Backorder minimization in multiproduct assemble-to-order systems

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1 IIE Tranacton (2005) 37, Copyrght C IIE ISSN: X prnt / onlne DOI: / Backorder mnmzaton n multproduct aemble-to-order ytem YINGDONG LU, 1 JING-SHENG SONG 2 and DAVID D. YAO 3 1 IBM T.J. Waton Reearch Center, Yorktown Heght, NY 10598, USA E-mal: yngdong@u.bm.com 2 The Fuqua School of Bune, Duke Unverty, Durham, NC 27708, USA E-mal: jong@duke.edu 3 Department of Indutral Engneerng and Operaton Reearch, Columba Unverty, New York, NY 10027, USA E-mal: yao@columba.edu Receved November 2002 and accepted December 2004 We conder a multproduct aemble-to-order ytem. Component are bult to tock wth nventory controlled by bae-tock rule, but the fnal product are aembled to order. Cutomer order of each product follow a batch Poon proce. The leadtme for replenhng component nventory are tochatc. We tudy the optmal allocaton of a gven budget among component nventore o a to mnmze a weghted average of backorder over product type. We derve eay-to-compute bound and approxmaton for the expected number of backorder and ue them to formulate urrogate optmzaton problem. Effcent algorthm are developed to olve thee problem, and numercal example llutrate the effectvene of the bound and approxmaton. 1. Introducton Realzng the compettve advantage of beng flexble, reponve a well a effcent, and enabled by advanced technologe, more and more manufacturng compane are reengneerng ther product degn and delvery procee to move toward ma cutomzaton (Pne, 1993). Th requre modularzng the producton proce o that the product can be quckly aembled from tandardzed component and module n dfferent confguraton, baed on what cutomer ndvdually requet. A a reult, a new knd of producton-nventory ytem ha emerged and becomng ncreangly popular. Th the Aemble-To-Order (ATO) ytem: nventore are kept only at the component level, and the fnal product are aembled only after cutomer order are receved. Th new ytem, n turn, preent challengng operatonal and ytem degn ue to manager. A each cutomer order typcally nvolve everal component n dfferent amount, the tockout of any component wll caue a delay n fulfllng the order. So, the optmal tock level of one component hould be determned n conjuncton wth thoe of other component to enure ther multaneou avalablty. Standard ngle-tem nventory plannng tool, whle utable for the ma-producton make-to-tock envronment, are no longer applcable. New plannng tool are needed to trke the optmal nventory-ervce tradeoff n ATO ytem. The current paper preent an effort toward th goal. More pecfcally, we conder an ATO ytem upportng multple type of demand, whch arrve at the ytem followng compound Poon procee. The component nventore are reuppled from outde uppler after random replenhment leadtme. For a gven component, the leadtme are ndependent, dentcally dtrbuted (..d.) random varable. The leadtme for dfferent component are alo ndependent but may have dfferent dtrbuton. Snce the form of the optmal nventory-control polcy for th ytem unknown, bae-tock polce are wdely adopted n practce. For th reaon, we aume the nventory of each component controlled by a bae-tock polcy. The ytem performance meaure we focu on the expected backorder for each product. Our objectve to mnmze a weghted average of backorder over all product type, ubject to a budget contrant on the component nventory. Through Lttle law (Wolff, 1989), th objectve relate drectly to the repone tme performance n fulfllng cutomer order. The optmzaton problem under tudy qute complex. Frt, t objectve functon noneparable and nondfferentable. Second, t evaluaton nvolve jont probablte, whch can be computatonally challengng. To deal wth the econd dffculty, we develop upper and lower bound on the backorder that nvolve margnal dtrbuton only and ue thee bound and approxmaton X C 2005 IIE

2 764 Lu et al. derved from them a urrogate objectve functon n the optmzaton problem. Th approach mlar n prt to that n Song and Yao (2002) for the ngle-product ATO ytem. However, even wth the mpler urrogate objectve functon the frt dffculty reman. Moreover, the bound for the multproduct model here are more nvolved than n the ngle-product cae and exhbt dfferent tructure, o the dea ued to develop the algorthm n Song and Yao do not apply. In partcular, the lower bound n the multproduct cae lead to an optmzaton problem known a the mn-maxum reource allocaton problem (ouvel et al., 2001), for whch known algorthm are lmted to the branch-andbound type. One man contrbuton of th paper the dentfcaton of a pecal property n our model, whch we call a tack tructure. Ung th property, we are able to develop a much more effcent greedy algorthm to olve the problem. Another nteretng fndng that a modfcaton of the upper bound approach n Song and Yao (2002) produce not only a good approxmaton of the average backorder but alo a urrogate optmzaton problem that bear the ame tack tructure a the one n the lowerbound approach. Several other author have worked on optmzaton model for ATO ytem. Thee tude are qute dfferent from