Asymptotically Optimal Inventory Control for AssembletoOrder Systems with Identical Lead Times


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1 Asymptotcay Optma Inventory Contro for AssembetoOrder Systems wth Identca ead Tmes Martn I. Reman Acateucent Be abs, Murray H, NJ 07974, Qong Wang Industra and Enterprse Systems Engneerng, Unversty of Inos at UrbanaChampagn, Urbana, I 61801, May 22, 2013 Optmzng mutproduct assembetoorder (ATO) nventory systems s a ongstandng dffcut probem. We consder ATO systems wth dentca component ead tmes and a genera b of materas. We use a reated twostage stochastc program (SP) to set a ower bound on the average nventory cost and deveop nventory contro poces for the dynamc ATO system usng ths SP. We appy the frststage SP optma souton to specfy a basestock repenshment pocy, and the secondstage SP recourse near program to make aocaton decsons. We prove that our poces are asymptotcay optma on the dffuson scae, so the percentage gap between the average cost from ts ower bound dmnshes to zero as the ead tme grows. Key words : AssembetoOrder, Inventory Management, Stochastc near Program, Stochastc Contro, Asymptotc Optmaty, Dffusons Scae. 1. Introducton The assembetoorder (ATO) nventory system, where mutpe components are used to produce mutpe products, s a cassca and much studed mode n nventory theory. Demand for the products s random, whe components are obtaned from an uncapactated supper after a determnstc (component dependent) ead tme. The components that are used n each product are specfed n the b of materas (BOM) matrx. A key assumpton s that assemby s nstantaneous, so that nventory s kept at the component rather than the product eve. Any unfufed demand s backogged and each backogged product ncurs a productdependent constant backog cost per unt of tme. Each component n nventory ncurs a componentspecfc nventory hodng cost per unt of tme. The objectve s to fnd a contro pocy, whch can convenenty be descrbed va repenshment (orderng from supper) and aocaton (assembng to meet demand) poces, that mnmzes the ongrun average expected tota cost. 1
2 2 As descrbed brefy beow, t has ong been recognzed that for systems wth mutpe products, exact optmzaton s dffcut, so most pror work has focused on optmzng wthn partcuar pocy casses and has yeded suboptma poces n genera. In ths paper we contnue the deveopment and anayss, begun n Doğru et a. 4, of an approach that uses the souton to a partcuar stochastc program (SP) as the bass for constructng contro poces for ATO nventory systems. Ths paper, smary to 4, restrcts attenton to the stuaton where a components have the same ead tme,. The SP that we use s a twostage stochastc near program wth compete recourse that s a partcuar reaxaton of the SP that arses n the context of a oneperod ATO mode, cf. Song and Zpkn 26. The frst stage decson s how many components to order. The second stage decson, taken after demand has been reazed, s how much of each product to assembe. The reaxaton, ntroduced n 4, was shown there to provde a ower bound on the achevabe cost n the dynamc ATO system under any feasbe pocy. It was suggested n 4 that a basestock pocy be used for repenshment, wth the optma frst stage souton of the SP used as basestock eves. In addton, t was shown n 4 that, for a smpe ATO structure known as the W mode (where 3 components are used to assembe 2 products  see Fgure 1), the souton of the second stage (recourse) P motvates, n a natura manner, a smpe prorty pocy for aocaton. Whe t was noted n 4 that there shoud be a way, n prncpa, to transate the souton of the recourse P nto an aocaton pocy, the deveopment of ths transaton was eft for future work. One contrbuton of ths paper s to f that gap. In partcuar, we dspay an P, whch s equvaent to the recourse P, that provdes a backog target, and ntroduce an assocated aocaton prncpe that aows the dentfcaton of aocaton poces that propery track the backog target. It was noted n 4 that the numerca experments carred out on the W system there were consstent wth asymptotc optmaty of our pocy as the ead tme grows, and ths asymptotc optmaty was conjectured there. The man contrbuton of ths paper s to show that ths asymptotc optmaty hods, not ony for the W mode, but for any ATO system where a components have dentca ead tmes. The rest of ths paper s organzed as foows. In the remander of ths secton we descrbe reated work and descrbe our SPbased approach as a four step process. We defne the nventory contro probem n Secton 2. We dscuss the use of the SP to set a ower bound on the nventory cost and to deveop nventory poces n Secton 3. We carry out the asymptotc anayss n Secton 4. Secton 5 contans most of the proofs. A cear future goa s to extend the resuts of ths paper to systems wth nondentca ead tmes. To ths end, we dscuss reated chaenges n Secton Reated Work There s a tremendous gap between optmzng nventory contro n oneproduct ATO systems and dong so n mutproduct ATO systems. The former s a competey soved probem. For
3 3 systems wth one component, Karn and Scarf 14 showed that the optma contro s a basestock pocy wth the basestock eve determned by a newsvendor mode. For systems wth mutpe components wth dfferent ead tmes, Rosng 23 found the optma contro pocy by drawng an anaogy wth the optma contro of mutecheon systems. For systems wth mutpe products, the search for an optma pocy s hndered by the aocaton decson that dstrbutes components to demands for dfferent products. To be optma, the aocaton pocy may n prncpe depend on the entre status of the ppene, whch ncudes not ony the tota numbers of ordered components that are yet to arrve, but aso the exact tmes when such repenshment orders were paced. Trackng ths nformaton woud requre a tremendous state space that grows exponentay wth the ead tme. Except n a few rare cases n whch speca parameter vaues aong wth smpe structure render the aocaton decson nconsequenta (Doğru et a. 4, u et a. 18, u et a. 15, and Reman and Wang 21), currenty there appears to be no computatonay feasbe way to obtan an optma pocy. As s dscussed n 4, where a more detaed dscusson of pror work s gven, to varous degrees most exstng approaches rey on a FrstInFrstOut (FIFO) scheme to smpfy aocaton decsons (see e.g, u and Song 16, u et a. 17, for the contnuous revew cases, and Agrawa and Cohen 1, Akçay and Xu 2, Hausman et a. 12, and Zhang 27 for dscrete revew cases.) Among these papers t s worth sngng out 16, whch fnds the optma basestock eves for FIFO aocaton wth an arbtrary BOM. There are a few departures from FIFO n recent deveopment. For nstance, no hodback poces proposed n u et a. 18 requre that a demands shoud be served when components are avaabe, and hence voate the defnton of FIFO, whch hods components for demands that arrve earer regardess of whether they can be served. However, optmzaton over ths cass of poces, ncudng the repenshment decson, s not consdered n 18. Another stream of reated work nvoves asymptotc anayss. Pambeck and Ward 20 consder an ATO producton/nventory system. Ther mode dffers from ours n that component producton s subject to capacty constrants. They consder a broader optmzaton probem where product prcng, producton capacty and nventory management are a subject to contro. Of specfc nterest to us here s ther nventory management, whch nvoves aocaton of components to products. They provde an asymptotcay optma (on dffuson scae) aocaton pocy for mnmzng tota dscounted cost n the socaed hghvoume asymptotc regme. Ther pocy uses a BgStep (cf. Harrson 8) nspred dscrete revew pocy. Three ponts about ths paper reatve to 20 are worth mentonng: (1) We consder ong run average rather than dscounted costs; (2) We use a contnuous revew pocy; and (3) Our resuts aso hod n the hgh voume mt and can be transated to that regme by hodng fxed and ettng the demand arrva rate grow.
