A 2-Approximation Algorithm for Stochastic Inventory Control Models with Lost Sales

Transcription

1 A 2-Aroximaion Algorihm for Sochasic Invenory Conrol Models wih Los Sales Resef Levi Sloan School of Managemen, MIT, Cambridge, MA, 02139, USA Ganesh Janakiraman IOMS-OM Grou, Sern School of Business, New York Universiy, New York, NY 10012, USA Mahesh Nagarajan Sauder School of Business,Universiy of Briish Columbia, Vancouver, BC V6T 1Z2, Canada In his aer, we describe he firs comuaionally efficien olicies for sochasic invenory models wih los sales and relenishmen lead imes ha admi wors-case erformance guaranees. In aricular, we inroduce dual-balancing olicies for los-sales models ha are conceually similar o dualbalancing olicies recenly inroduced for a broad class of invenory models in which demand is backlogged raher han los. Tha is, in each eriod, we balance wo oosing coss: he execed marginal holding coss agains he execed marginal los-sales cos. Secifically, we show ha he dual-balancing olicies for he los-sales models rovide a wors-case erformance guaranee of 2 under relaively general demand srucures. In aricular, he guaranee holds for indeenden (no necessarily idenically disribued) demands and for models wih correlaed demands such as he AR(1) model and he mulilicaive auo-regressive demand model. The olicies and he wors-case guaranee exend o models wih caaciy consrains on he size of he order and sochasic lead imes. Our analysis has several novel elemens beyond he balancing ideas for backorder models. Key words: Invenory, Aroximaion ; Dual-Balancing ; Algorihms; Los Sales MSC2000 Subjec Classificaion: rimary: 90B05, ; Secondary: 68W25, OR/MS subjec classificaion: rimary: invenory/roducion, aroximaion/heurisics ; Secondary: roducion/scheduling, aroximaion/heurisics 1. Inroducion In his aer, we address one of he fundamenal roblems in sochasic invenory heory, he single-iem, single locaion, eriodic-review, sochasic invenory conrol roblem wih los sales, which we refer o as he los-sales roblem. This roblem has challenged researchers and raciioners for over five decades as very lile is known abou he srucure of he oimal olicy, and here are no known rovably good heurisics even for he simles seings. We build on ideas firs roosed by Levi, ál, Roundy and Shmoys [5]. They roosed wha are called dual-balancing olicies for a class of invenory models where unsaisfied demand is backlogged raher han los. These olicies have wors-case erformance guaranees, ha is, for each insance of he roblem, he execed cos of he olicy is guaraneed o be a mos C imes he oimal execed cos (for some consan C). In his aer, we discuss he imlemenaion and he wors-case analysis of dual-balancing olicies alied o invenory models wih los sales. These models have mahemaical characerisics ha are very differen han he models in which excess demand is backlogged and hus require a fundamenally differen and novel wors-case analysis. In aricular, we shall describe he firs comuaionally efficien olicies for invenory models wih los sales ha have a wors-case erformance guaranee of 2. The analysis is based on several new ideas ha we believe will conribue o fuure research in his domain. Sochasic invenory heory rovides sreamlined models wih he following common seing. The goal is o coordinae a sequence of orders over a lanning horizon of finiely many discree eriods, aiming o suly a sequence of random demands over he lanning horizon wih minimum execed cos. The cos consiss of a er-uni ordering cos for ordering suly unis a he beginning of each eriod (wih or wihou caaciy consrains), a er-uni holding cos for carrying excess invenory from one eriod o he nex, and a er-uni enaly cos for no saisfying demand on ime. The dynamics of hese models is as follows. A he beginning of each eriod, before he demand in his eriod is observed, a non-negaive rocuremen order is laced wih an ouside sulier incurring a cos roorional o he number of unis ordered. This order will arrive and become available afer a lead ime of several eriods. The demand in ha eriod is hen observed and is saisfied o he maximum exen ossible from he curren invenory 1

