approxmaton 2.917.917.92.92.921.916 Lead Tme: Exponental (L), Base-Stock: Level 4 Product 1 Product 2 Product 3 Product 4 Product 5 Product 6 Smulaton.798.81.819.831.842.799 Lower Bound (Margnal).773.781.774.8.89.785 Lower Bound (Parwse).783.792.79.812.822.795 Normal Approxmaton 1.819.83.852.852.854.818 Normal Approxmaton 2.798.813.818.828.832.87 Lead Tme: Erlang(2, L/2): Base-Stock: Level 1 Product 1 Product 2 Product 3 Product 4 Product 5 Product 6 Smulaton.98.975.984.978.985.984 Lower Bound (Margnal).976.973.98.974.982.981 Lower Bound (Parwse).977.974.982.975.983.982 Normal Approxmaton 1.988.984.988.986.988.988 Normal Approxmaton 2.985.985.987.984.987.988 Lead Tme: Erlang(2, L/2): Base-Stock: Level 2 Product 1 Product 2 Product 3 Product 4 Product 5 Product 6 Smulaton.951.951.964.958.966.962 Lower Bound (Margnal).945.943.952.948.955.955 Lower Bound (Parwse).948.947.957.952.96.958 Normal Approxmaton 1.967.966.973.972.973.974 Normal Approxmaton 2.963.964.967.965.968.965

Lu, Song, and Yao / 33 Table 3. (Contnued). Lead Tme: Erlang(2, L/2): Base-Stock: Level 3 Product 1 Product 2 Product 3 Product 4 Product 5 Product 6 Smulaton.98.99.922.919.924.94 Lower Bound (Margnal).897.896.896.897.91.892 Lower Bound (Parwse).91.9.98.97.914.9 Normal Approxmaton 1.931.931.936.936.936.925 Normal Approxmaton 2.919.92.922.922.923.918 Lead Tme: Erlang(2, L/2): Base-Stock: Level 4 Product 1 Product 2 Product 3 Product 4 Product 5 Product 6 Smulaton.81.815.834.841.853.88 Lower Bound (Margnal).773.781.774.8.89.785 Lower Bound (Parwse).786.795.799.817.829.795 Normal Approxmaton 1.828.84.862.862.863.824 Normal Approxmaton 2.8.817.822.831.836.89 ton wth mean l and Erlang dstrbuton wth parameters l /2 2. Table 3 reports the numercal results from four dfferent approxmatons along wth smulaton. The results are also plotted n Fgure 3. The four approxmaton schemes are: () Lower Bound (Margnal), or LB1, whch s the lower bound n (2); () Lower Bound (Parwse), or LB2, whch s the parwse lower bound n (21), usng bvarate Posson dstrbutons; () Normal Approxmaton, or NA1, whch s the factorzed normal approxmaton gven n (19); and (v) Normal Approxmaton 2, or NA2, whch uses bvarate normal dstrbuton n the parwse lower bound. Fgure 3. Approxmatons vs. smulaton: the desktop example. Exponental, Level 1 Erlang, Level1.995.99.985.98.975.97.965.96 Sm LB1 LB2 NA1 NA2.99.985.98.975.97.965 Sm LB1 LB2 NA1 NA2 Exponental, Level 2 Erlang, Level 2.99.98.97.96.95.94.93 Sm LB1 LB2 NA1 NA2.99.98.97.96.95.94.93 Sm LB1 LB2 NA1 NA2 Exponental, Level 3 Erlang, Level 3.94.92.9.88.86 Sm LB1 LB2 NA1 NA2.94.92.9.88.86 Sm LB1 LB2 NA1 NA2 Exponental, Level 4 Erlang, Level 4.86.84.82.8.78.76.74.72 Sm LB1 LB2 NA1 NA2.9.85.8.75.7 Sm LB1 LB2 NA1 NA2

34 / Lu, Song, and Yao Fgure 4. Fll rate 1..95.9.85.8.75 The effect of leadtme varablty. Product Level 1, Exp Level 2, Exp Level 3, Exp Level 4, Exp Level 1, Erl Level 2, Erl Level 3, Erl Level 4, Erl From these results, we observe the followng: The factorzed normal approxmaton overestmates the fll rates n all cases. The parwse normal approxmaton also overestmates the fll rate n most (but not all) cases. Not surprsngly, the parwse lower bound outperforms the margnal lower bound n all cases. The parwse normal approxmaton also outperforms the factored normal approxmaton n all cases. For hgh fll rates (.95 and above), the parwse lower bound s better than the parwse normal approxmaton. In fact, n ths range, t s the best approxmaton among four. In ths case, the margnal lower bound s also very relable. For fll rates around the range of.8.85, the parwse normal approxmaton appears to be the best among all four. In the range of.9.95, the two parwse bounds are the best. Comparng the results wth dfferent leadtme dstrbutons, we also observe that the parwse approxmatons defntely capture more of the probablstc natures of the system such as the effect of leadtme dstrbutons, whle the margnal lower bound does not. Because the factorzed normal approxmaton and the margnal lower bound requre much less computatonal effort, and ther performance s reasonably good, they can be used n practce as quck estmates. Also, one of them s a (proven) lower bound, whle the other offers (n most cases) an upper bound. In Fgure 4 we plot the smulaton results reported n Table 3. The purpose s to llustrate the effect of leadtme varablty. Clearly, the leadtme varablty has more pronounced mpact on fll rates n the lower ranges. 