Backorder Minimization in Multiproduct Assemble-to-Order Systems

Transcription

1 Backorder Mnmzaton n Multproduct Assemble-to-Order Systems Yngdong Lu IBM T.J. Watson Research Center Yorktown Heghts, NY Tel. (914) , e-mal: yngdong@us.bm.com Jng-Sheng Song The Fuqua School of Busness Duke Unversty, Durham, NC Tel. (919) , e-mal: jssong@duke.edu Davd D. Yao Dept. of Industral Engneerng and Operatons Research Columba Unversty, New York, NY Tel. (212) , e-mal: yao@columba.edu January 2004; revsed September 2004, December 2004 Abstract We consder a multproduct assemble-to-order system. Components are bult to stock wth nventory controlled by base-stock rules, but the fnal products are assembled to order. Customer orders of each product follow a batch Posson process. The leadtmes for replenshng component nventory are stochastc. We study the optmal allocaton of a gven budget among component nventores so as to mnmze a weghted average of backorders over product types. We derve easy-to-compute bounds and approxmatons for the expected number of backorders and use them to formulate surrogate optmzaton problems. Effcent algorthms are developed to solve these problems, and numercal examples llustrate the effectveness of the bounds and approxmatons. 1

2 1 Introducton Realzng the compettve advantages of beng flexble, responsve as well as effcent, and enabled by advanced technologes, more and more manufacturng companes are reengneerng ther product desgn and delvery processes to move toward mass customzaton(see, e.g., Pne 1993). Ths requres modularzng the producton process so that the product can be quckly assembled from standardzed components and modules n dfferent confguratons, based on what customers ndvdually request. As a result, a new knd of producton-nventory system has emerged and s becomng ncreasngly popular. Ths s the assemble-to-order (ATO) system: nventores are kept only at the component level, and the fnal products are assembled only after customer orders are receved. Ths new system, n turn, presents challengng operatonal and system desgn ssues to managers. As each customer order typcally nvolves several components n dfferent amounts, the stockout of any component wll cause a delay n fulfllng the order. So, the optmal stock level of one component should be determned n conjuncton wth those of other components to ensure the smultaneous avalablty. Standard sngle-tem nventory plannng tools, whle sutable for the mass-producton make-to-stock envronment, are no longer applcable. New plannng tools are needed to strke the optmal nventory-servce tradeoff n ATO systems. The current paper presents an effort towards ths goal. More specfcally, we consder an ATO system supportng multple types of demand, whch arrve at the system followng compound Posson processes. The component nventores are resuppled from outsde supplers after random replenshment leadtmes. For a gven component, the leadtmes are ndependent, dentcally dstrbuted (..d.) random varables. The leadtmes for dfferent components are also ndependent but may have dfferent dstrbutons. Snce the form of the optmal nventory-control polcy for ths system s unknown, base-stock polces are wdely adopted n practce. For ths reason, we assume the nventory of each component s controlled by a basestock polcy. The system performance measure we focus on s the expected backorders for each product. Our objectve s to mnmze a weghted average of backorders over all product types, subject to a budget constrant on the component nventory. Through Lttle s Law (Wolff 1989), ths objectve relates drectly to the response tme performance n fulfllng customer orders. The optmzaton problem under study s qute complex. Frst, ts objectve functon s nonseparable and non-dfferentable. Second, ts evaluaton nvolves jont probabltes, whch can be computatonally challengng. To deal wth the second dffculty, we develop upper and lower bounds on the backorders that nvolve margnal dstrbutons only and use these bounds and approxmatons derved from them as surrogate objectve functons n the optmzaton problem. Ths approach s 2

3 smlar n sprt to that n Song and Yao (2002) for the sngle-product ATO system. However, even wth the smpler surrogate objectve functons the frst dffculty remans. Moreover, the bounds for the mult-product model here are more nvolved than n the sngle-product case and exhbt dfferent structures, so the deas used to develop the algorthms n Song and Yao do not apply. In partcular, the lower bound n the mult-product case leads to an optmzaton problem known as the mn-max-sum resource allocaton problem (see, e.g., Kouvels et al. (2001)), for whch known algorthms are lmted to the branch-and-bound types. One man contrbuton of ths paper s the dentfcaton of a specal property n our model, whch we call a stack structure. Usng ths property, we are able to develop a much more effcent greedy algorthm to solve the problem. Another nterestng fndng s that a modfcaton of the upper bound approach n Song and Yao (2002) produces not only a good approxmaton of the average backorder but also a surrogate optmzaton problem that bears the same stack structure as the one n the lower-bound approach. Several other authors have worked on optmzaton models for ATO systems. These studes are qute dfferent from ours n the detaled modellng assumptons and soluton approaches. For example, nstead of the contnuous-tme scheme consdered here, one stream of research employs a dscrete-tme (perodc-revew) model. Wthn ths framework, Hausman et al. (1998) assume that demands n dfferent tme perods are ndependent, and demand n each perod follows a multvarate normal dstrbuton. Component replenshment leadtmes are constant. The paper develops heurstc methods to solve the problem of maxmzng a lower bound on the fll rate, whch s a multvarate normal probablty, subject to a lnear budget constrant. Usng the same framework, Agrawal and Cohen (2001) study the problem of mnmzng the total expected component nventory costs subject to constrants on the order fll rates. Instead of FCFS allocaton rule, they assume a combnaton of FCFS and a far-share allocaton polcy. Cheng et al. (2002) also use the dscrete-tme formulaton and study the problem of mnmzng average component nventory holdng cost subject to product-famly dependent fll rate constrants. They assume..d. replenshment leadtmes and FCFS allocaton rule. An exact algorthm and a greedy heurstc are developed. Among artcles usng a contnuous-tme framework, Gallen and Wen (2001) assume..d. component leadtmes n a sngle-product assembly system. Demand s Posson wth rate λ. To keep the analyss tractable, the paper mposes a synchronzaton assumpton, namely, that components are assembled n the same sequence they are ordered n. (Order synchronzaton s not assumed n our model.) In the context of a repar shop, Cheung and Hausman (1995) assume..d. leadtmes and multvarate Posson demand model. Ths s a specal case of ours (the unt demand case). Unlke n our paper, they assume complete component cannbalzaton, whch makes sense n the repar shops but not n ATO systems. Under ths assumpton, the product backorders s the max- 3