2 2 : on-hand. A he end of he eriod, wo ossible coss are incurred: excess unis of invenory incur a roorional holding cos and unsaisfied unis of demand incur a roorional enaly cos. The goal is o find an ordering olicy ha minimizes he overall execed coss over he enire horizon. There are wo common assumions regarding unsaisfied demand a he end of a eriod. These differen assumions disinguish beween wo fundamenally differen classes of models. In he firs class of models, he assumion is ha unsaisfied unis of demand will say in he sysem, each incurring a er-uni enaly cos for each eriod unil i is saisfied. Tha is, unsaisfied demand is backlogged from eriod o eriod in a manner symmeric o excess invenory ha is carried from eriod o eriod. These are called invenory models wih backlogged demands. In he second class of models, which is he focus of his aer, each unsaisfied uni of demand is los, i.e., i incurs a one-eriod enaly cos and hen leaves he sysem. While hese wo classes of models are equivalen if he lead ime is equal o zero, ha is when he orders arrive insananeously, hey are fundamenally differen for any osiive lead ime. In aricular, he sae-of-he-ar knowledge on los-sales models wih lead imes is very limied comared o he well undersood models wih backlogged demands. Dynamic rogramming has been he mos dominan aradigm in sudying sochasic invenory models wih los sales and backlogged demand (see Zikin [15] and Secion 2.1 below for dynamic rogramming formulaions of he los-sales model). The oimizaion roblem is defined recursively over ime, using subroblems for each ossible sae of he sysem. In aricular, he ordering decision in each eriod is made based on he available informaion a he beginning, which includes he join condiional disribuion of fuure demands, addiional informaion ha may be available by ha eriod and he ieline vecor. The ieline vecor consiss of he invenory on-hand a he beginning of he eriod and he quaniies of he ousanding orders ha were laced in as eriods and have no ye arrived. Clearly, he ieline vecor has lengh equal o he lead ime, which suggess ha he sae sace of he corresonding dynamic rogram can grow exonenially fas wih he lengh of he lead ime. However, i urns ou ha in models wih backlogged demands, i is sufficien o consider only he sum of he invenory on-hand a he beginning of he eriod (or any backlogged demand) and he quaniies of he ousanding orders. This sum is usually called he invenory osiion of he sysem. The inuiion is ha since all unsaisfied demands are backlogged, he imac of he decision made in he curren eriod on he fuure coss deends only on he difference beween he invenory osiion of he sysem (afer ordering) and he cumulaive demand over he lead ime (o be realized). Moreover, he oimal olicies in he models wih backlogged demands have a simle form and are called sae-deenden basesock olicies. In each eriod, here is a arge invenory osiion level, referred o as he base-sock level, which is unaffeced by he secific ieline vecor. If a he beginning of he eriod he invenory osiion is below he arge base-sock level, we order u-o ha arge. If he invenory osiion a he beginning of he eriod is above he arge base-sock level, no order is laced. The oimal base-sock levels can be comued by solving he corresonding dynamic rogram. Since he invenory osiion is a sufficien saisic for he ieline vecor, he comuaional comlexiy of dynamic rograms for backorder models is insensiive o he lead ime and is almos solely dicaed by he comlexiy of he demand srucure. We refer he reader o [15, 10, 5] for roofs of he oimaliy of base-sock olicies and a discussion of he relevan lieraure regarding invenory models wih backlogged demands. In conras, in sysems wih los-sales sysems and a osiive lead ime, he imac of he decision made in each eriod on he fuure coss is caured hrough a comlicaed mahemaical exression ha deends on he secific sequence of boh he ousanding orders as well as he demands over subsequen eriods. Secifically, he oimal decision in each eriod deends on he enire ieline vecor and no only on he invenory osiion as is he case in models wih backlogged demands. As a resul, he oimal olicy in los-sales models is significanly more comlex and does no ake he simle form of a base-sock olicy and he invenory osiion is no a sufficien saisic for he ieline vecor. Moreover, he sae sace of he corresonding dynamic rogram grows exonenially fas wih he lead ime. Due o he aforemenioned difficulies, he lieraure on los-sales models is limied. Karlin and Scarf [4] have been he firs o sudy he oimal olicies for models wih los sales and osiive lead imes. They have considered a los-sales model wih discree finie and infinie horizon and wih indeenden, idenically disribued demands. They have shown ha a base-sock olicy canno be oimal. Furhermore, for he case where he lead ime is equal o one eriod, hey have arially characerized he srucure of he oimal olicy. Secifically, hey have shown ha he oimal ordering quaniy is a