8. CONCLUDING REMARKS We have studed a multproduct ATO system wth..d. leadtmes and ndependent base-stock controls at the components. Modelng the demand process as a multvarate compound Posson process, we have derved the ont dstrbuton of outstandng orders n terms of the generatng functon, along wth closed-form formulas for the frst and second moments of the ont dstrbuton. Examples were gven that shed lght on how product structures and leadtme dstrbutons affect correlaton. We have also establshed several comparson results, based on stochastc order relatons, whch yeld nsght to the mpact of varablty on system performance. In addton, we dscussed the value of advance demand nformaton. Fnally, based on the exact analyss, we developed several approxmaton schemes and bounds for the ont dstrbuton and related servce measures such as order fll rates. As mentoned n 2, sometmes the multple unts n a replenshment order can be treated as a sngle, ndvsble unt at the supply system. Therefore, the entre batch experences the same leadtme. In ths case, the arrval process to the supply subsystem s a multvarate Posson process (not compounded). Our analyss n 3 s stll vald, wth batch szes Q = 1 for all. As shown n (3), the ont congeston level at the supply subsystem, X, has a multvarate Posson dstrbuton. The net nventory of component can be expressed as X s Q n where Q n s an ndependent copy of Q. Thus, wth the ont dstrbutons of X and Q, one can compute the type-k order fll rate, ] X P Q n + QK s K and bounds based on smlar deas can also be developed. Further work s needed to develop computatonal effcent means to solve problems that seek the optmal nventoryservce trade-off n the type of multproduct ATO systems studed here. Along ths lne, several structural propertes of the system need to be further explored. These are the focus of our follow-up studes. APPENDIX Proof of Proposton 1. Consder any gven subset of components K. Let A K t be the cumulatve number of type-k arrvals up to tme t. Then A K t follows a Posson dstrbuton wth parameter K, and ts generatng functon s Ez AK t = e K tz 1 At any tme t, the number of obs n queue assocated wth type-k arrvals can be expressed as follows: AK t X K t = Q Kn k=1 1 k nt (33) where Q Kn denotes the nth ndependent copy of Q K, and t s the ndcator functon that the arrval to queue 1 k n

generated by the nth demand s stll n servce by tme t, n = 1 A K t. Let K n be the arrval epoch of the nth demand, n = 1 A K t. The event that the arrval to queue generated by nth arrval remans n servce by tme t s equvalent to that the processng tme for ths ob s longer than t n. That s, { 1 wp G c 1 k n t = t n K wp G t n K k = 1 Q Kn Hence, for any z, = 1 m,wehave m E = z XK t A K t K ] AK t K 1 K 2 K A K t QK1 Q KAK t G t n + z G c t K n ] Q Kn Observe that gven A K t, n K follows ndependent unform dstrbuton over t. Hence, ] m E z XK t A K t = 1 t t QK G 1 u + z 1 G c 1 u G mu + z m G c m u du ] AK t (34) Takng expectatons on both sdes wth respect to A K t, we obtan the generatng functon for X K t: K t z 1 z m t = exp K QK G 1 u + z 1 G c 1 u G mu Lettng t,wehave + z m G c m u 1 du ] K z 1 z m = exp K QK G 1 u + z 1 G c 1 u G mu + z m G c m u 1 du ] (35) Let X K = X1 K XK m be the random vector correspondng to the above generatng functon K.WehaveX K t X K n dstrbuton. As the arrval processes, A K t K, are ndependent across K, the generatng functon of Xt = X 1 t X m t s Lu, Song, and Yao / 35 t z 1 z m t = exp K QK G 1 u + z 1 G c 1 u G mu K + z m G c m u 1du ] (36) Lettng t, we obtan the desred expresson n (2). Proof of Proposton 2. To show (), note that the nequalty n (15) mples G c u du G c u du T T Ths can be drectly verfed va ntegraton by parts. (Also see Barlow and Proschan 1975, Theorem 7.3. From (14), we observe that E T X s a lnear combnaton wth postve coeffcents