our n the detaled modelng aumpton and oluton approache. For example, ntead of the contnuou-tme cheme condered here, one tream of reearch employ a dcrete-tme (perodc-revew) model. Wthn th framework, Hauman et al. (1998) aume that demand n dfferent tme perod are ndependent, and demand n each perod follow a multvarate normal dtrbuton. Component replenhment leadtme are contant. The paper develop heurtc method to olve the problem of maxmzng a lower bound on the fll rate, whch a multvarate normal probablty, ubject to a lnear budget contrant. Ung the ame framework, Agrawal and Cohen (2001) tudy the problem of mnmzng the total expected component nventory cot ubject to contrant on the order fll rate. Intead of a frt-come frt-erved (FCFS) allocaton rule, they aume a combnaton of FCFS and a far-hare allocaton polcy. Cheng et al. (2002) alo ue the dcrete-tme formulaton and tudy the problem of mnmzng average component nventory holdng cot ubject to productfamly-dependent fll rate contrant. They aume..d. replenhment leadtme and a FCFS allocaton rule. An exact algorthm and a greedy heurtc are developed. Among artcle ung a contnuou-tme framework, Gallen and Wen (2001) aume..d. component leadtme n a ngle-product aembly ytem. Demand Poon wth rate λ. To keep the analy tractable, the paper mpoe a ynchronzaton aumpton, namely, that component are aembled n the ame equence n whch they are ordered. (Order ynchronzaton not aumed n our model.) In the context of a repar hop, Cheung and Hauman (1995) aume..d. leadtme and a multvarate Poon demand model. Th a pecal cae of our (the unt demand cae). Unlke n our paper, they aume complete component cannbalzaton, whch make ene n the repar hop but not n ATO ytem. Under th aumpton, the product backorder are the maxmum of the component backorder, o ther backorder functon much mpler than and completely dfferent from our. Lu and Song (2002) tudy the multproduct unt demand ytem. Aumng there a backorder cot rate aocated wth each backlogged cutomer order, they formulate an uncontraned optmzaton model to mnmze the expected average holdng and backorder cot. Wang (1999) conder multvarate compound Poon demand but the upply proce for each component capactated and modeled a a ngle-erver queue. Applyng the aymptotc reult developed n Glaerman and Wang (1998), he examne the problem of mnmzng average nventory cot ubject to a fll-rate contrant. Our tudy appear to be the frt attempt to tackle optmzaton of multproduct backorder wth batch demand. We refer the reader to Song and Zpkn (2003) for a more detaled urvey of the tate-of-the-art reearch on ATO ytem. The ret of the paper organzed a follow. We tart wth model decrpton and ome prelmnare n Secton 2. In Secton 3, we develop the lower bound for backorder, a well a oluton approache to the lower-bound problem. We then turn to conderng upper bound n Secton 4. In Secton 5 we preent numercal tude to ae and compare the effectvene of the oluton approache developed n the prevou ecton. 2. Model and prelmnare Let I ={1, 2,...,m} denote the et of all component ndce. Cutomer order arrve at the ytem followng a tatonary Poon proce, denoted {A(t), t 0}, wth rate λ. Each order may requre everal component n dfferent amount multaneouly. For any ubet of component I,weay an order of type f t cont of potve unt of each component n and zero unt n I\. We aume that there a fxed probablty q that an order of type, q = 1. Thu, the type- order tream form acompound Poon proce wth rate λ = q λ. A type- order requet Q j unt of component j, and Q = (Q j, j ) follow a known dcrete probablty dtrbuton. We aume that each order type ndependent of the other order type and of all other event. We denote to be the et of all demand type, that, ={ I : q > 0}. Note that not necearly the et of all poble ubet of I. Th demand model qute general and cover a number of mportant pecal cae. For example, f Q j are determntc, then Q repreent the Bll-Of-Materal (BOM) of the type- product. If Q j = Qξj where ξj are contant

3 Backorder mnmzaton n multproduct aemble-to-order ytem 765 and Q a random varable, then (ξj )repreent the BOM of the type- product wherea Q the random quantty of th product that a cutomer demand. For each component, let denote the famly of ubet of that contan. Itclear that the demand proce for component form acompound Poon proce wth rate: λ = λ, and batch ze Q, the mxture of Q for all. (Throughout the paper we ue ubcrpt to ndcate component type and upercrpt for demand type.) The overall demand rate for component thu: µ = λ E ( Q ). Note, however, that thee compound Poon procee are no longer ndependent. Demand are flled on a FCFS ba. Upon the arrval of any demand, f there enough on-hand nventory for all the component requred by the demand, then the demand flled mmedately. In other word, the tme to aemble the component nto the end-product aumed to be neglgble. We alo aume complete backloggng for demand that cannot be flled mmedately. When a demand arrve and ome of t requred component are n tock but other are not, we can ether hp the n-tock component or put them ade a commtted nventory. However, a