3 : 3 decreasing funcion of he invenory on-hand a he beginning of he eriod, and is equal o zero ouside a secified inerval. Moreover, he rae of decrease (as a funcion of he invenory on-hand) is sricly smaller han one. Wih he addiional assumion ha demands are exonenially disribued, hey have also resened a seady-sae analysis of he dynamics of los-sales sysems ha use base-sock olicies. Moron [8] has exended he analysis of Karlin and Scarf o los-sales models wih deerminisic lead imes. He has shown ha he oimal ordering olicy is a funcion of he enire ieline wih he following characerisics: (a) here is a comac region around he origin (ha is, all he comonens of he ieline vecor are zero) such ha he order quaniy is sricly osiive if and only if he ieline vecor is in his region, (b) he order quaniy decreases a a rae sricly beween zero and one wih resec o each comonen of he ieline vecor and (c) he rae of decrease in he order quaniy er comonen is higher for comonens in he ieline ha are scheduled o arrive laer in ime. He has also derived uer and lower bounds on he robabiliy ha he oimal olicy will have enough invenory o mee demand in he eriod in which he curren order will arrive (a lead ime ahead). Furhermore, he has used hese bounds o derive uer and lower bounds on he oimal ordering olicy as a funcion of he curren ieline vecor. In a subsequen aer, Zikin [17] has used sae ransformaion echniques o esablish simler roofs for he srucure of oimal olicies in he los-sales models discussed in his aer. Moreover, he has exended he resuls of Karlin and Scarf and Moron o models wih caaciy consrains on he size of he order, Markov modulaed demands and sochasic lead imes (wih no order-crossing). Moron [9] has considered myoic olicies for los-sales models, in which, in each eriod, an order is laced ha minimizes he execed cos in he eriod in which his order arrives. There are oher aers on los-sales models like he ones by Nahmias [12] and Johansen [2] ha roose differen heurisics and resen comuaional resuls on he erformance of hese heurisics. The comuaional exerimens in all of hese aers are resriced o insances where he lead ime is shor, equal o one or wo eriods or o models wih exremely low demands. In a recen subsequen aer, Zikin [16] resens comuaional exerimens in which he ess he erformance of several heurisics, including he dual-balancing olicy described in his aer. He has focused on scenarios where he demands are indeenden and idenically disribued; more secifically, hey follow oisson and Geomeric disribuions. Using sae-reducion echniques, he is able o comue he oimal olicy for insances wih lead ime equal o 4. Comuing oimal olicies wih resec o insances wih longer lead imes seems very challenging. Moreover, o he bes of our knowledge, here is no heurisic for los-sales models ha has been shown o erform well over a large bed of es roblems of realisic size. Equally imoranly, none of hese aers rovides a wors-case analysis of he roosed heurisics. Moreover, Levi, ál, Roundy and Shmoys [5] have shown ha he myoic olicy for he los-sales model even wih lead ime equal o zero does no have wors-case erformance guaranees. Secifically, hey have shown a class of examles for which he myoic olicy is arbirarily more exensive han he oimal olicy. Reiman has considered a model wih coninuous ime and wih demand following a oisson rocess and comared base-sock olicies and olicies ha lace an order in a fixed frequency [13]. In his aer we build on he recen work of Levi, ál, Roundy and Shmoys [5] who have develoed wha are called dual-balancing olicies for a class of uncaaciaed sochasic invenory models wih backlogged demands. These ideas have been exended o caaciaed models [7] and muli-echelon models [6], again wih backlogged demands. These dual-balancing olicies are comuaionally efficien and have a worscase erformance guaranee of 2 for he resecive models under general assumions on he demand srucure and he cos arameers. The dual-balancing olicies are based on wo novel ideas: a marginal cos accouning aroach and cos balancing echniques. We noe ha he marginal cos accouning scheme is very differen han he sandard dynamic rogramming based cos accouning aroach radiionally used o analyze hese models. Using he marginal cos accouning aroach, he dual-balancing olicy is based on he reeaed use of cos balancing echniques. In each eriod, wo oosing (i.e. he holding and backlogging) coss are balanced. The wors-case analysis in he above hree aers is heavily based on he mahemaical roeries of models wih backlogged demands and uses a eriod-by-eriod amorizaion cos of he dual-balancing olicy wih he cos of he oimal olicy. Crucial o he analysis is he fac ha in backorder models, comaring he invenory osiions of any wo olicies in a eriod rovides sufficien informaion o analyze heir resecive erformance a lead ime ahead. In his aer, we describe a dual-balancing olicy for models wih los sales, which is conceually similar o he dual-balancing olicy for models wih backlogged demands. However, he above-menioned