4 4 The dscrete revew approach used n 20 was apped by u et a. 15 to a partcuar ATO system. They consdered the socaed N mode (2 components and 2 products) where the components are aowed to have dfferent (determnstc) ead tmes. They obtaned an asymptotcay optma dscrete revew pocy for mnmzng tota dscounted cost n the hgh voume asymptotc regme. Fnay, Godberg et a. 5 showed that a very smpe contro pocy (order a fxed amount n each perod) s asymptotcay optma for a ost saes mode as the ead tme grows wth a other parameters hed fxed The SPbased Approach The approach taken n ths paper can be summarzed n the foowng foursteps: 1. ntroduce a twostage SP whose optma souton s a ower bound on the average nventory cost that can be acheved by any feasbe nventory pocy; 2. sove the SP; 3. use the SP optma souton to desgn a dynamc contro pocy for the assocated ATO system; and 4. prove that the pocy deveoped n step 3 s asymptotcay optma on the dffuson scae. As we mentoned above, the SP that we need to sove s a twostage stochatc near program wth compete recourse. Hence whe an SP can n genera be dffcut to sove, ours s one of the easest types, and can be handed by standard methods such as Sampe Average Approxmaton (SAA), cf. Shapro et a. 25. In ths sense, our approach repaces an mpossby compcated dynamc contro probem by a sovabe optmzaton probem. Our approach s smar to the fourstep method ntroduced n Harrson 7 and used repeatedy snce then, e.g. n Harrson 8, Harrson and ópez 9, Harrson and Wen 11, Pambeck and Ward 20, and many others. In that approach the orgna probem nvoves contro of a stochastc processng network and the (asymptotc) ower bound s gven by a Brownan contro probem. It s worthwhe to descrbe the contrbutons of ths paper n terms of the four steps above. We resort to the use of the ower bound SP provded n 4 for Step 1. The SP s stated n terms of an nfmum, and ths nfmum may not be attaned. A transformed SP, wth a mnmum that s attaned, was provded n 4 for the W system. We provde, n Theorem 2, a transformed SP that works for a BOMs. Ths resut makes souton of the SP possbe, and hence fts n Step 2. As mentoned above, we provde both a repenshment and aocaton pocy based on the SP souton. Ths s Step 3. Fnay, our proof of asymptotc optmaty s Step 4. Athough from a superfca pont of vew t appears that a four steps have been competed for the case wth dentca ead tmes, t s worth pontng out that the appcaton of ths approach to rea ndustra scae probems has not yet been undertaken. In addton, the case wth nondentca
5 ead tmes remans. A ower bound SP was provded for ths case n Reman and Wang 21, where a repenshment pocy was aso provded. Some of the chaenges nvoved n provng asymptotc optmaty n ths case are descrbed n Secton The Inventory Contro Probem We consder an ATO system that has m products and n components. The b of materas s gven by the matrx A, of whch eement a j represents the amount of component j (1 j n) needed to assembe one unt of product (1 m), and the j th coumn, A j (1 j n), gves the use of component j by a products. We assume that a j (1 m, 1 j n) are nonnegatve ntegers. (We can thus hande any ratona quanttes by approprate defnton of a component.) We denote by ā the argest eement of A and a the smaest nonzero eement. As auded to n the Introducton, we assume that a components have the same repenshment ead tme. Athough ths paper focuses on contnuousrevew modes, our anayss can be extended to perodcrevew modes wth tte effort. There are three semnfnte tme ntervas that arse n our mode and anayss. Demand for products arrves over the nterva 0, ). Orders for components can be paced over the nterva, ). Fnay, we do our cost accountng over the nterva, ), whch we ca our optmzaton horzon. Demand s modeed by the vector process D = {D(t), t 0}, where D(t) = (D 1 (t),..., D m (t)), t 0, D (t) s the amount of demand for product (1 n) that arrves wthn the nterva 0, t, and D(0 ) = 0. (A sampe paths are taken to be rght contnuous.) We assume that D s compound Posson: the number of orders s a Posson process Λ = {Λ(t), t 0} wth EΛ(1) = λ, and there s an assocated..d. sequence of random vectors that gve order szes. A generc eement of ths sequence denoted by S = (S 1,..., S m ), where S s the order sze for product (1 m). Athough the order sze vectors are ndependent, the components S (1 m) can be dependent. We assume that S has a fnte (2 + δ) moment,.e., η ES 2+δ <, 1 m, where δ > 0 can be arbtrary sma. Thus demands arrvng per unt tme have fnte means µ = (µ 1,..., µ m ) where µ = λes, 1 m, and a fnte covarance matrx Σ. We denote the varances by σ, 1 m. 5
6 6 The demand that arrves at a partcuar tme t (f any) s denoted by d(t) D(t) D(t ), t 0, whe demand that arrves between two dstnct tme ponts s denoted by D(t 1, t 2 ) D(t 2 ) D(t 1 ), t 2 > t 1 0. Wth a sght abuse of notaton, et D(t) D(t, t), t, denote demand that arrves wthn the ead tme mmedatey before tme t. Snce the arrva process s compound Posson, D(t) have the same dstrbuton for a t. et D = (D 1,..., D m ) denote a random vector that has ths dstrbuton. We refer to D as the ead tme demand, and note that ED = µ and E(D ED)(D ED) = Σ. (1) As prevousy mentoned, the contro pocy can convenenty be descrbed va a repenshment pocy and an aocaton pocy. We denote the repenshment pocy by γ and the aocaton pocy by p, so the contro pocy s gven by (γ, p). A repenshment pocy γ gves rse to the process {R(t), t }, where R(t) = (R 1 (t),..., R n (t)), and R j (t) represents the amount of component j (1 j n) ordered durng, t. et R( ) = 0 and defne r(t) R(t) R(t ), t, R(t 1, t 2 ) R(t 2 ) R(t 1 ), t 2 > t 1, and R(t) R(t, t), t 0, to be orders paced at tme t, durng the perod (t 1, t 2, and wthn the ead tme mmedatey before tme t respectvey. Snce one cannot order a negatve quantty, each eement of R(t) s nondecreasng over t (t ). Reca that we aow repenshment orders startng at t =. An aocaton pocy p gves rse to the process {Z(t), t 0}, where Z(t) = (Z 1 (t),..., Z m (t)), and Z (t) s the amount of product (1 n) served durng 0, t. et Z(0 ) = 0 and defne z(t) Z(t) Z(t ), t 0, Z(t 1, t 2 ) Z(t 2 ) Z(t 1 ), t 2 > t 1 0, and Z(t) Z(t, t), t,