demand condered to be backlogged unle t can be atfed completely. When there are backorder, they are alo flled on a FCFS ba. We remark that FCFS a uboptmal allocaton rule. Among other thng, an optmal allocaton polcy need to take nto account the multaneou avalablty of component tock. However, due to the complexty of the ytem, the tructure of optmal or near-optmal allocaton rule not yet known. Becaue of t mplcty and eae of mplementaton, FCFS wdely ued acro ndutre. FCFS alo make the etmaton of product backorder more tractable. Alternatve approache have been dcued n the lterature, motly n a dcrete-tme formulaton, o that ome knd of myopc allocaton ued wthn each perod. When dealng wth order n dfferent perod, however, mot of thee approache ue FCFS. In our contnuou-tme ettng, there no order batchng, o myopc allocaton not applcable. See Song and Zpkn (2003) for detal. The nventory of each component controlled by an ndependent bae-tock polcy, wth: := the bae-tock level for component. That, upon each demand arrval, f the nventory poton (.e., the on-hand nventory plu on-order poton mnu backorder) of component le than, then order up to ; otherwe, do not order. Th type of polcy known to be optmal when there are no econome of cale n replenhment and when the component demand are ndependent. When component demand are correlated a n our cae, the optmal polcy hould be a coordnated polcy. Unfortunately, the prece form of the optmal polcy tll unknown. Hence, the ndependent bae-tock polce are wdely ued n practce becaue of ther mplcty. Snce we follow a bae-tock polcy and the demand arrve n batche, each component replenhment order compre everal unt. We aume that, for each unt of component, the replenhment leadtme are..d. random varable wth a common cumulatve dtrbuton G. Let L denote the generc random varable wth dtrbuton G and mean E [L = l. Denote G c = 1 G. Aume the leadtme are ndependent among the component; that, L ndependent of L j for any j. Forany tme t, let I (t)bethe net nventory of component at t, and X (t) bethe number of outtandng order of component at t. Then, by the nature of the bae-tock control, we have: I (t) = X (t), = 1,...,m. (1) It clear that X (t)equal to the number of job n ervce n an M/G / queue, wth arrval rate λ,for = 1,...,m. However, nce the arrval of any demand of type- generate multaneou arrval at all the component queue,, thee m queue are not ndependent. Lu et al. (2003) how that the jont dtrbuton of (X 1 (t),...,x m (t)) ha a teady-tate lmt (X 1,...,X m ) wth probablty generatng functon: ψ(τ 1,...,τ m ) [ m := E j=1 τ X j j [ = exp λ ( ( ψ Q G1 (u) + τ 1 G c 1 (u),...,g m(u) + τ m G c m (u)) 1 ) du, (2) where ψ Q (x 1,...,x m ) denote the generatng functon of Q = (Q ). We can ue ψ to derve the moment and covarance of the X. In partcular: θ := E [X = l λ E ( Q ) = µ l, (3) σ 2 := Var[X = θ + λ [ E (( Q ) 2 ) E ( Q ) [ G c j (u) 2 du. (4) 0 And for j, σ j := Cov[X, X j = λ E ( Q Q ) j G c (u)gc j (u)du. :,j 0 0 (5)

4 766 Lu et al. It can be verfed that σ 2 θ for all. Alo, σ j 0 and t zero f and only f there no demand type that requre both and j. For unt demand model,.e., Q j = 1, for all j, ψ(τ 1,...,τ m )the probablty generatng functon of a multvarate Poon dtrbuton. That, the margnal dtrbuton of X Poon wth mean λ l, = 1,...,m. In th paper, the performance meaure of prmary concern are: B = teady-tate number of backordered type- demand, for all. We now how how thee quantte depend on the jont dtrbuton of (X 1,...,X m ). Denote x + := max{0, x}. Note that: B = teady-tate number of backordered component = [X +, depend on the margnal dtrbuton of X only. So t eay to compute. To dentfy B,tmportant to fgure out the correpondng component backlog that contrbute to t. Let B be the teady-tate number of backlogged unt of component that are due to type- demand. We have that: B ( B λ E [ Q ) = 1 k J λ J E [ B ( λ E [ Q ) Q J = 1 k, µ (6) where the 1 k (p) are..d. Bernoull random varable wth parameter p. Inthe cae of unt demand, the number of backorder of type : B { } = max B, where B = B ( λ ) 1 k. (7) λ When demand arrve n batche, however, we mut frt tranlate B nto batche. Let: { } n B Q, := nf n : Q (k) B, (8) where Q (k), k = 1,...,n, are ndependent cope of Q. Then, B = max BQ,. (9) Let c be the unt cot of component. Let w 0be aweghtng factor for the average backorder for product. Itmeaure the relatve mportance of ervce provded to type- order. Denote := ( 1,..., m ) and Z+ m the m- dmenonal non-negatve nteger vector pace. We are ntereted n the followng optmzaton problem: (P) mn w E [B ().t. c c m m C, Z+ m. (10) where C > 0gven contant, repreentng the total nventory budget. In ndutral applcaton, nventory budget often apple to afety tock only. Snce nventory can be dvded nto work-n-proce (WIP) and afety tock, and the WIP part ndependent of the bae-tock level, the contrant n the above problem formulaton content wth practce. (Alo ee Sherbrooke (1992)). We note that, ntead of a fxed budget on the target afety tock level, another commonly een formulaton to mnmze average nventoryholdng and backorder-penalty