4 4 : analysis for models wih backlogged demand is no alicable o models wih los sales. In aricular, he invenory osiion does no rovide sufficien informaion o comare he coss of differen olicies. In addiion, a eriod-by-eriod amorizaion of he cos of he dual-balancing olicy wih he cos of he oimal olicy does no seem useful. To overcome hese difficulies, we describe a fundamenally differen analysis which is based on wo novel ideas. Raher han a eriod-by-eriod comarison, we use a global amorizaion of he los-sales coss of he dual-balancing olicy wih he los-sales coss of he oimal olicy. In addiion, we inroduce a new conce called he runcaed invenory osiion which generalizes he aforemenioned noion of invenory osiion. As we have already menioned, he invenory osiion in a cerain eriod is defined o be he sum of he on-hand invenory a he beginning of he eriod lus all ousanding orders. The runcaed invenory osiion is defined o be he sum of he on-hand invenory lus all he ousanding orders ha have been ordered by a cerain eriod, ossibly earlier han he curren eriod. In oher words, he runcaed invenory osiion accouns for he on-hand invenory and all he ousanding orders ha will arrive by a cerain eriod. The new conce of runcaed invenory osiion is used o comare wo olicies in a los-sales sysem. Our main resul is ha he dual-balancing olicy for he los-sales model has a wors-case erformance guaranee of 2. The wors-case analysis holds for models wih relaively general demand srucures. For examle, i holds under he assumion ha he demands in differen eriods are indeenden, no necessarily idenically disribued (see Secion 3 below for deails). Moreover, he analysis also holds in many models in which he demands in differen eriods are correlaed; secifically, i holds in he mulilicaive auoregressive demand model and he AR(1) model, which are commonly used in he lieraure. Finally, he olicy and he wors-case analysis exends o models wih sochasic lead imes (under he assumion of no-crossing of orders ) and o models wih caaciy consrains on he size of he order in each eriod. We noe ha he dual-balancing olicy can be comued efficienly in mos if no all of he realisic scenarios. As an examle, we focus aenion on he case where he demands are indeenden inegervalued random variables wih bounded suor, and rovide a deailed analysis of he running ime of he dual-balancing olicy. Dynamic rogramming aroach seems o be comuaionally inracable, since he running ime grows exonenially fas in he lead ime. In conras, we show ha he dual-balancing olicy can be comued in ime olynomial in he number of eriods and he lengh of he suor of he demands. The res of he aer is organized as follows. In Secion 2, we describe he los-sales model and a dynamic rogramming formulaion of he model. In Secion 3, we describe a dual-balancing olicy for los-sales models and he new wors-case analysis under he assumion ha he demands in differen eriods are indeenden. In Secion 4, we discuss relaed comuaional issues of he dual-balancing olicy. Finally, in Secion 5 we describe several imoran exensions of he dual-balancing olicy and he wors-case analysis o models wih caaciy consrains on he size of he order, models wih sochasic lead imes and o models wih demand srucures ha allow correlaion beween demands in differen eriods. 2. The Los-Sales Model In his secion, we rovide he mahemaical formulaion of he los-sales model and inroduce some of he noaion used hroughou he aer. As a general convenion, we disinguish beween a random variable and is realizaion using caial leers and lower case leers, resecively. Scri fon is used o denoe ses. We consider a finie lanning horizon of T eriods numbered = 1,..., T (noe ha and T are boh deerminisic). There is a sequence of sochasic demands ha occur over he lanning horizon, which are denoed by D 1,..., D T, all of which have finie mean. We firs assume ha demands in differen eriods are indeenden of each oher, hough no necessarily idenically disribued. In Secion 5 we shall show ha his assumion can be relaxed o allow several imoran srucures of correlaion beween demands in differen eriods. As ar of he model, we assume ha a he beginning of each eriod s, here is an observed informaion se ha is denoed by f s. The informaion se f s conains all of he informaion ha is available a he beginning of ime eriod s. More secifically, he informaion se f s consiss of he realized demands (d 1,..., d s 1 ) over he inerval [1, s) (in general f s can conain addiional informaion ha became available by ime eriod s). The informaion se f s in eriod s is one secific realizaion in he se of all