of T Gc u du, for T T ; hence, we have the desred nequalty. To show the orderngs n () we frst consder unt arrvals. In ths case, X follows a multvarate Posson dstrbuton, whch s known (Johnson et al. 1996, Kawamura 1979) to be the lmtng dstrbuton of a sequence of multvarate bnomal varates Y n. The generatng functon of Y n s ( 1 n z 1 z m = a 1 = 1 a m = p a1 a m z a 1 1 z a m m ) n (37) wth np a1 a m w a1 a m as n, where the lmts w a1 a m relate to X s va the followng: EX = 1 1 a 1 = a 1 = a =1 a =1 1 w a1 a 1 1a +1 a m a m = 1 1 1 EX X = a =1 1 w a1 a 1 1a +1 a 1 1a +1 a m a m = In general, for any T 1 m,wehave ] 1 1 E X = w a1 a m T a 1 = 1T a m = mt where T = { T 1 T c In partcular, we can choose p a1 a m = w a1 a m /n to construct each Y n. Also, from (37), we see that Y n s the summaton of ndependent and dentcally dstrbuted multvarate Bernoull varates Z n, whch have the followng common probablty generatng functon: 1 1 Z n z 1 z m = p a1 a m z a 1 1 z a m m a 1 = a m =

36 / Lu, Song, and Yao The above argument apples to the tlde system as well. Hence, smlarly, we have multvarate bnomal varates Ỹ n, and the correspondng Z n. From the nequalty establshed n Part (), we can see that for each T, 1 1 1 1 p a1 a m p a1 a m a 1 = 1T a m = mt a 1 = 1T a m = mt Notce that for the multvarate Bernoull varates, Z n and Z n, the above mples that for any vector u = u 1 u m wth u 1, wehave PZ n u P Z n u That s, Z n uo Z n Because ths orderng s preserved under..d. summaton (e.g., Shaked and Shanthkumar 1994, Theorem 4.B.1), we have Y n uo Y n. Lettng n, we obtan the desred X uo X. Next, consder batch arrvals. To be specfc, wrte PQ K = r = 1 m= pr K 1 r m. Each vector r 1 r m corresponds to a fxed bll of materals. We can decompose the arrval process for type-k demands nto several ndependent Posson arrval processes, one for each confguraton r 1 r m. In other words, X = r 1 r m X r 1 r m, where X r 1 r m s the number of outstandng orders generated by correspondng demand and has the followng generatng functon: r 1r m z 1 z m ] m =E Xr 1rm z ] ] m =exp p r1 r m G u+z G c ur 1 du On the other hand, consder another generatng functon: ] m r exp p r1 r m G u+z k G c ]du u 1 (38) k=1 It s not dffcult to see that the random vector that corresponds to the above generatng functon s the lmt of a sequence of multvarate bnomal varates. Ths mples that the generatng functons of the latter wll converge to the generatng functon n (38). Lettng z k = z for all k = 1 r, we obtaned the desred result. As for the lower-orthant order, notce that for any T 1 m ( 1T a 1 = mt a m = meanwhle, p a1 a m ) n = n 1T mt n 1T mt 1T mt n Now, the convex order mples that, for any U 1 m, the followng holds: ] ] G u 1 du G u 1 du U U (The above can agan be drectly verfed va ntegraton by parts, or see Barlow and Proschan 1975, Theorem 7.4.) Hence, we have, for any T, ( 1T mt =exp K K 1T mt ( =exp K K G u 1 T K T K Consequently, for n large enough, we have n 1T mt n 1T mt or equvalently, ( 1T mt a 1 = a m = p a1 a m ) n ( 1T a 1 = mt a m = ] du ) ] ) G u 1 du p a1 a m ) n whch mples that Z n lo Z n, and hence X lo X by the same argument as n the case of upper-orthant orderng. Fnally, for the comparson result wth respect to the varablty of batch szes, the above argument also apples, once we observe that a convex order on the batch sze, along wth (14), wll guarantee E T X E X T, whch s the startng pont of the above argument. Proof of Proposton 3. Consder demand type K, wth the correspondng arrval process beng A K t; the assocated outstandng orders of component, X K t follows the expresson n (33). Now, let A K t, = 1 m, be..d. copes of A K t, and wrte AK t ˆX K t = Q Kn k=1 1 k n t Wrte ˆX K t = ˆX 1 Kt ˆX m K t. Note the followng: For any a 1 a m,wehave PA K t a 1 A K t a m ] m = P A K t mna PA K t a That s, A K t A K t lo A K 1 t AK mt (39) For each = 1 m, defne a functon y Q Kn ] h y = 1 1 k n t a k=1 Obvously, h y s nonnegatve and decreasng (.e., nonncreasng) n y. Hence, followng Shaked and Shanthkumar (1994 Theorem 4.G.1) we know that the lower-orthant orderng n (39) mples ] ] m m m E h A K t E h A K t = Eh A K t