cot. Solvng th type of problem wll be requre a dfferent et of technque from thoe developed here; ee Lu and Song (2002) for the untdemand cae. A ponted out n the Introducton, the objectve functon n Equaton (10) relate to a delay objectve va Lttle law: Suppoe we want to mnmze: w E [W (), where W denote the delay (repone tme) n fllng type- order, and w the aocated cot. Then, we can wrte: w E[W () = w λ E [B (), hence, we can et w = w /λ n Equaton (10). Obvouly, the objectve functon n (P) very dffcult to evaluate due to the nvolvement of the jont dtrbuton of (X 1,...,X m ) and other ntrcate relaton, uch a the max operaton. In the ret of the paper we develop two approxmate objectve functon whch nvolve only the margnal dtrbuton of X.Wefurther relax the nteger requrement of the decon varable and treat the anon-negatve real value, a they are nvolved n bound and approxmaton. In many applcaton, the are large value; hence, gnorng the ntegralty approprate. The reult two urrogate problem, and we develop effcent oluton approache to olve them. 3. The lower-bound approach We now develop an approxmaton of E [B that ue margnal dtrbuton of X only. One key tep to apply Jenen nequalty to a max operaton, reultng n a lower bound. Therefore, we call th a lower-bound approach. Note that from Equaton (8) we know that B Q, a toppng tme wth repect to {Q (1), Q (2),...,n}, and Q (k) B, B Q, a.. Takng expectaton on both de and applyng Wald dentty on the left-hand de yeld: E [ Q [ Q, [ E B E B. (11)

5 Backorder mnmzaton n multproduct aemble-to-order ytem 767 Now takng expectaton on both de of Equaton (6), and applyng Wald dentty to the ummaton on the rghthand de, we have: E [ B λ E [B E [ Q =. (12) µ Hence, makng ue of Jenen nequalty along wth Equaton (11) and (12), we obtan: [ E [B = E max = max BQ, { λ E [B µ max E [ B Q, E [ B max E [ Q }. (13) We now ue the lower bound n Equaton (??) a a urrogate for the objectve functon n Equaton (??). Defne: b ( ) = E [B ( ) = E [X +. Then, the reultng lower-bound problem can be expreed a: { } (LB) mn w λ b ( ) max, (14) µ.t. c c m m C; Z+ m. Clearly, the objectve functon n (LB) much eaer to evaluate becaue only the margnal dtrbuton are nvolved. Two extreme cae of th problem have been tuded n the lterature. The frt cae that the component demand are ndependent,.e., q = 0f not a ngleton. In th cae the problem reduce to: m mn w b ( ), =1.t. c c m m C: Z+ m. A nce feature of th problem that the objectve functon a eparable convex functon. Sherbrooke (1992) preent agreedy algorthm to olve th problem. The econd extreme cae a ngle-product ATO ytem,.e., ={1,...,m} the only demand type and the BOM fxed. We can alo redefne the unt of each component, o that we obtan a unt demand model. In th cae, the problem become: mn max{b 1 ( 1 ),...,b m ( m )}, (15).t. c c m m C; Z m +. Ung the property that each b ( )decreang and convex, Song and Yao (2002) how that a greedy algorthm can guarantee optmalty. The general cae, however, much more complcated and le tructured becaue we need to deal wth the um of maxmum over overlappng ubet. To ncreae tractablty, n the followng we further relax the nteger requrement n (LB), and treat each a a non-negatve real varable. Let R m + denote the m-dmenonal non-negatve real vector pace. The relaxed problem then: { } (LB ) mn w λ b1 ( 1 ) max, (16) µ.t. c c m m C; R m +. The reultng objectve value a lower bound for (LB ), whch n turn tll a lower bound on the orgnal problem. Although the objectve functon convex, t not dfferentable, o the conventonal tool for convex program are not applcable. In the ret of th ecton we wll dentfy pecal tructure of the problem and how that we can tranform the problem nto one that mnmze the maxmum over non-overlappng et, whch n turn can be converted nto a et of ubproblem wth lnear objectve functon and ordered varable. The prce we pay that the orgnal lnear contrant functon now become a non-lnear decreang functon. Nonethele, we wll how that each of thee ubproblem greedly olvable, o we can obtan the overall optmal oluton effcently. Once we obtan a oluton for (LB ), we round t to the nearet nteger oluton a an approxmate oluton for the orgnal problem The tack tructure A change of varable wll better reveal the tructure of (LB ). Note that b ( ) = E [X + trctly decreang convex n R 1 +. Th mple that t nvere b 1 ( )well defned and alo decreang and convex. Snce lm b ( ) = 0, we denote b 1 (0) =+.Now,for each = 1, 2,...,m, defne: z := b ( ), or = b 1 (µ z ):= h (z ). (17) µ Let z := (z 1,...,z m ). Then, the problem. (LB )equvalent to: (LB ) mn w λ max z (18) z.t. c h (z ) C; z R m +. Suppoe z = (z1,...,z m ) an optmal oluton to Equaton (18). Denote u := max z.obvouly, f then u u.wthout lo of generalty, rename the product type n acendng order of u : u 1 u 2. We now argue that we can aume that all z, 1 have the ame value. Th becaue f there ext an 0 1 uch that: z 0 < u 1 := max zj, j 1 then, reagnng z 0 the value u 1 wll not detroy the optmalty of z : the objectve value wll reman the ame and the contrant wll tll be atfed becaue h 0 ( )adecreang functon.