5 : 5 ossible realizaions of he random vecor F s = (D 1,..., D s 1 ). This se is denoed by F s. We consider only non-aniciaory olicies, ha is, in making a decision in eriod s, a feasible olicy can use only he observed informaion se f s. In each eriod s = 1,..., T, a non-negaive rocuremen order is laced from an ouside sulier, incurring a er-uni ordering cos c s. The order laced in eriod s will arrive and become available only afer a osiive lead ime, denoed by L. We assume ha L is a known osiive ineger, ha is, an order laced in eriod s will arrive a he beginning of eriod s + L. (In Secion 5, we will consider models where he lead imes are sochasic.) We now describe he dynamics of he los-sales model. A he beginning of each eriod s, as a funcion of he observed informaion se f s, we observe he join (condiional) disribuion of fuure demands (if demands in differen eriods are indeenden of each oher, hen he join disribuion is fixed regardless of he observed informaion se). A he beginning of eriod s, he sysem is characerized hrough he ieline vecor. The ieline vecor is denoed by s and consiss of L comonens. The L h comonen is he invenory on hand (or on-hand invenory) available a he beginning of eriod s afer he order laced L eriods ago in eriod s L has arrived and before he demand in eriod s is realized. We denoe he invenory on-hand a he beginning of eriod s by I s. The oher L 1 comonens of he ieline vecor are he ousanding orders ha have been laced in revious eriods and have no ye arrived. Secifically, he j h comonen of s is equal o Q s j, he size of he order laced j eriods ago, i.e., in eriod s j (for j = 1,..., L 1). Tha is, s = (Q s 1,..., Q s+1 L, I s ). Observe ha a he beginning of eriod s all he comonens of he ieline vecor are known deerminisically. We nex secify he sequence of evens in each eriod s: (i) The order of q s L unis laced in eriod s L arrives and he on-hand invenory is hus i s = (i s 1 d s 1 ) + +q s L. Observe ha (i s 1 d s 1 ) + is he invenory on-hand a he end of eriod s 1. (ii) Following a given olicy, q s unis are ordered (q s 0), and his incurs a cos of c s q s. Nex he demand in eriod s is realized and is saisfied o he maximum exen ossible from he invenory on-hand. Since unsaisfied demand is los and leaves he sysem, he on-hand invenory decreases by min{d s, i s }. In addiion, we observe a new informaion se f s+1 F s+1. (iii) A he end of he eriod, coss are incurred. If (i s d s ) > 0 hen we incur a oal holding cos of h s (i s d s ) (his means ha here is excess invenory ha needs o be carried o ime eriod s + 1). On he oher hand, if (i s d s ) < 0 we incur a oal los-sales enaly cos of s (d s i s ) (his means ha in eriod s here is unsaisfied demand ha is los). For ease of exosiion, we firs assume ha he cos arameers are saionary, ha is, for each = 1,..., T, we have h = h > 0, = > 0 and c = c 0. We furher assume ha c = 0. (The wors-case analysis resened below holds for any c > 0.) We will show ha in fac he analysis allows us o have ime-deenden holding coss arameers and non-increasing ordering and los-sales enaly arameers. In aricular, he analysis holds for models wih saionary cos arameers and discoun facor. The goal is o find an ordering olicy ha minimizes he overall execed holding coss and los-sales enaly coss over he enire horizon [1, T ]. We consider only olicies ha are non-aniciaory, i.e., a ime s, he informaion ha a feasible olicy can use consiss only of f s. Thus, for each feasible olicy, given an informaion se f s, he ieline vecor a ime eriod s and he order quaniy in eriod s are known deerminisically. 2.1 Dynamic rogramming Formulaion In his secion, we discuss he dynamic rogramming formulaion of he los sales model and discuss he associaed difficulies in he analysis. Observe ha in a los-sales model he cos in eriod s deends on he invenory on-hand a he beginning of he eriod, ha is, he execed cos in eriod s is equal o E[h(I s D s ) + + (D s I s ) + ].