7 7 to be demand served at tme t, durng the perod (t 1, t 2, and wthn the ead tme mmedatey before tme t respectvey. Snce one can ony serve a postve amount of demand, Z(t) s eementwse nondecreasng over tme. The event sequence at any tme s as foows: arrva of new demands, recept of prevousyordered components, aocaton of avaabe components to serve demands, and pacement of new orders. Not a events happen at each tme. Unsatsfed demands are backogged and unused components stay n nventory. et B(t) = (B 1 (t),..., B m (t)) and I(t) = (I 1 (t),..., I n (t)) be the backog and nventory eves at t (t 0) after these events. Then B(t) = B(t ) + d(t) z(t) (2) and I(t) = I(t ) + r(t ) Az(t), (3) where we defne B(0 ) = 0 and I(0 ) = 0. Between two dstnct tme ponts 0 t 1 < t 2, B(t 2 ) = B(t 1 ) + D(t 1, t 2 ) Z(t 1, t 2 ) (4) and I(t 2 ) = I(t 1 ) + R(t 1, t 2 ) AZ(t 1, t 2 ). (5) Specazng these condtons to t 1 = t and t 2 = t, B(t) = B(t ) + D(t) Z(t) (6) and I(t) = I(t ) + R(t ) AZ(t). (7) For dscussons beow, we defne B (t) and I (t) as backog and nventory eves at tme t (t 0) after demand arrva and components recept but before the aocaton of components. Hence, B(t) = B (t) z(t), t 0, I(t) = I (t) Az(t), t 0. We defne Q(t) AB (t) I (t) = AB(t) I(t), t 0, (8) as the component shortage at tme t: there are more components j onhand than the amount needed to cear the exstng backog f Q j (t) 0 and ess than enough f Q j (t) > 0 (1 j n). et h j be the cost of hodng a unt of component j (1 j n) n nventory per unt of tme and b be the cost of keepng a unt of demand for product (1 m) n backog per unt of tme. Wthout oss of generaty, we assume that h j > 0 (1 j n) and b > 0 (1 m). (We
8 8 can remove a components wth zero hodng costs and a products wth zero backog costs from the mode wth no effect on the tota cost.) Satsfyng a unt of demand for product (1 m) removes from the system a cost of n c = b + a j h j perunt of tme. We refer to c as the unt nventory cost of product (1 m). et b = (b 1,..., b m ), h = (h 1,..., h n ) and c = (c 1,..., c m ). j=1 Hence the tota expected nventory pus backog cost at tme t s h I(t) + b B(t). The goa of nventory management s to deveop a pocy (γ, p) to mnmze the foowng ongrun average expected tota nventory cost: C γ,p = m sup T 1 T T + E b B(t) + h I(t) dt. (9) Note that the ntegra n (9) s over the nterva, T +, whch s consstent wth our optmzaton horzon, rather than 0, T as n equaton (8) of 4. The dfference between these ntervas becomes mmatera when the mt T s taken. To be feasbe, the pocy cannot serve more demand than the amount arrved or the amount aowed by the avaabe suppy of requred components,.e., for any 0 t 1 < t 2, Z(t 1, t 2 ) B(t 1 ) + D(t 1, t 2 ), (10) AZ(t 1, t 2 ) I(t 1 ) + R(t 1, t 2 ). (11) The pocy needs to be nonantcpatng,.e., r(t) and z(t) can depend ony on nformaton avaabe at t, gven by B(0 ), I(0 ), {D(s), 0 s t},{z(s), 0 s < t}, and {R(s), s < t}. We concude ths secton by showng two ATO systems (see Fgure 1) that have been commonystuded n the terature. In the W system (e.g., 4), two products are assembed from three components, and both products use the same amount of the common component 0. In the M system (e.g., 16, 19), two components are used to bud three products where the mdde product 0 uses the same amount of each component as the sde product that uses that component excusvey. 3. Stochastc Program: ower Bound and Pocy Deveopment We present the ower bound SP of 4 aong wth an aternate ower bound SP n Secton 3.1 and use the aternate ower bound SP to desgn the nventory pocy n Secton 3.2.
9 9 Fgure 1 Two Sampe ATO Systems 3.1. The SP ower Bound The oneperod ATO mode consdered n Song and Zpkn 26 corresponds to the foowng SP: mn{h y + b ED Eφ(y; D)}, y 0 φ(y; D) = max{c z z D, Az y}. (12) z 0 Ths s aso a smpe verson of Newsvendor network of Harrson and van Meghem 10. Ths SP nvoves mnmzng the nventory cost for a partcuar pont n tme n the absence of any past hstory. Hence the repenshment pocy reduces to a onetme decson, y = (y 1,..., y n ) where y j s the order quantty of component j (1 j n). The aocaton decson aso smpfes to the choce of a vector z = (z 1,..., z m ) where z s the amount of product demand (1 m) to serve. Demand s gven by a random vector D. Cost parameters b, h, and c are anaogous to those n Secton 2. Components are aocated after observng a demands, an arrangement that s mpossbe for a dynamc ATO system where demand arrvas contnue nto the ndefnte future. Athough ntutvey t may seem that the SP (12) provdes, at each pont n tme, a reaxaton of the dynamc nventory contro probem, t was shown n Doğru et a. 4 that ths s not the case. In partcuar, n the dynamc system, past decsons may ead to backogs of ower vaue products, aowng the manager to dvert components ordered to cear these backogs to serve more vauabe new arrvas f needed. The SP n (12) has no such fexbty. Doğru et a. 4 ntroduced the foowng reaxed verson of (12) where the nta backog (α) can be chosen optmay: C = nf Φ(α) (13) α 0
10 10 Φ(α) nf {h y + b E(α + D) Eφ(y; α + D)}, y 0 φ(y; α + D) = max{c z z α + D, Az y}. (14) z 0 By Theorem 2.1 n 4, f D has the same dstrbuton as the ead tme demand, then C n (14) s a ower bound on the cost objectve n (9). Beow s a restatement of ther resut. Theorem 1. et γ, p be any feasbe repenshment and aocaton poces. et C γ,p be the resutng ongrun average tota expected nventory cost as defned n (9). et C be gven by (14). Then C C γ,p. (15) It s easy to verfy that Φ(α) s nonncreasng n α. Due to the unbounded support of D, the nfmum n (13) may not be attaned at fnte vaues of α and y even though C s fnte. To dea wth ths ssue we consder an aternatve reaxaton of (12). Instead of optmzng the nta backog eve, we aow y and z to be negatve, gvng rse to the foowng SP: C = nf C(y) y R n C(y) h y + b ED Eϕ(y; D), { } ϕ(y; D) max c z z D, Az y. (16) z R m The absence of nonnegatvty constrants n the above formuaton aows the manager to carry out an nfeasbe feat of undong past orderng and aocaton decsons that she may regret after observng new demands. Thus t s not surprsng that C s aso a ower bound on C γ,p. Moreover, the foowng resut, whose proof s n Secton 5, shows that C attans the same vaue as C, but unke the atter SP, the souton s attaned at a fnte optma vaue, y (possby not unque). Theorem 2. There exsts y R n such that Moreover, for any y such that C(y ) = C, C(y ) = C = C. where M s a constant that depends ony on A, b, h, and ED. y j M, 1 j n, (17) Foowng the theorem, we denote C as the cost ower bound and formaze the SP as C = mn y R n C(y) C(y) h y + b ED Eϕ(y; D) ϕ(y; D) max z Rm{c z z D, Az y}. (18) Ths SP has compete recourse. We w use the recourse probem ϕ(y; D) to desgn an aocaton pocy and the optma souton y to specfy a repenshment pocy.