6 768 Lu et al. Smlarly, note that u 2 := max j 2 z j the econd mallet of the u value. For any 2 \ 1,fz < u 2, then we can ncreae t value to u 2 wthout changng the objectve value or volatng the contrant o to preerve the optmalty of z. Contnung th argument, we can conclude that there ext an optmal oluton z to Equaton (18) that atfe a tack-lke tructure a formalzed below. Suppoe ={ 1,..., n },.e., there are total of n product (or, demand) type. Then we can wrte the et of all component a: I = n l=1 l := {1,...,m}. Conder a partcular order (permutaton) of the n ubet, π = (π 1,π 2,...,π n ). Form n new ubet nductvely a follow: π 1 = π1, π l = πl \( π 1 π l 1 ), l = 2,...,n. (19) Then, clearly, ( π 1,..., π n )apartton of I (wherea ( π1,..., πn )not). Note that f n > m then ome π l wll be empty. Propoton 1. There ext an optmal oluton z to Equaton (18) atfyng the followng tack-lke tructure. Suppoe π = (π 1,π 2,...,π n ) a permutaton uch that u πl := max j πl zj ncreang n l,.e: u π1 u π2 u πn. Let ( π 1,..., π n ) be the partton of I pecfed n Equaton (19). Then, we have: z j = u πj, for all l = 1,...,n. j π l 3.2. The optmzaton algorthm To mplfy notaton, denote, for each permutaton π = (π 1,π 2,...,π n ): λ π l := λ π l, w π l := w π l, uπl := u πl, u π = (u πl,...,u πn ). Furthermore, baed on Propoton 1, denote: g l (z) := π l c h (z). We can then rewrte the optmzaton problem n Equaton (18) a follow: n (LB ) mn mn w π l λ π l u πl, (20) π u π l=1 n ( ).t. g l uπl 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 contrant, a (LB π ). We can olve (LB π )forall π; the one that yeld the mallet objectve value then the optmal oluton to the orgnal problem (LB ). For each π, (LB π )equvalent to a problem of mnmzng a eparable convex functon wth a ngle lnear contrant, hence, t greedly olvable va a margnal allocaton algorthm. To decrbe th algorthm, reformulate the problem wth yet another tranformaton of varable: y 1 := u π1 ; y l := u πl u πl 1, l = 2,...,n. Hence, u πl = y 1 + +y l, l = 2,...,n. Alo denote y := (y 1,...,y n ); v l := w π l λ πl + +w π n λ π n, l = 1,...,n. Then, (LB π ) become the followng: n (LB π ) mn v l y l, (22) y l=1 n.t. g l (y 1 + +y l ) C, y R n +. l=1 The margnal allocaton cheme work a follow: Start wth y = 0; hence, the zero objectve value. In each tep, allow an ncreae of >0ofthe objectve value, wth beng aprepecfed contant. Th correpond to allowng y k to ncreae by an amount /v k,for each k = 1,...,n. (Examnng thee vertex pont uffcent becaue g l ( )convex.) Th, n turn, correpond to the decreae of the left-hand de of the contrant by the followng amount: n [g l (y 1 + +y k + +y l ) l=k g l (y 1 + +y k + /v k + +y l ). (Recall, g l ( ) adecreang functon, for all l.) Therefore, elect the k that yeld the larget uch decreae, and update y k to y k + /v k.repeat th procedure untl the left-hand de of the contrant reduced to C Heurtc Although for each π, (LB π )greedly olvable, we tll need to olve n! uch problem. One heurtc to do parwe mprovement: tart from any permutaton, ay, (1,...,n). Compare th wth all other permutaton that reult from nterchangng 1 wth l 1. Th nvolve olvng n margnal allocaton problem lke (LB π )above. Suppoe wappng 1 and π 1 gve the mallet objectve value. Next, fx π 1 at the frt poton, apply the ame parwe nterchange to the remanng n 1 element of the permutaton, and o forth. Th heurtc requre olvng n(n 1)/2 (a oppoed to n!) margnal allocaton problem.

7 Backorder mnmzaton n multproduct aemble-to-order ytem 769 An even mpler heurtc to apply the margnal allocaton cheme that olve (LB π ) drectly to the problem (LB ) n Equaton (20). The detal are a follow: Start wth π = (1,...,n), and et uπl = 0forall l. Suppoe the current oluton (uπl ), correpondng to ome permutaton π = (π l ). For each k = 1,...,n, do: (a) ncreae u πk to u πk + /(w π k λ π k ); rearrange the u value, and let π k denote the reultng permutaton; (b) evaluate the left-hand de of the contrant n Equaton (21). Select k a the one correpondng to the mallet value n (b) above (.e., the one that yeld the larget decreae n the left-hand de of the contrant). Update u πk to u πk + /(w π k λ π k ); and π to π k. Repeat the above untl the left-hand de of the contrant n Equaton (21) reduced to C. Remark 1. Note that wth any gven permutaton the orgnal dcrete lower-bound urrogate problem (LB) a eparable dcrete convex programmng, whch can be ealy olved by a greedy algorthm. So the heurtc algorthm above can alo be appled to olve the problem n (LB). 