4 mum of the component backorders, so the bockorder functon s much smpler than and completely dfferent from ours. Lu and Song (2002) study the multproduct unt demand system. Assumng there s a backorder cost rate assocated wth each backlogged customer order, they formulate an unconstraned optmzaton model to mnmze the expected average holdng and backorder costs. Wang (1999) consders multvarate compound Posson demand but the supply process for each component s capactated and modeled as a sngle-server queue. Applyng the asymptotc results developed n Glasserman and Wang (1998), he examnes the problem of mnmzng average nventory cost subject to a fll-rate constrant. Our study appears to be the frst attempt to tackle optmzaton of mult-product backorders wth batch demand. We refer the reader to Song and Zpkn (2003) for a more detaled survey of the state-of-the-art research on ATO systems. The rest of the paper s organzed as follows. We start wth model descrpton and some prelmnares n??. In??, we develop the lower bounds for backorders, as well as soluton approaches to the lower-bound problem. We then turn to consderng upper bounds n??. In?? we present numercal studes to assess and compare the effectveness of the the soluton approaches developed n the prevous sectons. 2 Model and Prelmnares Let I = {1, 2,, m} denote the set of all component ndces. Customer orders arrve at the system followng a statonary Posson process, denoted {A(t), t 0}, wth rate λ. Each order may requre several components n dfferent amounts smultaneously. For any subset of components K I, we say an order s of type K f t conssts of postve unts of each component n K and 0 unts n I \K. We assume that there s a fxed probablty q K that an order s of type K, K qk = 1. Thus, the type-k order stream forms a compound Posson process wth rate λ K = q K λ. A type-k order requests Q K j unts of component j K, and Q K = (Q K j, j 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 K to be the set of all demand types, that s, K = {K I : q K > 0}. Note that K s not necessarly the set of all possble subsets of I. Ths demand model s qute general and covers a number of mportant specal cases. For example, f Q K j are determnstc, then Q K represents the bll-of-materals (BOM) of the type-k product. If Q K j = Qξj K where ξj K are constant and Q s a random varable, then (ξj K) represents the BOM of the type-k product whle Q s the random quantty of ths product that a customer demands. For each component, let K denote the famly of subsets of K that contan. It s clear that 4

5 the demand process for component forms a compound Posson process wth rate λ = K K λ K and batch sze Q, the mxture of Q K for all K K. (Throughout the paper we use subscrpts to ndcate for component types and superscrpts for demand types.) The overall demand rate for component s thus µ = K K λ K E(Q K ). Note, however, that these compound Posson processes are no longer ndependent. Demands are flled on a frst-come-frst-served (FCFS) bass. Upon the arrval of any demand, f there s enough on-hand nventory for all the components requred by the demand, then the demand s flled mmedately. In other words, the tme to assemble the components nto the end-product s assumed 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. We remark that FCFS s a suboptmal allocaton rule. Among other thngs, an optmal allocaton polcy needs to take nto account the smultaneous avalablty of component stocks. However, due to the complexty of the system, the structure of optmal or near-optmal allocaton rule s not yet known. Because of ts smplcty and ease of mplementaton, FCFS s wdely used across ndustres. FCFS also makes the estmaton of product backorders more tractable. Alternatve approaches have been dscussed n the lterature, mostly n a dscrete-tme formulaton, so that some knd of myopc allocaton s used wthn each perod. When dealng wth orders n dfferent perods, however, most of these approaches use FCFS. In our contnuous-tme settng, there s no order batchng, so myopc allocaton s not applcable. See Song and Zpkn for detals. 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 onorder poston mnus 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 5

6 the optmal polcy s stll unknown. Hence, the ndependent base-stock polces are wdely used n practce because of ther smplcty. Snce 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 E[L ] = l. Denote G c = 1 G. Assume the leadtmes are ndependent among the components; that s, L s ndependent of L j for any j. 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) It s clear that X (t) s equal to the number of jobs n servce n an M/G / queue, wth arrval rate λ, for = 1,..., m. However, snce the arrval of any demand of type K generates smultaneous arrvals at all the component queues, K, these m queues are not ndependent. Lu et al. (2003) show that the jont dstrbuton of (X 1 (t),..., X m (t)) has a steady-state lmt (X 1,..., X m ) wth probablty generatng functon m ψ(τ 1,..., τ m ) := E[ j=1 τ X j j ] [ ] = exp λ K (ψ QK (G 1 (u) + τ 1 G c 1(u),..., G m (u) + τ m G c m(u)) 1)du, (2) K K 0 where ψ QK (x 1,..., x m ) denotes the generatng functon of Q K = (Q K ) K. We can use ψ to derve the moments and covarances of X s. In partcular, θ := E[X ] = l λ K E(Q K ) = µ l, (3) K K σ 2 := Var[X ] = θ + λ K [E((Q K ) 2 ) E(Q K )] [G c j(u)] 2 du. (4) K K 0 And for j, σ j := Cov[X, X j ] = K:,j K λ K E(Q K Q K j ) 0 G c (u)g c j(u)du. (5) It can be verfed that σ 2 θ for all. Also, σ j 0 and t s zero f and only f there s no demand types that requre both and j. 6