11 Pocy Deveopment We frst show how the SP above naturay eads us to propose (as n 4) a basestock repenshment pocy, and then focus our efforts on the more compex deveopment of an aocaton pocy, whch was eft open n 4. In (18), the optma frststage souton, y, specfes the amounts of components to order for servng demand D. Snce D s the ead tme demand, the souton can be naturay mtated by a basestock pocy n an ATO system by ettng y be the basestock eves. If y s not unque, t does not matter whch one we seect, as ong as t s used consstenty to set basestock eves. A basestock pocy keeps the nventory poston of each component,.e., the tota nventory (onhand and n ppene) n excess of the amount needed to cear exstng backogs, at a constant eve. Wthout oss of generaty, we assume that the optma nventory postons y are reached at t = 0. To keep nventory postons at that eve at a subsequent tmes, R(t) = AD(t), t,.e., amounts ordered durng the past ead tme are the same as the amounts needed to satsfy demands that arrved durng that perod. The quantty R(t) s aso the tota nventory n the ppene at tme t. Thus y = I(t) + R(t) AB(t), t, (19) whch gves rse to the foowng expresson of the onhand nventory: I(t) = y + AB(t) AD(t), t. (20) Combnng (20) wth (8) yeds the foowng smpe expresson for the component shortage process under a basestock pocy: Q(t) = AD(t) y, t. (21) The aocaton outcome n (18) typcay cannot be exacty repcated n nventory contro of ATO systems. Components n the atter case cannot wat to be aocated unt after a demands have been observed. Nor can prevousyaocated components be cawed back. Nevertheess, the recourse P of (18) does suggest the foowng approach: gven Q(t) as component shortage at tme t (t 0), we set backog target B (t) by sovng B (t) = arg mn{c B B 0, AB Q(t)}, t 0. (22) Ths P s equvaent to the recourse P n (18): denote B by D z and Q(t) by AD y, mn B R m{c B B 0, AB Q(t)} = mn m{c (D z) D z 0, A(D z) AD y } z R = c D max z Rm{c z z D, Az y }. (23)
12 12 The P (23) optmzes the aocaton of components whe the P (22) optmzes the aocaton of component shortage. The vaue of ths transformaton w become obvous when we gve the ntuton for provng asymptotc optmaty n Secton 4.3. To use ths resut to nduce an asymptotcay optma pocy and to avod wde fuctuatons of the backog target n response to sma perturbatons of component shortage, we need B (t) to be unformy pschtz contnuous wth respect to Q(t). If the optma souton of (22) s a sngeton for a possbe vaues of Q(t), then the condton foows from Hoffman s emma 13 (aso see Theorem 10.5 of Schrjver 24). If the optma souton s not unque, these resuts ony mpy that the souton sets are pschtz contnuous, and we sove the foowng quadratc programmng probem to seect the souton wth mnmum Eucdean norm: mn { B AB Q(t), c B B 0 ψ } (24) where ψ s the optma objectve vaue of (22) and B denotes Eucdean norm of B. In genera pschtz contnuty of sets does not mpy pschtz contnuty of the mnmum norm seecton, cf. Aubn and Frankowska 3. However, for the speca case of (24), Theorem 4.1.d n Han et a. 6 shows that the optma souton s unque and pschtz contnuous Usng the backog targets, we estabsh the foowng Verfcaton emma, whch presents a suffcent condton for achevng exact optmaty under the aforementoned basestock pocy. emma 1. (Verfcaton emma) Any nventory pocy that uses a basestock repenshment pocy wth basestock eves y s optma f the resutng backog eves satsfy c EB(t) = c EB (t) for a t. (25) Athough ths emma s not qute a speca case of the emma 2 n 21 because we are usng a basestock pocy here, the proof s mmedate from (8), (20), (23), and (25): b EB(t) + h EI(t) = c EB(t) + h (y AED(t)) = c EB (t) + h y h (AED) = b ED + h y Emax{c z z D, Az y }, z.e., the ntegrand n (9) reaches the ower bound gven by (18) at a t. The emma suggests that n an dea stuaton, product backogs shoud aways stay at ther targets. Whe the condton cannot aways be satsfed n genera, t motvates us to defne the foowng prncpe for desgnng mpementabe poces that myopcay am to keep backog eves cose to ther targets.
13 Aocaton Prncpe: Backog eves exceedng ther targets, B (t) > B (t) (1 m, t 0), are not aowed to persst f the requred components are a avaabe,.e., the aocaton pocy must yed (B (t) B (t)) + mn j:a j >0 {(I j(t) a j + 1) + } = 0, 1 m. (26) In addton, any product whose backog eve s beow or at the target s not served,.e., z (t) (B (t) B (t)) +, 1 m. (27) Specazng ths prncpe to the W system, when c 1 > c 2, the optma target n (22) becomes B (t) = arg mn{c 1 B 1 + c 2 B 2 B 0, B Q (t), B 1 + B 2 Q 0 (t), = 1, 2} (28) = (Q + 1 (t), Q + 2 (t) (Q 0 (t) Q + 1 (t)) + ). Hence to meet ths target, product 1 shoud have no backog f there s no shortage of the sde component 1, whch mpes exacty the same prorty rue as n 4: gve the common component to product 2 ony f the same component cannot be used to serve product 1. The aocaton prncpe aso dctates that when a product has a strcty postve backog that does not exceed ts target, ts demand s not served even f a requred components are avaabe. Hence reservaton ( hod back ) n some stuatons s a feature of our prncpe. Such a stuaton can happen n the M system, e.g., when c 0 > c 1 + c 2. The aocaton prncpe s not a specfc pocy, but rather a condton that may be satsfed by many dfferent poces. We show beow that, couped wth a basestock pocy that uses the correct basestock eves, any aocaton pocy that satsfes the aocaton prncpe s asymptotcay optma. In systems wth more than two products sharng one component, the defnton n (8) and the P n (22) ndcate that when the backog of one product s strcty beow ts target, backogs of other products may a exceed ther targets. In ths case, the aocaton prncpe may aow a famy of poces that dffer n ther seectons of the atter products to serve frst. We can aso reax the aocaton prncpe to admt more poces. For nstance, t s shown n 4 that n the W system wth a short ead tme, f c 1 vasty exceeds c 2, then wthhodng some common component 0 from product 2 when there s no component shortage can resut n a ower nventory cost than the myopc prorty pocy that serves as much demand as possbe. To aow the former stuatons, we can generaze (26) to (B (t) B (t)) + mn j:a j >0 {(I j(t) a j + 1 w j ) + } = 0, 1 m, (29) where w j s the amount of component j (1 j n) wthhed from product (1 m). As we show n Secton 4.3, f w j (1 m; 1 j n) remans a constant or grows sowy enough n, poces that satsfy (29) w aso be asymptotcay optma. 13