4. The upper-bound approach We now wtch to ung an upper-bound approach to derve urrogate, whch nclude an upper bound and ome approxmaton derved from upper bound, for the objectve functon n the backorder mnmzaton problem n Equaton (10). The man dea the followng nequalty: for a et of non-negatve varable, x 0, we have max{x } a + (x a) +, (23) whch hold for any a, and can be drectly verfed. Hence, mnmzng the rght-hand de above, wth repect to a, yeld an upper bound for max {x }. Note, however, that when a < 0, the rght-hand de become: a + (x a) x. So, n mnmzng the rght-hand de of Equaton (23), we only need to conder a 0. When (x )avector of non-negatve random varable, the upper bound ha the advantage of nvolvng only the margnal dtrbuton. Hence, applyng th to Equaton (9), we have: B := max { B Q, } a + ( Q, B a ) +, a.., (24) where a a parameter. Therefore, ung the above upper bound a a urrogate objectve, and wrtng a := (a ) (and := ( ) I a before), we have the followng optmzaton problem: E ( B Q, a ) +, (25) mn (a,) w a + w.t. c c m m C, R m +. In the above problem, mnmzng over a not a mportant a mnmzng over,athe former meant to mprove on the upper bound. (In fact, a we hall llutrate below, we could very well forgo the mnmzaton over a by fxng t at a pecfc value.) Hence, we propoe to ue a common (calar) a for all product type, and then optmze a. Th way, the problem become: (UB) mn (a,) a w + w.t. c c m m C, 0. E ( B Q, a ) +,(26) For fxed a, (UB) a eparable convex programmng, whch greedly olvable. We can then conduct a lne earch on a. Alternatvely, electng a = E(B Q, ), we have: B { ( Q,)} max E B + E [ B Q, E ( B Q, ) +, a.. Takng expectaton on both de yeld: E [B { ( Q,)} max E B + E [ B Q, E ( B Q, ) +. (27) Note that from Equaton (8) we have: (B Q, 1) + Q (k), a.., B where Q (k) are ndependent cope of Q and 0 Q (k) = 0. Applyng Wald dentty yeld: E [ B E [ Q E [ B Q, 1 +. (28) It reaonable to ue B /E [Q toapproxmate B Q,. Th relatonhp exact for the unt demand model. Conequently, we can ue E [B /E [Q toapproxmate E [B Q,. Apply thee approxmaton to Equaton (27), we obtan: { [ } E B E [B max E [ Q + E [ B E ( ) B + E [ Q. (29) We now go one tep further by replacng the ummaton over n the rght-hand de of Equaton (29) wth the maxmum over, reultng n: { [ E B E [B max E [ Q + E [ B E ( ) B + } E [ Q { λ = max E [ B E [ B + E ( B ) + } µ E [ Q. (30)

8 770 Lu et al. Ung the above approxmaton n the objectve of (P) yeld the followng urrogate problem: { λ (APP) mn w max E [B ( ) µ + E [ B ( ) E B ( ) + } E [ Q, (31).t. c c m m C, 0 R m +. Note that although the orgnal dea wa to ue an upper bound to urrogate the orgnal objectve functon, we took two tep of approxmaton to reve the upper-bound functon to acheve Equaton (30). Therefore, the objectve functon n (APP) no longer an upper bound. In the ret of th ecton we hall how that (APP) exhbt the ame tructure a n (LB ). The followng propoton tell u that E [B ( ) EB ( ) + decreang n. So, the objectve functon n (APP) ha the ame property, and therefore t nvere functon unquely defned. Propoton 2. For any convex functon f (x), E f [B E(B ) decreang n.inpartcular, Var(B ) decreang n, for each and each ; and Var(B ) decreang n, for all. Proof. We frt how that Var(B )decreang n.recall, B = [X +. Hence, t uffce to how that: E [(X ) + 2 E [(X 1) + 2 [E(X ) + 2 [E(X 1) + 2. (32) We have, (X ) + (X 1) + = 1(X + 1), (X ) + + (X 1) + = (2X 2 + 1)1(X + 1). (Note that we are treatng the X, a well a the anteger valued.) Hence: E{[(X ) + 2 [(X 1) + 2 } = E [2X 2 + 1)1(X + 1), wherea [E(X ) + 2 [E(X 1) + 2 = [E(X ) + + E(X 1) + [E (X ) + E(X 1) +, = E [(2X 2 + 1)1(X + 1)P (X + 1). Hence, the nequalty n Equaton (32) hold. Next, wrte B,n a B condtonng upon X = n,for n = 0, 1,...From Equaton (6), nce X ndependent of the Bernoull varable nvolved, t uffce to how that E f [B,n E(B,n ) ncreang n n. Denote M n := B,n E(B,n ). From Equaton (6), clearly, {M n } formamartngale (Ro, 1996). Hence, the convexty of f mple that {f (M n )} a ubmartngale, whch, n turn, mple that E f (M n )ncreang n n, and hence decreang n,whch what dered. Lettng f (x) = x 2,wehaveE(Mn 2) = Var(B,n ). Note that we can wrte: Var ( B ) [ ( ) [ ( ) = E Var B X + Var E B X. (33) The frt term on the rght-hand de above decreang n, nce Var(B,n )ncreang n n gven X = n a argued above. Followng Equaton (6), the econd term equal to (λ E [Q /λ )Var(B ); hence, t alo decreang n,fromwhat ha already been hown for Var(B ). Therefore, we can conclude that Var(B )decreang n. Fnally, oberve from Equaton (6) that B ( )/ (λ E [Q ) and [X + 1 k (1/µ )have the ame frt two moment and the latter ndependent of.wereplace the former wth the latter a an approxmaton. Th lead to the followng problem: { } E (APP) mn w λ [B ( ) max + r( ), (34) µ where r( ) = E.t. c c m m C, R m +. [ [X = + ( ) 1 1 k E [X + +. µ Followng mlar argument n the proof of Propoton 2, we can how that r( )alo a decreang functon of. Therefore, (APP ) ha the ame tructure a the lowerbound problem n the lat ecton. So, the ame optmzaton and heurtc algorthm can be appled here. 5. Numercal example In th ecton, we preent reult from a et of numercal experment, n whch the heurtc algorthm derved n the prevou ecton are teted for ther effcency n fndng the (near-)optmal oluton to the backorder mnmzaton problem. A mentoned before, what complcate the multproduct problem the number of overlappng ubet that conttute dfferent type of product, not the number of component. For ntance, f one et of product hare no common component wth another et of product, then we can mply decompoe the orgnal ytem nto two eparate ytem, each of whch cover one et of product. An extreme cae that each product cont of only a ngle component, reultng n a eparable convex problem whch greedly olvable (ee Sherbrooke (1992)). For th purpoe, n our experment we chooe example that are complex n the ene that there are many common component hared by dfferent product. We conder an ATO ytem wth x component and the followng x product type: {2, 5}, {3, 5}, {1, 2, 5}, {1, 3, 6}, {1, 3, 4, 5}, {1, 3, 4, 6}. µ