7 For unt demand model,.e., Q K j = 1, for all j K, ψ(τ 1,, τ m ) s the probablty generatng functon of a multvarate Posson dstrbuton. That s, the margnal dstrbuton of X s Posson wth mean λ l, = 1,, m. In ths paper, the performance measures of prmary concern are: B K = steady-state number of backordered type-k demand, for all K. We now show how these quanttes depend on the jont dstrbuton of (X 1,..., X m ). Denote x + := max{0, x}. Note that B = steady-state number of backordered component = [X s ] + depends on the margnal dstrbuton of X only. So t s easy to compute. To dentfy B K, t s mportant to fgure out the correspondng component backlogs that contrbute to t. Let B K be the steady-state number of backlogged unts of component that are due to type K demand. We have ( ) B B K λ K E[Q K = 1 ] k J K λ J E[Q J ] k=1 B = k=1 ( λ K E[Q K 1 ] ) k, (6) µ where 1 k (p) s are..d. Bernoull random varables wth parameter p. In the case of unt demand, the number of backorders of type K s B K = max K {BK }, where B K = B k=1 ( ) λ K 1 k. (7) λ When demand arrves n batches, however, we must frst translate B K nto batches. Let B Q,K := nf{n : where Q K (k), k = 1,..., n, are ndependent copes of QK. Then, n Q K (k) B K }, (8) k=1 B K = max K BQ,K. (9) Let c be the unt cost of component. Let w K 0 be a weghtng factor for the average backorder for product K. It measures the relatve mportance of servce provded to type K orders. Denote s := (s 1,..., s m ) and Z m + the m-dmensonal nonnegatve nteger vector space. We are nterested n the followng optmzaton problem: (P ) mn s w K E[B K (s)] s.t. c 1 s c m s m C, s Z+ m. (10) K K 7

8 where C > 0 s gven constant, representng the total nventory budget. In ndustral applcatons, nventory budget often apples to safety stock only. Snce nventory can be dvded nto WIP and safety stock, and the WIP part s ndependent of the base-stock level, the constrant n the above problem formulaton s consstent wth practce. (Also see Sherbrooke (1992).) We note that, nstead of a fxed budget on the target safety stock levels, another commonly seen formulaton s to mnmze average nventory-holdng and backorder-penalty cost. Solvng ths type problems wll requre a dfferent set of technques from those developed here; see Lu and Song (2002) for the unt-demand case. As ponted out n the Introducton, the objectve functon n (??) relates to a delay objectve va Lttle s Law: Suppose we want to mnmze w K E[W K (s)], K K where W K denotes the delay (response tme) n fllng type K orders, and w K s the assocated cost. Then, we can wrte w K E[W K (s)] = wk λ K E[BK (s)]; hence, we can set w K = w K /λ K n (??). Obvously, the objectve functon n (P) s very dffcult to evaluate due to the nvolvement of the jont dstrbuton of (X 1,..., X m ) and other ntrcate relatons, such as the max operaton. In the rest of the paper we develop two approxmate objectve functons whch nvolve only the margnal dstrbutons of X. We further relax the nteger requrement of the decson varables and treat s s as non-negatve real values, as they are nvolved n bounds and approxmatons. In many applcatons, s s are large values; hence, gnorng the ntegralty s approprate. The result s two surrogate problems, and we develop effcent soluton approaches to solve them. 3 The Lower-Bound Approach We now develop an approxmaton of E[B K ] that uses margnal dstrbuton of X only. One key step s to apply Jensen t nequalty to a max operaton, resultng n a lower bound. Therefore, we call ths s a lower-bound approach. and Note that from (??) we know that B Q,K B Q,K k=1 s a stoppng tme wth respect to {Q K (1), QK Q K (k) B K, Takng expectatons on both sdes and applyng Wald s dentty on the left-hand sdes yelds a.s. (2),, n}, E[Q K ]E[B Q,K ] E[B K ]. (11) 8

9 Now takng expectatons on both sdes of (??), and applyng Wald s dentty to the summaton on the rght-hand sde, we have E[B K ] = λk E[B ]E[Q K ] µ. (12) Hence, makng use of Jensen s nequalty along wth (??) and (??), we obtan E[B K E[B K { ] = E[max K BQ,K ] max K E[BQ,K ] λ K } ] max K E[Q K ] = max E[B ]. (13) K We now use the lower bound n (??) as a surrogate for the objectve functon n (??). Defne b (s ) = E[B (s )] = E[X s ] +. Then, the resultng lower-bound problem can be expressed as: (LB) mn s K K w K λ K max K {b (s ) µ } (14) s.t. c 1 s c m s m C; s Z m +. Clearly, the objectve functon n (LB) s much easer to evaluate because only the margnal dstrbutons are nvolved. Two extreme cases of ths problem have been studed n the lterature. The frst case s that the component demands are ndependent,.e., q K = 0 f K s not a sngleton. In ths case the problem reduces to mn s m w b (s ) =1 s.t. c 1 s c m s m C; s Z m +. A nce feature of ths problem s that the objectve functon s a separable convex functon. Sherbrooke (1992) presents a greedy algorthm to solve ths problem. The second extreme case s a sngle-product assemble-to-order system,.e., K = {1,..., m} s the only demand type and the BOM s fxed. We can also redefne the unt of each component, so that we obtan a unt demand model. In ths case, the problem becomes mn s max{b 1 (s 1 ),..., b m (s m )} (15) s.t. c 1 s c m s m C; s Z m +. Usng the property that each b (s ) s decreasng and convex, Song and Yao (2002) shows that a greedy algorthm can guarantee optmalty. The general case, however, s much more complcated and less structured because we need to deal wth the sum of maxmums over overlappng subsets. To ncrease tractablty, n the followng 9 µ