14 14 4. Asymptotc Anayss For the purposes of our asymptotc anayss, we ntroduce a famy of ATO systems ndexed by the ead tme. A parameters other than are hed fxed, whe. (As was noted n the Introducton, our resuts aso appy to the hghvoume mt consdered n 20, where the ead tme stays constant whe the order arrva rate λ ncreases.) et C denote the ong run average cost for our pocy, and et C denote the ower bound, both for ead tme. Then our man resut (Theorem 4) states that C m = 1. (30) C In Theorem 3 we show that 1 2 C converges to a fnte postve constant. Thus (30) s equvaent to C C m = 0, (31) and (31) s actuay what we show. The scang of the costs and varous stochastc processes n ths system s bascay that of a (functona) centra mt theorem. Ths refects a smpe fact: As the ead tme grows, the tota demand over a ead tme (when propery centered and scaed) converges to a normay dstrbuted random varabe. Observe that (31) s more strngent than the fudscae asymptotc optmaty crteron. The defnton of the atter s the same as n (31) except that s repaced by. A pocy that s optma ony on the fud scae may have an average cost that dffers from the exact optmum by a quantty on the order of (or possby arger). In ths case, the rato n (30) may not converge to unty. Asymptotc optmaty on the fud scae s an easy crteron that s satsfed by every exstng approach that we are aware of. However, for each pocy that we can fnd n the terature, there s aways a counterexampe, constructed rather easy, showng that t s not asymptotcay optma on the dffuson scae. For exampe, the numerca experments conducted n 4 ndcated that (30) does not hod for the pocy deveoped n 16. To ndcate ther dependence on, the varous random varabes and processes w assume the same defntons as before except that they may have a superscrpt () or be assgned an argument to specfy the system under dscusson. Some varabes are centered (.e., takng the dfference from ts mean). As a rue of our notaton, we use X (attachng a tde) to denote a scaed but not centered vaue of varabe X, and ˆX (attachng a hat) to denote a centered and scaed vaue of X. For nstance, anaogous to the defnton of D n Secton 2, D () denotes the random vector that has the same dstrbuton as demands arrvng over a ead tme n system. Its centered and scaed verson s gven by ˆD () D() µ. (32)
15 15 Foowng (1), E ˆD () = 0 and E ˆD () ( ˆD () ) = Σ. Furthermore, for the nterest of our dscusson, we scae wthout centerng demand arrvas, orders paced, and demand served at tme t (t 0), between tmes t 1 and t 2 (t 2 > t 1 0), and wthn a ead tme mmedatey precedng tme t (t 0): d () (t) d() (t), D() (t 1, t 2 ) D() (t 1, t 2 ), D() (t) D() (t), r () (t) r() (t), R() (t 1, t 2 ) R() (t 1, t 2 ), R() (t) R() (t), and z () (t) z() (t), Z() (t 1, t 2 ) Z() (t 1, t 2 ), Z() (t) Z() (t), respectvey. We aso scae the backog and nventory eves at each tme by B () (t) B() (t) and Ĩ() (t) I() (t), t 0, respectvey. Combnng these two quanttes gves rse to the scaed verson of the shortage process,.e., A B () (t) Ĩ() (t) = Q () (t) Q() (t) t 0. The next two subsectons present some premnary emmas. In Secton 4.1, we dscuss propertes of scaed (and sometmes aso centered) versons of demand processes. In Secton 4.2, we address propertes of our nventory pocy. The dscusson cumnates n Secton 4.3 where we prove that our pocy s asymptotcay optma on the dffuson scae The Demand Process We frst consder demands arrvng at a gven pont n tme, that s, order szes. As ncreases, even though the dstrbuton of the order sze S does not change, the maxmum order sze over a arrvas that occur wthn a ead tme w ncrease. However, the foowng emma shows that ts expected vaue s neggbe on the eadng order of average cost ( ) that s of nterest to us. emma 2. Under the assumpton that S has fnte (2 + δ) moment (δ > 0), E d () (τ) sup t 1 τ t 3λ 1 2+δ (1 + η ) δ 2(2+δ), 1 m. (33) Besdes the sze of snge arrvas, we are aso nterested n the tota demand arrvng between two dstnct tme ponts, D () (t 1, t 2 ), 0 t 1 < t 2. Ther centered and scaed vaues are ˆD () (t 1, t 2 ) D() (t 1, t 2 ) (t 2 t 1 )µ, 0 t 1 < t 2,
16 16 where E ˆD () (t 1, t 2 ) = 0 and E ˆD () (t 1, t 2 ) ˆD () (t 1, t 2 ) = (t 2 t 1 )Σ. Athough we don t expcty use ths fact, t shoud be noted that a functona centra mt theorem ndcates that when, ˆD(0, t) converges to a Brownan moton. In the foowng emma, we estabsh upper bounds on the centeredandscaed verson of the expected maxmum demand that occurs wthn certan subntervas of a ead tme. Note that athough the resut s stated wth respect to the tme nterva 0, 1, due to statonarty ths resut hods for any nterva of the form t, t + 1 for t 0. emma 3. For a = 1,..., m, E sup 1/4 τ 1 E ( where κ can be any postve constant. sup 0 τ 1/4 () ˆD (0, τ) () ˆD (0, τ) τκ ) + (1 + σ 2 ) 1/8, (34) σ2 κ 1/4, (35) We aso center and scae demand over a ead tme, ˆD () (t) ˆD () (t 1, t), t 1. The dstrbuton of ˆD (t) s the same as ˆD () n (32) Inventory Pocy Foowng our repenshment pocy n each system, we sove the foowng SP: mn y R m C() (y) (36) where C () (y) = h y + b (µ) Eϕ(y; D () ) ϕ(y; D () ) = max z R n{c z z D(), Az y} and use the optma souton y () to set basestock eves. Aternatvey, we can center and scae decson varabes by ettng to transform (36) nto ŷ = y Aµ and ẑ = z µ mn ŷ R m Ĉ() (ŷ) (37) where Ĉ() (ŷ) = h ŷ Eϕ(ŷ; ˆD () ) ϕ(ŷ; ˆD () ) = max ẑ R n{c ẑ ẑ ˆD (), Aẑ ŷ}. One may easy verty that Ĉ () (ŷ) = C() ( 1/2 ŷ + Aµ ). (38)
17 17 Therefore the optma souton(s) of (37), ŷ (), s reated to that of (36) as foows ŷ () = y() Aµ. (39) Here y () sets basestock eves and ŷ () gves ther dfferences from mean demands, measured on the scae. The former grow wthout bound as the ead tme ncreases whe by the emma beow, whose proof s n Secton 5, the atter are bounded regardess how ong the ead tme s. emma 4. There exsts a constant M such that for a > 0, ŷ () j M, 1 j n. (40) We can smary specfy the aocaton prncpe for the th system. Operatng at the ead order, for each, we sove the foowng (scaed) verson of (22) B () (t) = arg mn{c B B 0, AB Q () (t)}, t 0, (41) to set the backog target. Reca that n (22), we appy a mnmum norm seecton to keep the optma souton B (t) unformy pschtz contnuous wth respect to Q(t). Here we contnue the same approach to make B () (t) unformy pschtz contnuous wth respect to Q () (t). As a consequence, the foowng emma (whose proof s n Secton 5) shows that we may use demand fuctuatons to bound the changes of backog target over tme. emma 5. There exsts a constant g that depends ony on A and c, such that for any t 2 > t 1 1, () B (t 2 ) () B (t 1 ) g =1 () () ˆD (t 1, t 2 ) ˆD (t 1 1, t 2 1), 1 m, (42) and for t 1, () B (t) () B (t ) g =1 () () d (t) d (t 1), 1 m. (43) 4.3. Asymptotc Optmaty To prove the asymptotc optmaty of our pocy, we frst appy the centra mt theorem to show that 2 1 C () converges to a fnte postve constant, so that (30) s satsfed f our pocy s asymptotcay optma on the dffuson scae,.e., t satsfes (31). We then ntroduce an asymptotc verson of the Verfcaton emma to present a suffcent condton for the atter optmaty, foowed by a proof that our pocy satsfes ths condton. We ntroduce the mt stochastc program as foows. et ξ = (ξ 1,..., ξ m )