6 6 : Noe ha he decision made in eriod s of how many unis o order affecs only he coss over he inerval [s + L, T ] (recall ha he order laced in eriod s will arrive a he beginning of eriod s + L). Moreover, he imac of he decision in eriod s is caured hrough he effec i has on he invenory on-hand a he beginning of eriod s + L. Unforunaely, in los-sales models, here is no racable way o caure he imac of he decision in eriod s on he invenory on hand in eriod s + L. In aricular, he imac of he decision made in eriod s on he invenory on-hand a he beginning of eriod s+l deends on he secific sequence of boh he ousanding orders a he beginning of eriod s and he realized demands over he inerval [s, s + L). Thus, he mahemaical exressions of he dynamics of he los-sales model are quie involved. As we have already seen, for each = 1,..., T, I +1 = (I D ) + + Q +1 L, (1) which imlies ha he invenory on-hand in eriod s + L deends on he decision of how many unis o order in eriod s hrough a comlicaed recursive exression. Thus, he resuling dynamic rogram formulaion deends on he enire observed ieline vecor. Le V s ( s, f s ) = V s ((q s 1,..., q s+1 L, i s ), f s ) be he oimal execed cos over he inerval [s + L, T ] given an observed ieline vecor s and an observed informaion se f s. The recursion in he los-sales model is V s ((q s 1,..., q s+1 L, i s ), f s ) = min q s 0 {E[h(I s+l(q s ) D s+l ) + + (D s+l I s (q s )) + f s ] + (2) E[V s+1 ((q s, q s 1,..., q s+2 L, (i s D s ) + + q s+1 L ), F s+1 ) f s ]}, where I s+l (q s ) is he invenory on-hand in eriod s + L assuming ha in eriod s, we have ordered q s unis. I is readily verified ha he sae sace of he above dynamic rogram grows exonenially fas in he lengh of he lead ime L even in simle cases where he demands in differen eriods are assumed o be indeenden and idenically disribued. This imlies ha solving he above dynamic rogram is likely o be inracable exce for cases wih very small lead imes. Moreover, he dynamic rogram does no rovide much insigh on he srucure of he oimal olicies and his a main reason why heoreical research on los sales models is limied. 3. Dual-Balancing olicy for he Los Sales Model In his secion, we shall describe a dualbalancing olicy for he los-sales model, and hen resen a wors-case analysis ha holds under relaively general assumions on he demand disribuions D 1,..., D T. We shall show ha under hese assumions, he dual-balancing olicy has a wors-case erformance guaranee of 2. Tha is, he execed cos of he olicy is guaraneed o be a mos wice he execed cos of an oimal olicy. In his secion, we describe he dual-balancing olicy and is wors-case analysis in he case where demands in differen eriods are assumed o be indeenden of each oher, hough no necessarily idenically disribued. In Secion 5 we discuss several imoran exensions of he dual-balancing olicy and he wors-case analysis o more general models. 3.1 Dual-Balancing olicy The olicy for he los-sales model is conceually similar o he one roosed by Levi, ál, Roundy and Shmoys for he model wih backlogged demand [5]. Tha is, in each eriod s, condiioned on he observed informaion se f s, we balance he (condiional) execed marginal holding cos incurred by he unis ordered in ha eriod over he enire horizon agains he (condiional) execed los-sales enaly cos incurred a lead ime ahead in eriod s + L. For a given olicy, le Hs be he marginal holding cos incurred by he unis ordered in eriod s over he enire horizon, and le Π s be he los-sales enaly cos incurred in eriod s + L. The cos of olicy can hen be exressed as T L C( ) = (Hs + Π s ), s=1 ignoring he marginal holding cos of unis ordered before eriod 1 and he los-sales enaly coss over he inerval [1, L] ha are idenical for every olicy. However, he exressions of Hs and Π s are differen in he los-sales model, and are significanly more comlex comared o he corresonding exressions in he models wih backlogged demands. Recall ha I is he on-hand invenory in eriod afer he order laced in eriod L has arrived and before he demand in eriod has occurred. We have already seen ha, for each = 1,..., T 1, I +1 = (I D ) + + Q +1 L, (3)