10 we further relax the nteger requrement n (LB), and treat each s as a nonnegatve real varable. Let R m + denote the m-dmensonal nonnegatve real vector space. The relaxed problem s then, (LB ) mn s K K w K λ K max K {b (s ) µ } (16) s.t. c 1 s c m s m C; s R m +. The resultng objectve value s a lower bound for (LB), whch s n turn stll a lower bound on the orgnal problem. Although the objectve functon s convex, t s not dfferentable, so the conventonal tools for convex programs are not applcable. In the rest of ths secton we wll dentfy specal structures of the problem and show that we can transform the problem nto one that mnmzes the maxmums over non-overlappng sets, whch n turn can be converted nto a set of subproblems wth lnear objectve functons and ordered varables. The prce we pay s that the orgnal lnear constrant functon now becomes a non-lnear decreasng functon. Nonetheless, we wll show that each of these subproblems s greedly solvable, so we can obtan the overall optmal soluton effcently. Once we obtan a soluton for (LB ), we round t to the nearest nteger soluton as an approxmate soluton for the orgnal problem. 3.1 The Stack Structure A change of varable wll better reveal the structure of (LB ). Note that b (s ) = E[X s ] + s strctly decreasng convex n s R 1 +. Ths mples that ts nverse b 1 ( ) s well defned and also decreasng and convex. Snce lm s b (s ) = 0, we denote b 1 (0) = +. Now, for each = 1, 2,..., m, defne z := b (s ) µ, or s = b 1 (µ z ) := h (z ). (17) Let z := (z 1,..., z m ). Then, the problem (LB ) s equvalent to (LB) mn w K λ K max z (18) z K K K s.t. c h (z ) C; z R m +. Suppose z = (z 1,..., z m) s an optmal soluton to (??). Denote u K := max K z. Obvously, f K K then u K u K. Wthout loss of generalty, rename the product types n ascendng order of u K : u K1 u K2. We now argue that we can assume that all z, K 1 have the same value. Ths s because f there exsts an 0 K 1 such that z 0 < u K1 := max j K 1 z j, 10

11 then, reassgnng z 0 the value u K1 wll not destroy the optmalty of z : the objectve value wll reman the same whle the constrant wll stll be satsfed because h 0 ( ) s a decreasng functon. Smlarly, note that u K2 := max j K2 zj s the second smallest of the u K values. For any K 2 \ K 1, f z < u K2, then we can ncrease ts value to u K2 wthout changng the objectve value or volatng the constrant so to preserve the optmalty of z. Contnung ths argument, we can conclude that there exsts an optmal soluton z to (??) that satsfes a stack-lke structure as formalzed below. Suppose K = {K 1,..., K n },.e., there are total of n product (or, demand) types. Then we can wrte the set of all components as I = n l=1 K l := {1,..., m}. Consder a partcular order (permutaton) of the n subsets, π = (π 1, π 2,..., π n ). subsets nductvely as follows: Form n new K π 1 = K π1, K π l = K πl \(K π 1 K π l 1 ), l = 2,..., n. (19) Then, clearly, (K π 1,..., K π n ) s a partton of I (whereas (K π1,..., K πn ) s not). n > m then some K π l s wll be empty. Note that f Proposton 1 There exsts an optmal soluton z to (??) satsfyng the followng stack-lke structure. Suppose π = (π 1, π 2,..., π n ) s a permutaton such that u πl := max j Kπl zj s ncreasng n l,.e., u Kπ1 u Kπ2 u. Kπ n Let (Kπ 1,..., Kπ n ) be the partton of I specfed n (??). Then, we have, z j = u Kπl, j K π l for all l = 1,..., n. 3.2 The Optmzaton Algorthm To smplfy notaton, denote, for each permutaton π = (π 1, π 2,..., π n ), λ π l := λ K π l, w π l := w K π l, uπl := u Kπl, u π = (u π1,..., u πn ). Furthermore, based on Proposton??, denote g l (z) := c h (z). Kπ l 11

12 We can then rewrte the optmzaton problem n (??) as follows. (LB ) mn π s.t. mn u π n w π l λ π l u πl (20) l=1 n g l (u πl ) C, (21) l=1 u π1 u πn, u π R n +. Hence, gven the permutaton π, denote the nner mnmzaton part of the above problem, ncludng the constrants, as (LB π ). We can solve (LB π ) for all π; the one that yelds the smallest objectve value s then the optmal soluton to the orgnal problem (LB ). For each π, (LB π ) s equvalent to a problem of mnmzng a separable convex functon wth a sngle lnear constrant, hence, t s greedly solvable va a margnal allocaton algorthm. To descrbe ths algorthm, reformulate the problem wth yet another transformaton of varables: y 1 := u π1 ; y l := u πl u πl 1, l = 2,..., n. Hence, u πl = y y l, l = 2,..., n. Also denote y := (y 1,..., y n ); Then, (LB π ) becomes the followng: (LB π ) v l := w π l λ π l + + w πn λ πn, l = 1,..., n. mn y s.t. n v l y l (22) l=1 n g l (y y l ) C, y R n +. l=1 The margnal allocaton scheme works as follows: Start wth y = 0; hence, the zero objectve value. In each step, allow an ncrease of > 0 of the objectve value, wth a prespecfed constant. Ths corresponds to allowng y k to ncrease by an amount /v k, for each k = 1,..., n. (Examnng these vertex ponts s suffcent because g l ( ) s convex). Ths, n turn, corresponds to the decrease of the left hand sde of the constrant by the followng amount: n [g l (y y k + + y l ) g l (y y k + /v k + + y l )]. l=k (Recall, g l ( ) s a decreasng functon, for all l. ) Therefore, select the k that yelds the largest such decrease, and update y k to y k + /v k. Repeat ths procedure untl the left hand sde of the constrant s reduced to C. 12