18 18 be a normaydstrbuted random vector wth mean 0 and covarance Σ. For y R n, et where Ĉ(y) h y Eϕ(y; ξ) ϕ(y; ξ) max z Rm{c z z ξ, Az y}. Note that for gven (y, ξ), ϕ(y; ξ) s aways feasbe, so Ĉ nf Ĉ(y) y R m s aso a twostage SP wth compete recourse. et Ĉ C. Theorem 3. (See Secton 5 for proof) The optma vaue of the scaed SP converges to that of the mt SP: m Ĉ = Ĉ. (44) From the above t s cear that our pocy satsfes (30) f t s asymptotcay optma on the dffuson scae,.e. satsfes (31). Beow we present a suffcent condton for our approach to satsfy (dffusonscae) asymptotc optmaty condton (31). We refer to our concuson as an Asymptotc Verfcaton emma to hghght that t s a parae to the Verfcaton emma n Secton 3.2. In both cases, the condton s gven by the excesses of backog eves at each tme over ther targets. Whe the exact condton n emma 1 cannot hod n genera, ts asymptotc verson n the emma beow (whose proof s n Secton 5) s much easer to satsfy. emma 6. Asymptotc Verfcaton emma Any famy of nventory poces that use basestock repenshment wth basestock eves y () n the system wth ead tme s asymptotcay optma f { m sup E B () (t) E B } () (t) = 0. (45) t 1 Usng the above emma, we can prove asymptotc optmaty of a pocy by showng that the dfference between the expected backog eves and ther targets s neggbe for a products and at a tmes. Such condton can be satsfed by poces that satsfy the aforementoned aocaton prncpe under the morebroady defned condton (29). The specazaton of the atter condton to the th system s (B () (t) B () (t)) + mn j:a j >0 {(I() j (t) a j + 1 w () j ) + } = 0, 1 m. (46) For the pocy to be asymptotcay optma, we show that t s suffcent that m w () j = 0, 1 m, 1 j n. (47)
19 Theorem 4. et {(γ (), p ()), > 0} denote a famy of poces that use basestock repenshment wth basestock eves y () and use an aocaton pocy that satsfes the aocaton prncpe aong wth (47). Then m 19 C γ (),p () C = 1. (48) Before presentng the proof, an ntutve expanaton s n order. We refer to the postve dfference of a backog eve from ts target as the excess, and the negatve dfference as the defct. The openng part of the proof makes a smpe but usefu observaton: the defnton of backog target (24) and the requrement of the aocaton prncpe (46) (wth condton (47)) mpy that no product w have a nontrva amount of excess uness some other product has a nontrva amount of defct. Hence our concuson hods f the expected defct of any product never passes a neggbe eve,.e., when condton (51) n the proof beow appes. To prove the condton, we frst note that the backog targets, B (), are on the same order ( ) as the component shortage, Q (). Demand arrva rates (wth tme scaed, but not space) are on the order of. Snce products wth a defct are not served, as ncreases, the probabty that a product has a defct persstng for more than a ead tme becomes asymptotcay neggbe. So do the expected defcts, a stuaton shown by the dscusson of Case 1 n (53)(56). For sampe paths where the defct perssts for ess than a ead tme, taken care of n (57)(63), emma 3 s used to show that, agan, the expected defct s asymptotcay neggbe. The proof of the above arguments s factated by statonary demand processes and the basestock repenshment pocy. The atter renders the past states rreevant after a ead tme, so we can deveop a bound on the expected defct over the fnte tme nterva 0, 2 and appy the bound unformy to the nfnte tme horzon. Proof Foowng emma 6, we prove the resut by showng that (45) hods. For gven and t 1, et { S + (t) = : { S (t) = : Snce B () (t) s feasbe for (24), () B (t) > () B (t) < } () B (t), 1 m, } () B (t), 1 m. A j B () (t) Q () (t) = A j B () (t) Ĩ() (t). Therefore for a j = 1,..., n, S + (t) a j ( () B (t) () B (t)) Ĩ() j (t) + a j ( S (t) () () B (t) B (t)). (49)
20 20 Observe that for every S + (t), there exsts some j such that a j > 0 and S + (t) a j ( If not, then (49) mpes that () () B (t) B (t)) a ( j S (t) () B (t) B () (t)) 1 a j w() j. (50) B () (t) > B () (t) and I () j (t) a j + 1 w () j > 0 for a j such that a j > 0, whch voates the aocaton prncpe (46). Snce a terms on the efthand sde of (50) are strcty postve, the nequaty mpes that 1 a j w() j () < a j ( B (t) B () (t)) + 1 a j w() j S (t) a j ( () B (t) () B (t)). Usng (47), m 1 a j w() j = 0, and we prove the theorem by showng that for any ɛ > 0, f s arge enough, then for any S (t), For a gven S (t), et E () () B (t) B (t) < ɛ for a t 1. (51) t () = sup{τ : 0 τ t and () B (τ) < () B (τ)} We can wrte E () B (t) () B (t) = E( () B (t) () B (t))1(t () < t 1) + E( () B (t) and prove (51) by consderng the two stuatons on the rghthand sde separatey. In cases where t () < t 1, B() (τ) demand for product s served durng ths ead tme. Therefore et j() be any j such that B () (t) () B (t))1(t () t 1) (52) () B (τ) for a τ (t 1), t. Under our pocy, no () D (t) = a 1 () j() Q j() (t)+ max j:a j >0 {a 1 j ˆD () (t) + µ. (53) () Q j (t) + } B () (t) 0, (54) where the frst nequaty hods because otherwse B () (t) s not optma for (41). From (20), Q () j() (t) = A j() B () (t) Ĩ() j() (t) = A j() D () (t) y () j() / = A j() ˆD () (t) ŷ () j(). (55)
21 21 Defne β = ā/a and ȳ () mn = mn,j { ŷ() j /a j }. Then β 1 and ȳ () mn s fnte by emma 4. From (54) and (55), () B () Q j (t) + (t) max max j:a j >0 a j j:a j >0 A j ˆD () (t) ŷ () j + Appyng the above nequaty and usng (53), ( () () E ( B (t) B (t))1(t () < t 1) E β ( E ( β + 1) =1 a j ( β + 1) β =1 () ˆD (t) ȳ () () mn + ˆD =1 ( E =1 ( β + 1) 2 m ȳ () mn + µ where the ast nequaty comes from Chebyshev s nequaty. If t () t 1, we frst estabsh that B () (t) () B (t) ( + () B (t) B () B () (t () ) () ˆD (t) ȳ mn. () (t) ) + µ () ˆD (t) (ȳ () mn + µ ) ˆD () (t) ȳ() mn + µ m( β + 1) ) + ) + σ, 2 (56) =1 (t () )) ( () B (t) () B (t () )) () B (t () ) (57) by consderng the foowng two scenaros. In scenaro one, demand for product s served at tme t (). Ths mpes that the product s preaocaton backog exceeds ts target. Foowng (27) and the defnton of t (), B () (t () ) = () B (t () ), so (57) hods. In scenaro two, demand for product s not served at tme t (), so B () (t () ) = B () (t () ) + () d (t () ). By the defnton of t () and because () d (t () ) 0, B () (t () ) < B () (t () ) () B (t () ), whch aso eads to (57).