9 Backorder mnmzaton n multproduct aemble-to-order ytem 771 The overall demand arrval rate λ = 4, whch plt nto the x product type a follow: q 25 = 0.10, q 35 = 0.40, q 125 = 0.15, q 136 = 0.10, q 1345 = 0.20, q 1346 = The component leadtme follow ndependent exponental dtrbuton wth mean: (l ) =1 6 = (1, 1, 1, 1, 2, 2). We conder two et of cot data: c (1) = (1, 1, 1, 1, 1, 1) and c (2) = (2, 2, 3, 2, 1, 1), wth repectve weght w (1) = (1, 1, 1, 1, 1, 1) and w (2) = (1, 1, 1.2, 1.5, 1.2, 1.5), and varou nventory budget level: C = 20, 24, 32 for c (1) ; and C = 30, 40, 50 for c (2). In Table 1 and 2, we ummarze the oluton obtaned by: () an exhautve earch va mulaton; () the lowerbound approach along wth the heurtc algorthm n Secton 3.3; () the upper-bound approach baed on (UB); and (v) the approxmaton approach baed on (APP ). For each approach, we report the optmal bae-tock level, the correpondng objectve value, and the reultng weghted average backorder. Note that whle the lat two meaure are equal for the mulaton-baed optmal oluton, they are dfferent for the other three approache. For ntance, the objectve value correpondng to () necearly a lower bound of the objectve value of (); wherea the weghted average backorder correpondng to () obtaned from ung the optmal bae-tock level derved from the lower-bound approach (to evaluate analytcally the orgnal objectve). We alo report the percentage lo of optmalty of each approxmate approach. In all of our numercal example, the leadtme follow exponental dtrbuton; the mpact of dfferent dtrbutonal aumpton upon the performance of thee ATO ytem can be found n [11. It alo ponted out n [11 that the dfference caued by dtrbutonal aumpton not gnfcant epecally for hgh performance ytem. Therefore, we beleve that thee aumpton wll not affect the accuracy of our approxmaton. In Table 3 and 4, we repeat the ame example wth λ = 8, and modfed budget level: C = 30, 36, 45 for c (1) ; and C = 40, 50, 60 for c (2). The reult ndcate that oluton from the urrogate problem are very cloe to the true optmal oluton n all cae. Overall, the approxmaton approach (APP )appear to be the mot effectve method. We now ue thee example to hed ome lght on how the optmal component nventory level dffer from a tandard ngle-tem approach ued n practce. Recall that the ngletem nventory theory ugget that we hould tock enough of each component to cover the leadtme demand plu ome afety tock to protect agant varablty (of both demand and leadtme). Accordng to Equaton (1), (3) and (4), Table 1. Reult obtaned for the ytem contanng the x product type q 25 = 0.10, q 35 = 0.40, q 125 = 0.15, q 136 = 0.10, q 1345 = 0.20 and q 1346 = Alo, L = (1,1,1,1,2,2) and the overall arrval rate equal to four Coeffcent Weght Budget Algorthm 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

10 772 Lu et al. Table 2. Reult obtaned for the ytem contanng x product type q 25 = 0.10, q 35 = 0.40, q 125 = 0.15, q 136 = 0.10, q 1345 = 0.20 and q 1346 = Alo, L = (1,1,1,1,2,2) and the overall arrval rate equal to four Coeffcent Weght Budget Algorthm 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 3. Reult obtaned for the ytem contanng x product type q 25 = 0.10, q 35 = 0.40, q 125 = 0.15, q 136 = 0.10, q 1345 = 0.20 and q 1346 = Alo, L = (1, 1, 1, 1, 2, 2) and the overall arrval rate equal to eght Coeffcent Weght Budget Algorthm 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

11 Backorder mnmzaton n multproduct aemble-to-order ytem 773 Table 4. Reult obtaned for the ytem contanng x product type q 25 = 0.10, q 35 = 0.40, q 125 = 0.15, q 136 = 0.10, q 1345 = 0.20, and q 1346 = Alo, L = (1,1,1,1,2,2) and the overall arrval rate equal to eght Coeffcent Weght Budget Algorthm 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 5. Optmal and approxmate afety factor Arrval rate Coeffcent Weght Budget k 1 k 1 k 2 k 2 k 3 k 3 k 4 k 4 k 5 k 5 k 6 k 6 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)