13 3.3 Heurstcs Although for each π, (LB π ) s greedly solvable, we stll need to solve n! such problems. One heurstc s to do parwse mprovement: Start from any permutaton, say, (1,..., n). Compare ths wth all other permutatons that result from nterchangng 1 wth l 1. Ths nvolves solvng n margnal allocaton problems lke (LB π ) above. Suppose swappng 1 and π 1 gves the smallest objectve value. Next, fx π 1 at the frst poston, apply the same parwse nterchange to the remanng n 1 elements of the permutaton, and so forth. Ths heurstc requres solvng n(n 1)/2 (as opposed to n!) margnal allocaton problems. An even smpler heurstc s to apply the margnal allocaton scheme that solves (LB π ) drectly to the problem (LB ) n (??). The detals are as follows: Start wth π = (1,..., n), and set u πl = 0 for all l. Suppose the current soluton s (u πl ), correspondng to some permutaton π = (π l ). For each k = 1,..., n, do: (a) ncrease u πk to u πk + /(w π kλ π k); rearrange the u values, and let π k denote the resultng permutaton; (b) evaluate the left hand sde of the constrant n (??) Select k as the one correspondng to the smallest value n (b) above (.e., the one that yelds the largest decrease n the left hand sde of the constrant). Update u πk to u πk + /(w π k λ π k ); and π to π k. Repeat the above untl the left hand sde of the constrant n (??) s reduced to C. Remarks Note that wth any gven permutaton the orgnal dscrete lower bound surrogate problem (LB) s a separable dscrete convex programmng, whch can be easly solved by a greedy algorthm. So the heurstc algorthm above can also be appled to solve the problem n (LB). 4 The Upper-Bound Approach We now swtch to usng an upper bound approach to derve surrogates, whch ncludes an upper bound and some approxmatons derved from upper bounds, for the objectve functon n the backorder mnmzaton problem n (??). The man dea s the followng nequalty: for a set of non-negatve varables, x 0, we have max{x } a + (x a) +, (23) 13

14 whch holds for any a, and can be drectly verfed. Hence, mnmzng the rght hand sde above, wth respect to a, yelds an upper bound for max {x }. Note, however, when a < 0, the rght hand sde becomes a + (x a) x. So, n mnmzng the rght hand sde of (??), we only need to consder a 0. When (x ) s a vector of non-negatve random varables, the upper bound has the advantage of nvolvng only the margnal dstrbutons. Hence, applyng ths to (??), we have B K := max{b Q,K } a K + (B Q,K a K ) +, a.s., (24) K where a K s a parameter. Therefore, usng the above upper bound as a surrogate objectve, and wrtng a := (a K ) K K (and s := (s ) I as before), we have the followng optmzaton problem: E(B Q,K a K ) + (25) mn (a,s) K K w K a K + K K w K K s.t. c 1 s c m s m C, s R m +. In the above problem, mnmzng over a s not as mportant as mnmzng over s, as the former s meant to mprove on the upper bound. (In fact, as we shall llustrate below, we could very well forgo the mnmzaton over a by fxng t at a specfc value.) Hence, we propose to use a common (scalar) a for all product type K, and then optmze a. Ths way, the problem becomes, (U B) mn (a,s) a w K + w K K K K K K E(B Q,K a) + (26) s.t. c 1 s c m s m C, s 0. For fxed a, (UB) s a separable convex programmng, whch s greedly solvable. We can then conduct a lne search on a. Alternatvely, selectng a = E(B Q,K ), we have: { } B K max E(B Q,K ) + K K Takng expectatons on both sdes yelds E[B K ] max K Note that from (??) we have { E(B Q,K ) [B Q,K } + K E(B Q,K )] +, a.s. E[B Q,K E(B Q,K )] +. (27) B K (B Q,K 1) + k=1 Q K (k), a.s., 14

15 where Q K (k) are ndependent copes of QK and 0 k=1 QK (k) = 0. Applyng Wald s dentty yelds E[B K ] E[Q K ]E[B Q,K 1] +. (28) It s reasonable to use B K /E[Q K ] to approxmate BQ,K. Ths relatonshp s exact for the unt demand model. Consequently, we can use E[B K ]/E[Q K ] to approxmate E[BQ,K ]. Apply these approxmatons to (??), we obtan { E[B E[B K K ] max ] } K E[Q K ] + E[B K E(B K )] + E[Q K K ]. (29) We now go one step further by replacng the summaton over n the rght-hand sde of (??) wth the maxmum over, resultng n { E[B E[B K K ] max ] K E[Q K ] + E[BK E(B K )] + } E[Q K ] { λ K = max K µ E[B K ] + E[BK E(B K )] + E[Q K ] }. (30) Usng the above approxmaton n the objectve of (P ) yelds the followng surrogate problem: (AP P ) mn s K { λ w K K max E[B (s )] + E[BK (s ) EB K (s )] + } K µ E[Q K ] s.t. c 1 s c m s m C, s R m +. Note that although the orgnal dea was to use an upper bound to surrogate the orgnal objectve functon, we took two steps of approxmatons to revse the upper bound functon to acheve (??). Therefore the objectve functon n (AP P ) s no longer an upper bound. In the rest of ths secton we shall show that the (AP P ) exhbts the same structure as n (LB ). The followng proposton tells us that E[B K (s ) EB K (s )] + s decreasng n s. So, the objectve functon n (AP P ) has the same property, and therefore ts nverse functon s unquely defned. Proposton 2 For any convex functon f(x), Ef[B K E(B K )] s decreasng n s. In partcular, Var(B K ) s decreasng n s, for each and each K ; and Var(B ) s decreasng n s, for all. Proof. We frst show that Var(B ) s decreasng n s. Recall, B = [X s ] +. Hence, t suffces to show E[(X s ) + ] 2 E[(X s 1) + ] 2 (31) [E(X s ) + ] 2 [E(X s 1) + ] 2. (32) 15