22 22 We now consder the rghthand sde of (57), startng from the ast term. From emma 2 and (43), there exsts some constant θ 1 such that E sup { t 1 τ t () () B (τ) B (τ ) } For the frst two terms, appyng (42) wth t 1 = t () and t 2 = t, B () (t) B () (t () ) g =1 ˆD () (t (), t) Under our nventory pocy, no product demand s served durng (t (), t, so B () (t) B () (t () ) et τ = t t (). Then 0 τ 1, and and for a = 1,.., m, µ (t t () D () (t (), t) = ) = µ τ, ˆD() ˆD () δ θ 1 2(2+δ). (58) () ˆD (t () 1, t 1). (59) (t (), t) + µ (t t () ). (60) (t (), t) = d () ˆD (1, 1 + τ), ˆD () (t (), t) ˆD () (t () 1, t 1) = d () ˆD (1, 1 + τ) () ˆD (0, τ). Thus appyng (59), (60) and above equvaences n dstrbuton, E ( () B (t) B () (t () )) ( () B (t) () B (t () )) E sup V ()+ (τ) 0 τ 1 (61) where V () (τ) g =1 et κ = µ /(2mg + 1) and defne () ˆD (1, 1 + τ) u () (τ) = () () ˆD (0, τ) + ˆD (1, 1 + τ) µ τ. () ˆD (0, τ) τκ and v () () (τ) = ˆD (1, 1 + τ) τκ, 1 m. Then and thus V () (τ) g E sup 0 τ 1 V ()+ (τ) g =1 =1 u () (τ) + g =1 v () E sup u ()+ (τ) + (g + 1) 0 τ 1 (τ) + v () (τ), =1 E sup v ()+ (τ). (62) 0 τ 1
23 23 Snce 0 τ 1, E sup 0 τ 1 u ()+ (τ) = E E sup 0 τ 1 ( ˆD ()+ sup 0 τ 1/4 (0, τ) ) + τκ () ˆD (0, τ) + E sup 1/4 τ 1 ( () ˆD (0, τ) ) + τκ (1 + σ 2 ) 1/8 + σ2 κ 1/4, 1 m, (63) where the ast nequaty s a drect appcaton of emma 3. Snce u () (τ) = d v () (τ) (1 m, 0 τ 1), we concude from (57), (58), (61), (62), and (63) that E ( () B (t) () B (t))1(t () t 1) θ 1 2(2+δ) δ + θ 2 1/8 + θ 3 1/4, where θ 1, θ 2 and θ 3 are constants. Ths nequaty and (56) prove (51), and thus the theorem. 5. Proofs Proof of Theorem 2 We frst prove that f y optmzes (16), then where M max {A j ED + (h 1 j 1 j n y j M, 1 j n, (64) ζ 1 j )b ED} and ζ j mn {b /a j }. :a j >0 Suppose that n (16), for gven y and D, z (y; D) optmzes ϕ(y; D). Hence the objectve vaue satsfes C(y) = h Ey Az (y; D) + b ED z (y; D). (65) For smpcty, denote z (y ; D) by z (1 j n). Snce Az y and z D, for a j = 1,...n, C(y) b ED z :a j >0 a j b a j ED z ζ j A j ED z ζ j (A j ED y j ), C(y) h Ey Az h j (y j A j Ez ) h j (y j A j ED). (66) Observe that the objectve vaue of (16) under a feasbe souton y = 0 and z(y; D) = 0 s b ED. The above nequates ndcate that a necessary condton for C(y) b ED s that y j A j ED b ED ζ j and y j A j ED + b ED h j, whch cannot hod f y j > M for some j = 1,.., n. Thus to be optma, y must satsfy (64) We now prove C = C. By repacng z wth z + α and y wth y + Aα, we transform (14) nto { } C = nf nf {h y + b ED Eφ (y, α; D)} (67) α 0 y Aα where φ (y, α; D) max {c z z D, Az y }. z α
24 24 Snce any feasbe souton for (67) s aso feasbe for (16), C C. To prove C C, defne G 1 (y, α) Eφ (y, α; D) = E max{c z z D, Az y}, z α G 2 (y) Eϕ(y; D) = E max z Rm{c z z D, Az y}. et α (k) = (k,..., k) and denote, as a feasbe souton to (67), y (k) j y j ( k a j ). Snce y s bounded by (64), f k s suffcenty arge, y (k) = y. Thus we ony need to prove that because f t hods, then =1 m G 1(y, α (k) ) G 2 (y ), (68) k C m k { h y + b ED G 1 (y, α (k) ) } h y + b ED G 2 (y ) = C. Gven that z s the optma souton that yeds ϕ(y ; D), z k (1 m) f A j D y j + ka, 1 j n. (69) Otherwse, f there exsts some ( = 1,.., m) such that z < k, then under (69) and because z D, no capacty constrant that nvoves z s bndng. Thus ncreasng z s feasbe and strcty mproves c z, whch contradcts the defnton of z. Defne the event Ω k0 {ω : D (ω) k 0, = 1,.., m}, where k 0 = ka M ām. We et k be suffcenty arge so that k 0 0 (whch s feasbe because M does not depend on k). Snce y j M (1 j n), for a ω Ω k0, A j D ka M ām Therefore the probabty that (69) does not hod satsfes a j ka M y j + ka, 1 j n. =1 P{ j {1,..., n} : A j D > y j + ka} P{ω / Ω k0 }. Snce z k (1 m) s feasbe for (67) on Ω k0 and z D, G 1 (y, α (k) ) G 2 (y ) c Ez 1(ω / Ω k0 ) c ED 1(ω / Ω k0 ), (70) =1
25 25 where 1() s the ndcator functon. Snce for a = 1,.., m, ED 1(ω / Ω k0 ) k 0 m =1 P{D k 0 } + ED 1(D k 0 ), we can prove the rghthand sde of (70) converges to zero and thus (68) s true by showng that whch hods for a = 1,..., m because ED 1(D k 0 ) = m ED 1(D k 0 ) = 0, 1 m, k 0 and ED 2 (1 m ) s fnte by assumpton. k 0 P{D x}dx ED 2 k 0 x 2 dx = ED 2 /k 0 Proof of emma 2 et s k be the k th (k = 1, 2,...) reazaton of S and We frst prove that for a k = 1, 2,..., s max (k) = max{s 1, s 2,..., s k }. (k) (1 + η )k 2(2+δ) δ. (71) k Es max et F,d (x) be the compmentary CDF of S (1 m). Then by Chebyshev s Inequaty, P{s max (k) > kx} = 1 1 F,d ( k kx) 1 1 et = δ/(2(2 + δ)) and use the above Es max (k) = k 0 k + ( 1 1 kη ( kx) 2+δ P{s max (k) > x}dx = k η ( kx) 2+δ ) = η k δ/2 x (2+δ). k η k δ/2 x (2+δ) dx < k + η k δ/2+ (1+δ) = (1 + η )k δ 2(2+δ). 0 k P{s max (k) > kx}dx To use (71) to prove the emma, et Λ () be the Posson random varabe wth mean λ (1 m), et p () k = P{Λ () = k}. Then E sup t 1 τ t sup t 1 τ t λ d () (τ) = d () (τ) = d s max (Λ () ), 1 m. k=0 p () k Es max (k) + k= λ +1 p () k Es max (k). (72)
26 26 For a k λ, Es max (k) Es max ( λ ). Thus λ k=0 p () k Es max (k) λ Esmax ( λ ). ( λ ) By appyng (71) and observng that λ 2 λ (1 m) For a k > λ, Es max λ ( λ ) λ(1 + η )( λ ) δ λ p () k = (λ)k e λ k! Usng (71) and observng that k 2(2+δ) (λ) k= λ +1 p () k δ Es max λ(1 + η )(λ/2) 2(2+δ) 2(2+δ) δ 2(1 + η )λ 1 2+δ δ 2(2+δ). (73) λ (λ) k 1 λ k (k 1)! e λ = k p() k 1. 2(2+δ) δ (k) λ The emma foows from (72), (73), and (74). when k > λ, k= λ +1 (1 + η ) λ p () Es max k 1 (k) k p () k 1 k 2(2+δ) δ k= λ +1 (1 + η )λ 1 2+δ δ 2(2+δ). (74) () Proof of emma 3 Snce ˆD (0, ) s a martngae (wth respect to the ftraton generated by () {D(t), t 0}) and x s a convex functon, ˆD (0, ) s a submartngae (1 m). We appy Doob s nequaty to prove both resuts. For (34), = For (35), E sup 0 τ 1/4 { 1/8 0 1/8 + P () ˆD (0, τ) sup 0 τ 1/4 E 1/8 () ˆD (0, τ) > x } () ( ˆD (0, 1/4 )) 2 x 2 dx = 1/8 (1 + σ 2 ) 1 m. E sup 1/4 τ 1 ( () ˆD (0, τ) ) + τκ E { dx + P 1/8 = sup 1/4 τ 1 1/4 κ P { sup 0 τ 1/4 ( sup 1/4 τ 1 () ˆD (0, τ) > x ) + () ˆD (0, τ) 1/4 κ () ˆD (0, τ) > x } } dx dx