12 774 Lu et al. we have: = θ + k σ, (35) where θ the average leadtme demand for component ; σ, the correpondng tandard devaton; and k, the afety factor, a potve parameter ndcatve of the level of protecton agant uncertanty or varablty. When there are multple tem (component), one mut addre how to chooe k for each. What often followed n practce the o-called equal fractle rule: havng the ame k value acro all component. Th equvalent to projectng the ame level of fll rate acro. Should th be the cae here? To anwer th queton, n Table 5 we report the optmal afety factor aocated wth the optmal bae-tock level n the prevou four table. From thee afety factor, we can derve the component fll rate from a normal dtrbuton table. For ntance, k = 1.00 correpond to a fll rate of (1) = 84.13%, where (x) denote the dtrbuton functon of the tandard normal varate; k = 2 correpond to a fll rate of (2) = 97.72%, and o forth. A can be oberved from the table, the optmal afety factor vary wdely among component. In ome cae, when the budget level relatvely low, the afety factor of certan component are even negatve. Thee obervaton ugget that the equal fractle rule n general not effectve n ATO ytem. Next to each optmal afety factor k,wealo report the approxmate afety factor k derved from the approxmate (APP ) oluton. Clearly, thee example ndcate that ung the approxmate approach we can come up wth cloe-to-optmal afety factor. To ummarze, among the three urrogate problem and algorthm developed here, both (LB) and (UB) have a computatonal advantage and can provde a good predcton of the bac trend of the ytem behavor under varou change. On the other hand, (APP ), wth more computatonal effort than thoe of (LB) and (UB), provde a good approxmaton to both the optmal oluton and the objectve value. (In cae where (APP ) doe not produce the bet reult, the relatve error of the optmal objectve are below 3%.) Reference Agrawal, M. and Cohen, M. (2001) Optmal materal control and performance evaluaton n an aembly envronment wth component commonalty. Naval Reearch Logtc, 48, Cheng, F., Ettl, M., Ln, G.Y. and Yao, D.D. (2002) Inventory-ervce optmzaton n confgure-to-order ytem. Manufacturng Servce Operaton Management, 4, Cheung,.L. and Hauman, W. (1995) Multple falure n a mult-tem pare nventory model. IIE Tranacton, 27, Glaerman, P. and Wang, Y. (1998) Leadtme-nventory tradeoff n aemble-to-order ytem. Operaton Reearch, 46, Gallen, J. and Wen, L. (2001) A mple and effectve component procurement polcy for tochatc aembly ytem. Queueng Sytem, 38, Hauman, W.H., Lee, H.L. and Zhang, A.X. (1998) Order repone tme relablty n a mult-tem nventory ytem. European Journal of Operatonal Reearch, 109, ouvel, P., arabat, S. and Yu, G. (2001) A mn-max-um reource allocaton problem and t applcaton. Operaton Reearch, 49, Lu, Y. and Song, J.S. (2005) Order-baed cot optmzaton n aembleto-order ytem. Operaton Reearch, 53, Lu, Y., Song, J.S. and Yao, D. D. (2003) Order fll rate, leadtme varablty, and advance demand nformaton n an aemble-to-order ytem. Operaton Reearch, 51, Pne, B.J. (1993) Ma Cutomzaton: The New Fronter n Bune Competton, Harvard Bune School Pre, Cambrdge, MA. Ro, S.M. (1996) Stochatc Procee, 2nd edn., Wley, New York, NY. Shanthkumar, J.G. and Yao, D.D. (1991b) Bvarate characterzaton of ome tochatc order relaton. Advance n Appled Probablty, 23, Sherbrooke, C.C. (1992) Optmal Inventory Modelng of Sytem, Wley, New York, NY. Song, J.S. and Yao, D.D. (2002). Performance analy and optmzaton n aemble-to-order ytem wth random leadtme. Operaton Reearch, 50, Song, J.S. and Zpkn, P. (2003). Supply chan operaton: aembleto-order ytem, n Supply Chan Management, De ok, T. and Grave, S. (ed.), North-Holland, Amterdam, The Netherland, Ch. 11. Wang, Y. (1999) Near-optmal bae-tock polce n aemble-to-order ytem under ervce level requrement. Preprnt, MIT Sloan School, MIT, Cambrdge, MA. Wolff, R. (1989) Stochatc Modelng and the Theory of Queue,Prentce Hall, Englewood Clff, NJ. Bographe Yngdong Lu a reearch taff member at the IBM T.J. Waton Reearch Center. H reearch nteret nclude appled probablty and tochatc model. Jng-Sheng (Jeannette) Song a Profeor at the Fuqua School of Bune, Duke Unverty. She obtaned her Ph.D. from Columba Unverty. Prevouly, he ha alo held faculty poton at the Unverty of Calforna at Irvne and Columba Unverty. She tude the degn and management of producton, nventory, and logtc ytem and coordnaton mechanm n upply chan. Her reearch ha been upported by the Natonal Scence Foundaton and ha appeared n everal leadng academc journal. She erve on the Edtoral Board of Management Scence, Operaton Reearch, Manufacturng & Servce Operaton Management, IIE Tranacton, and Naval Reearch Logtc. Davd Yao receved h Ph.D. degree from the Unverty of Toronto n 1983, and tarted h academc career at Columba Unverty, where he became a full profeor n He ha performed extenve reearch and conultng work n tochatc network, emconductor manufacturng, and upply chan management. He the author/co-author of over 160 refereed publcaton, three book and fve edted volume. He a recpent of numerou honor and award, an IEEE Fellow, and a holder of four US patent n manufacturng operaton and upply-chan logtc. He the Stochatc Model Area Edtor of Operaton Reearch, and Edtor-n-Chef of Foundaton and Trend n Stochatc Sytem. Contrbuted by the Supply Chan/Producton-Inventory Sytem Department