16 We have, (X s ) + (X s 1) + = 1(X s + 1), (X s ) + + (X s 1) + = (2X 2s + 1)1(X s + 1). (Note that we are treatng X s, as well as s s as nteger valued.) Hence, E{[(X s ) + ] 2 [(X s 1) + ] 2 } = E[(2X 2s + 1)1(X s + 1)]; where as [E(X s ) + ] 2 [E(X s 1) + ] 2 = [E(X s ) + + E(X s 1) + ][E(X s ) + E(X s 1) + ] = E[(2X 2s + 1)1(X s + 1)]P(X s + 1). Hence, the nequalty n (??) holds. Next, wrte B K,n as BK condtonng upon X s = n, for n = 0, 1,... From (??), snce X s ndependent of the Bernoull varables nvolved, t suffces to show that Ef[B,n K E(BK,n )] s ncreasng n n. Denote M n := B K,n E(BK,n ). From (??), clearly, {M n} forms a martngale (e.g., Ross 1996). Hence, the convexty of f mples that {f(m n )} s a submartngale, whch, n turn, mples that Ef(M n ) s ncreasng n n, and hence decreasng n s, whch s what s desred. Lettng f(x) = x 2, we have E(Mn) 2 = Var(B,n K ). Note that we can wrte Var(B K ) = E[Var(B K X )] + Var[E(B K X )]. (33) The frst term on the rght hand sde above s decreasng n s, snce Var(B,n K ) s ncreasng n n gven X s = n as argued above. Followng (??), the second term s equal to (λ K E[Q K ]/λ )Var(B ); hence, t s also decreasng n s, from what has already been shown for Var(B ). Therefore, we can conclude that Var(B K ) s decreasng n s. Fnally, observe from (??) that B K (s )/(λ K E[Q K ]) and [X s ] + k=1 1 k (1/µ ) have the same frst two moments and the latter s ndependent of K. We replace the former wth the latter as an approxmaton. Ths leads to the followng problem (AP P ) mn s K { w K λ K E[B (s )] } max + r(s ) K µ (34) s.t. c 1 s c m s m C, s R m +. 16

17 where [X s ] + ( ) 1 r(s ) = E[ 1 k E[X s ] + ] +. µ µ k=1 Followng smlar arguments n the proof of Proposton??, we can show that r(s ) s also a decreasng functon of s. Therefore, (AP P ) has the same structure as the lower-bound problem n the last secton. So, the same optmzaton and heurstc algorthms can be appled here. 5 Numercal Examples In ths secton, we present results from a set of numercal experments, n whch the heurstc algorthms derved n the prevous sectons are tested for ther effcency n fndng the (near- )optmal solutons to the backorder mnmzaton problems. As mentoned before, what complcates the multproduct problem s the number of overlappng subsets that consttute dfferent types of products, not the number of components. For nstance, f one set of products share no common components wth another set of products, then we can smply decompose the orgnal system nto two separate systems, each of whch covers one set of products. An extreme case s that each product conssts of only a sngle component, resultng n a separable convex problem whch s greedly solvable (see Sherbrooke). For ths purpose, n our experment we choose examples that are complex n the sense that there are many common components shared by dfferent products. We consder an ATO system wth sx components and the followng sx product types, {2, 5}, {3, 5}, {1, 2, 5}, {1, 3, 6}, {1, 3, 4, 5}, {1, 3, 4, 6}. The overall demand arrval rate s λ = 4, whch splts nto the sx product types as follows: q 25 = 0.10, q 35 = 0.40, q 125 = 0.15, q 136 = 0.10, q 1345 = 0.20, q 1346 = The component leadtmes follow ndependent exponental dstrbutons wth means (l ) 6 =1 = (1, 1, 1, 1, 2, 2). We consder two sets of cost data: c (1) = (1, 1, 1, 1, 1, 1) and c (2) = (2, 2, 3, 2, 1, 1), wth respectve weghts w (1) = (1, 1, 1, 1, 1, 1) and w (2) = (1, 1, 1.2, 1.5, 1.2, 1.5), and varous nventory budget levels: C = 20, 24, 32 for c (1) ; and C = 30, 40, 50 for c (2). In Tables 1 and 2, we summarze the solutons obtaned by () an exhaustve search va smulaton, () the lower-bound approach along wth the heurstc algorthm n??, () the upper-bound approach based on (UB), and (v) the approxmaton approach based on (AP P ). For each approach, we report the optmal base-stock levels, the correspondng objectve value, and the resultng 17