27 27 1/4 κ () E ( ˆD (0, 1)) 2 x 2 dx = σ2 κ 1/4, 1 m. Proof of emma 4 The proof s smar to the frst part of the proof of Theorem 2. Foowng the same reasonng that ead to (66) and observng that E ˆD () = 0, Ĉ () (ŷ) ζ j ŷ j and Ĉ () (ŷ) h j ŷ j, 1 j n. Snce ŷ () = 0 and ẑ () = 0 ˆD () (where the mnmum s taken componentwse) are feasbe for (37), Ĉ () (ŷ () ) c E0 ˆD () ) c E ˆD (), and thus and the emma hods because ŷ () j c E ˆD () h j ζ j, 1 j n, E where σ 2 (1 m) are fnte. () ˆD 2 + E( Proof of emma 5 For t 2 > t 1 1, () ˆD ) max{σ11, 2..., σmm}, 2 B () (t 2 ) = B () (t 1 ) + D () (t 1, t 2 ) Z () (t 1, t 2 ), Ĩ () (t 2 ) = Ĩ() (t 1 ) + R () (t 1 1, t 2 1) A Z () (t 1, t 2 ). Under a basestock pocy, R () (t 1 1, t 2 1) = A D () (t 1 1, t 2 1). The above three equatons mpy that Q () (t 2 ) Q () (t 1 ) = A B () (t 2 ) Ĩ() (t 2 ) (A B () (t 1 ) Ĩ() (t 1 )) = A( D () (t 1, t 2 ) D () (t 1 1, t 2 1)) = A( ˆD () (t 1, t 2 ) ˆD () (t 1 1, t 2 1)). Snce B () (t) s pschtz contnuous n Q () (t), there exsts ϑ such that for a = 1,..., m, n () () () () () () B (t 2 ) B (t 1 ) ϑ Q j (t 2 ) Q j (t 1 ) g ˆD (t 1, t 2 ) ˆD (t 1 1, t 2 1) (75) j=1 where g s a constant, so (42) hods. The proof of (43) s smar to the above, usng =1 Q () (t) Q () (t ) = A d () (t) r () (t 1) = A( d () (t) d () (t 1)).
28 28 Proof of Theorem 3 Ths resut foows from Theorem 2.2 n Robnson and Wets 22. Specfcay, ˆD () ( > 0) weaky converges to ξ. The recourse probem ϕ(ŷ; D ˆ () ) s aways feasbe and by Hoffman s emma, contnuous n both ŷ and ˆD (). Snce ϕ(ŷ; ˆD () ) c ˆD (), and ˆD () ( > 0) have the same fnte covarance matrx Σ, ϕ(ŷ; ˆD () ) s unformy ntegrabe over { ˆD (), > 0}. The functon h ŷ () s contnuous. By emma 4, the optma souton ŷ () ( > 0) s contaned n a fnte set. Thus a condtons of Theorem 2.2 n 22 are satsfed. Ther concuson (a) mmedatey eads to (44). Proof of emma 6 et {(γ(), p()), > 0} denote a famy of poces that use basestock repenshment wth basestock eves y (). Appyng (20) to the th system, I () (t) = y () + AB () (t) AD () (t), t 1. Scang I () (t) n the above and observng that ED () (t) = µ, b EB () (t) + h EI () (t) = c E B () (t) + h ŷ () t 1. (76) Appyng (20) to the shortage process n the th system, Q () (t) = AB () (t) I () (t) = AD () (t) y () d = AD () y (). Because of ths equvaence n dstrbuton, n the th system, (23) yeds c EB () (t) = c ED () z (), (77) where z () s the optma souton of the second (maxmzng) P n (23), whch s aso the recourse probem of the ower bound SP (18). Appyng (76) to scaed nventory cost n the th system, C γ(),p() { 1 T +1 = m T T 1 Foowng (77) and (39), Therefore } { b EB () (t) + h EI () (t) 1 dt = h ŷ () + m T T C () = h Ey() Az () + b ED () z () = h ŷ () + c E B () (t). C γ(),p() C () { 1 = m T T and the emma foows as a consequence. T +1 1 ( c E B () (t) E B ) } () (t) dt, T +1 1 } c E B () (t)dt.
29 29 6. Concusons and Open Probems We have deveoped an nventory pocy for ATO systems wth a genera BOM and dentca component ead tmes. We have proved that our pocy s asymptotcay optma on the dffuson scae, and consequenty, as the ead tme grows, the percentage dfference between the average nventory cost of our pocy and ts ower bound converges to zero. Snce the nventory cost of an ATO system and the compexty of ts optmzaton grows wth the ead tme, our resut addresses stuatons where an optma nventory pocy s most needed and most dffcut to deveop. Whe ths paper addresses ATO modes wth dentca ead tmes, we conjecture that a smar SPbased approach can aso ead to asymptotcay optma poces for genera systems wth nondentca ead tmes. The frst step has aready been competed n 21, whch shows that the optma souton of a partcuar K + 1 stage SP s a ower bound on the average nventory cost of ATO systems wth K 1 dfferent repenshment ead tmes. Athough a repenshment pocy, based on the souton of the K + 1 stage SP s suggested n 21, ths pocy s qute compcated. It seems cear, based on the optmaty of ths repenshment pocy n certan contexts (ncudng that of Rosng 23, where t reduces to hs), that when ead tmes are not dentca, a sensbe repenshment pocy shoud set nventory postons for components wth shorter ead tmes based on the avaabty of components wth onger ead tmes. Such a pocy woud n genera not be of basestock form. Our anayss n ths paper rees on propertes of basestock poces. In partcuar, Q(t) has a smpe expresson n terms of the demand process under a basestock pocy (cf. (21)) that woud no onger hod f the repenshment pocy were not basestock. We eave the task of deang wth ths sgnfcant ssue to future work. (Note that we woud aso need to prove sutaby generazed versons of Theorems 2 and 3, aong wth emmas 4 and 6.) On the other hand, n terms of an aocaton pocy t seems that our targetbased aocaton pocy shoud be transferabe to systems wth nondentca ead tmes. In addton to the ssue of provng asymptotc optmaty, there are ssues that arse n makng the SPbased approach computatonay feasbe for ndustra sze probems, both wth dentca and nondentca ead tmes. Gven the amount of efforts requred, we eave these tasks to future research. Acknowedgments We thank JongSh Pang for drectng us to Theorem 4.1 n 6. References 1 N. Agrawa and M. Cohen, Optma matera contro n an assemby system wth component commonaty, Nava Research ogstcs (NR), 48 (2001), pp Y. Akçay and S. Xu, Jont nventory repenshment and component aocaton optmzaton n an assembetoorder system, Management Scence, 50 (2004), pp
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