18 weghted average backorder. Note that whle the last two measures are equal for the smulaton based optmal soluton, they are dfferent for the other three approaches. For nstance, the objectve value correspondng to () s necessarly a lower bound of the objectve value of (); whereas the weghted average backorder correspondng to () s obtaned from usng the optmal base-stock levels derved from the lower-bound approach (to evaluate analytcally the orgnal objectve). We also report the percentage loss of optmalty of each approxmate approach. In all of our numercal examples, the leadtmes follow exponental dstrbutons, the mpact of dfferent dstrbutonal assumpton upon the performance of these assemble-to-order systems can be founded n [?]. It s also ponted out n [?] that the dfference caused by dstrbutonal assumpton s not sgnfcant especally for hgh performance systems. Therefore, we beleve that these assumptons wll not affect the accuracy of our approxmatons. In Tables 3 and 4, we repeat the same example wth λ = 8, and modfed budget levels: C = 30, 36, 45 for c (1) ; and C = 40, 50, 60 for c (2). [INSERT TABLE 1-4 HERE] The results ndcate that solutons from the surrogate problems are very close to the true optmal soluton n all cases. Overall, the approxmaton approach (AP P ) appears to be the most effectve method. We now use these examples to shed some lght on how the optmal component nventory levels dffer from a standard sngle-tem approach used n practce. Recall that the sngle-tem nventory theory suggests that we should stock enough of each component to cover the leadtme demand plus some safety stock to protect aganst varablty (of both demand and leadtme). Accordng to (1), (3) and (4), we have s = θ + k σ, (35) where θ s the average leadtme demand for component ; σ, the correspondng standard devaton; and k, the safety factor, a postve parameter ndcatve of the level of protecton aganst uncertanty or varablty. When there are multple tems (components), one must address how to choose k for each. What s often followed n practce s the so-called equal fractle rule havng the same k value across all components. Ths s equvalent to projectng the same level of fll rate across. Should ths be the case here? To answer ths queston, n Table 5 we report the optmal safety factors assocated wth the optmal base-stock levels n the prevous four tables. From these safety factors, we can derve the component fll rates from a normal dstrbuton table. For nstance, k = 1.00 corresponds to a fll rate of Φ(1) = 84.13%, where Φ(x) denotes the dstrbuton functon of the standard normal varate; 18

19 k = 2 corresponds to a fll rate of Φ(2) = 97.72%, and so forth. As can be observed from the table, the optmal safety factors vary wdely among components. In some cases, when the budget level s relatvely low, the safety factors of certan components are even negatve. These observatons suggest that the equal fractle rule n general s not effectve n ATO systems. Next to each optmal safety factor k, we also report the approxmate safety factor k derved from the approxmate (AP P ) solutons. Clearly, these examples ndcate that usng the approxmate approach we can come up wth close-to-optmal safety factors. [INSERT TABLE 5 HERE] To summarze, among the three surrogate problems and algorthms developed here, both (LB) and (UB) have a computatonal advantage and can provde a good predcton of the basc trend of the system behavor under varous changes. On the other hand, (AP P ), wth more computatonal effort than those of (LB) and (UB), provdes a good approxmaton to both the optmal soluton and the objectve value. (In cases where (AP P ) does not produced the best result, the relatve errors of the optmal objectves are below 3%.) References [1] Agrawal, m. and Cohen, M., Optmal Materal Control and Performance Evaluaton n an Assembly Envronment wth Component Commonalty. Naval Research Logstcs 48 (2001), [2] Chang, C.S., Shanthkumar, J.G., and Yao, D.D., Stochastc convexty and stochastc majorzaton, n Stochastc Modelng and Analyss of Manufacturng Systems (Chapter 5), D.D. Yao (ed.) Sprnger-Verlag, New York, [3] Cheng, F., Ettl, M., Ln, G.Y., and Yao, D.D., Inventory-Servce Optmzaton n Confgure-to-Order Systems, Manufacturng & Servce Operatons Management 4 (2002), [4] Cheung, K.L. and Hausman, W., Multple Falures n a Mult-Item Spare Inventory Model, IIE Transactons, 27 (1995), [5] Glasserman, P. and Wang, Y., Leadtme-Inventory tradeoffs n Assemble-to-Order Systems, Operatons Research, 46 (1998), [6] Gallen, J. and Wen, L., A Smple and Effectve Component Procurement Polcy for Stochastc Assembly Systems, Queueng Systems, 38 (2001), [7] Hausman, W.H., Lee, H.L. and Zhang, A.X., Order Response Tme Relablty n a Mult-Item Inventory System, European J. of Operatonal Research, 109 (1998), [8] Kouvels, P., Karabat, S. and Yu, G. A Mn-Max-Sum Resource Allocaton Problem and ts Applcatons, Operatons Research, 49 (2001),

20 [9] La, T. and Robbns, H., Maxmally Dependent Random Varables, Proc. Nat. Acad. Sc., 73 (1976), [10] Lu, Y. and Song, J.S. Order-Based Cost Optmzaton n Assemble-to-Order Systems, Preprnt, IBM T.J.Watson Research Center, Operatons Research, to appear. [11] Lu, Y., Song, J.S. and Yao, D. D. Order Fll Rate, Leadtme Varablty, and Advance Demand Informaton n an Assemble-to-Order System. Operatons Research 51 (2003), [12] Pne, B.J. Mass Customzaton: The New Fronter n Busness Competton. Harvard Busness School Press, Boston, [13] Ross, S.M., Stochastc Processes, 2nd ed., Wley, New York, [14] Shaked, M. and Shanthkumar, J.G., Stochastc Orders and Ther Applcatons, Academc Press, New York, [15] Shanthkumar, J.G. and Yao, D.D., Strong Stochastc Convexty and Its Applcatons, J. Appl. Prob. 28 (1991), [16] Shanthkumar, J.G. and Yao, D.D., Bvarate Characterzaton of Some Stochastc Order Relatons, Adv. Appl. Prob. 23 (1991a), [17] Sherbrooke, C.C., Optmal Inventory Modelng of Systems, Wley, New York, [18] Song, J.S., Order-Bases Backorders and Ther Implcatons n Mult-Item Inventory Systems Management Scence 48 (2002), [19] Song, J.S. and Yao, D.D., Performance Analyss and Optmzaton n Assemble-to-Order Systems wth Random Leadtmes, Operatons Research 50 (2002), [20] Song, J.S. and Zpkn, P. Supply Chan Operatons: Assemble-to-Order Systems, Chapter 11 of Supply Chan Management, T. De Kok and S. Graves, eds., n Handbooks n Operatons Research and Management Scence, Vol. XXX, North-Holland, [21] Wang, Y. Near-Optmal Base-Stock Polces n Assemble-to-Order Systems under Servce Levels Requrements, preprnt, MIT Sloan School, [22] Wolff, R., Stochastc Modelng and the Theory of Queues, Prentce Hall, Englewood Clffs, New Jersey,

21 Coeffcents Weght Budget Algorthms s1 s2 s3 s4 s5 s6 Objectve Ave. BackOrder Rel. Err. (%) (1,1,1,1,1,1) (1,1,1,1,1,1) C=20 Optmal Lower Bound Upper Bound Approxmaton C=24 Optmal Lower Bound Upper Bound Approxmaton C=32 Optmal Lower Bound Upper Bound Approxmaton (2,2,3,2,1,1) (1,1,1,1,1,1) C=30 Optmal Lower Bound Upper Bound Approxmaton C=40 Optmal Lower Bound Upper Bound Approxmaton C=50 Optmal Lower Bound Upper Bound Approxmaton Table 1: System: Sx product: q25=0.1,q35=0.4, q125=0.15, q136=0.1,q1345=0.2, q1346=0.05 Sx tems: L=(1,1,1,1,2,2) Overall Arrval rate=4

22 Coeffcents Weghts Budget Algorthms s1 s2 s3 s4 s5 s6 Objectve Ave. BackOrder Rel. Err. (%) (1,1,1,1,1,1) (1,1,1.2,1.5,1.2,1.5) C=20 Optmal Lower Bound Upper Bound Approxmaton C=24 Optmal Lower Bound Upper Bound Approxmaton C=32 Optmal Lower Bound Upper Bound Approxmaton (2,2,3,2,1,1) (1,1,1.2,1.5,1.2,1.5) C=30 Optmal Lower Bound Upper Bound Approxmaton C=40 Optmal Lower Bound Upper Bound Approxmaton C=50 Optmal Lower Bound Upper Bound Approxmaton Table 2: System: Sx product: q25=0.1,q35=0.4, q125=0.15, q136=0.1,q1345=0.2, q1346=0.05 Sx tems: L=(1,1,1,1,2,2) Overall Arrval rate=4

23 Coeffcents Weght Budget Algorthms s1 s2 s3 s4 s5 s6 Objectve Ave. BackOrder Rel. Err. (%) (1,1,1,1,1,1) (1,1,1,1,1,1) C=30 Optmal Lower Bound Upper Bound Approxmaton C=36 Optmal Lower Bound Upper Bound Approxmaton C=45 Optmal Lower Bound Upper Bound Approxmaton (2,2,3,2,1,1) (1,1,1,1,1,1) C=40 Optmal Lower Bound Upper Bound Approxmaton C=50 Optmal Lower Bound Upper Bound Approxmaton C=60 Optmal Lower Bound Upper Bound Approxmaton Table 3: System: Sx product: q25=0.1,q35=0.4, q125=0.15, q136=0.1,q1345=0.2, q1346=0.05 Sx tems: L=(1,1,1,1,2,2) Overall Arrval rate=8

24 Coeffcents Weghts Budget Algorthms s1 s2 s3 s4 s5 s6 Objectve Ave. BackOrder Rel. Err. (%) (1,1,1,1,1,1) (1,1,1.2,1.5,1.2,1.5) C=30 Optmal Lower Bound Upper Bound Approxmaton C=36 Optmal Lower Bound Upper Bound Approxmaton C=45 Optmal Lower Bound Upper Bound Approxmaton (2,2,3,2,1,1) (1,1,1.2,1.5,1.2,1.5) C=40 Optmal Lower Bound Upper Bound Approxmaton C=50 Optmal Lower Bound Upper Bound Approxmaton C=60 Optmal Lower Bound Upper Bound Approxmaton Table 4: System: Sx product: q25=0.1,q35=0.4, q125=0.15, q136=0.1,q1345=0.2, q1346=0.05 Sx tems: L=(1,1,1,1,2,2) Overall Arrval rate=8

25 ArrvalRate Coeffcents Weght Budget k1 k1' k2 k2' k3 k3' k4 k4' k5 k5' k6 k6' 4 (1,1,1,1,1,1) (1,1,1,1,1,1) (2,2,3,2,1,1) (1,1,1,1,1,1) (1,1,1,1,1,1) (1,1,1.2,1.5,1.2,1.5) (2,2,3,2,1,1) (1,1,1.2,1.5,1.2,1.5) (1,1,1,1,1,1) (1,1,1,1,1,1) (2,2,3,2,1,1) (1,1,1,1,1,1) (1,1,1,1,1,1) (1,1,1.2,1.5,1.2,1.5) (2,2,3,2,1,1) (1,1,1.2,1.5,1.2,1.5) Table 5: Optmal and approxmate safety factor