7 : 7 (where Q j for j 0 are given as an inu). Observe ha (I D ) + is he invenory on-hand a he end of eriod and Q +1 L is he order arriving a he beginning of eriod + 1. Assuming wihou loss of generaliy ha suly unis are consumed on a firs-ordered-firs-consumed basis, we conclude ha he Q s unis ordered in eriod s will be consumed only afer all he (I s+l 1 D s+l 1 ) + unis ha were on-hand a he beginning of eriod s + L (jus before he order laced in eriod s has arrived) are consumed. This leads o he following exression for he marginal holding cos incurred by he Q s unis ordered in eriod s: T Hs = h(q s (D [s+l,] (Is+L 1 D s+l 1 ) + ) + ) +. (4) Similarly, we exress =s+l Π s = (D s+l I s+l) + = (D s+l (Q s + (I s+l 1 D s+l 1 ) + )) +, (5) where he second equaliy follows from Equaion (3) above. Equaions (4) and (5) can be easily adaed o caure ime-deenden cos arameers. In addiion, we can incororae a linear ordering cos c s Q s ino Equaion (4) above. For each s = 1,..., T L and an observed informaion se f s F s, define he funcions ls B (qs B ) = E[Hs B (qs B ) f s ] and πs B (qs B ) = E[Π B s (qs B ) f s ]. As in he dual-balancing olicy for he model wih backlogged demand [5], in each eriod s, condiioned on he observed informaion se f s, we order qs B = q s o balance ls B (q s) = E[Hs B (q s) f s ] = πs B (q s) = E[Π B s (q s) f s ]. I is readily verified ha, condiioned on f s and he resuling ieline vecor B s, he funcions ls B and πs B deend only on qs B. Moreover, ls B is an increasing (convex) funcion of qs B, which is equal o 0 if qs B = 0 and goes o infiniy as qs B goes o infiniy. In addiion, πs B is a (convex) decreasing funcion of qs B, which admis a non-negaive value for qs B = 0 and is going o 0 as qs B goes o infiniy. If fracional orders are allowed he funcion ls B and πs B are coninuous and hus q s is well defined. (Laer we shall discuss he case where orders are resriced o be inegral, and demands are ineger-valued random variables.) The inuiion behind he idea of reeaedly balancing he funcions π B s and l B s above is ha in he los-sales model here are wo underlying oosing risks, he risk of under ordering incurring los-sales enaly cos and he risk of over ordering incurring holding coss. Balancing hese wo risks seems o be very effecive and comuaionally aracive. Surrisingly, his idea works significanly beer han minimizing he sum of he wo funcions. We also noe ha he dual-balancing olicy can be imlemened in an on-line manner. Tha is, he decision made in each eriod is no affeced by any fuure decision of he olicy, bu only by he currenly observed informaion se. This seems like an essenial roery if one wishes o avoid he burden of solving huge dynamic rograms. However, unlike myoic olicy, which in each eriod aims o minimize only he execed cos a lead ime ahead, he dual-balancing olicy does look ahead make use of available informaion abou he fuure evoluion of he sysem. Inegral orders and ineger-valued demands. Nex we discuss he case where he demands are ineger-valued random variables and he order quaniy in each eriod is resriced o be an ineger. We briefly describe a randomized dual-balancing olicy using ideas idenical o ones used in [5, 7]. In his case, he funcions ls B (qs B ) and πs B (qs B ) are iniially defined only for ineger values. Their iecewise linear inerolaions reserve he monooniciy (and convexiy) roeries discussed in Secion 3. The roblem is ha he balancer q s is likely o be fracional. Insead we consider he wo consecuive inegers qs 1 q s qs. 2 I is clear ha q s = λqs 1 + (1 λ)qs 2 for some 0 < λ < 1. We now order qs 1 wih robabiliy λ and qs 2 wih robabiliy 1 λ. 3.2 Analysis - Indeenden Demands Given he dual-balancing olicy for he los-sales model, we define Z o be he random balanced cos in eriod, i.e., Z = E[H B F ] = E[Π B F ] (for each = 1,..., T L). Using an idenical roof o he one in [5], we obain he following lemma. Lemma 3.1 The execed cos of he dual-balancing olicy is equal o wice he sum of execaions of he Z variables, i.e., E[C(B)] = 2 T L =1 E[Z ]. The wors-case analysis of he dual-balancing olicy in models wih backlogged demand [5] is based on a eriod by eriod amorizaion of he cos of he dual-balancing olicy agains he oimal olicy.

8 8 : This is done by comaring he resecive invenory osiions of he he wo olicies, in each eriod [5]. In conras, i is well-known [15] ha looking only on he invenory osiion is no sufficien o make oimal decisions in los-sales models. Similarly, unlike he analysis of he models wih backlogged demands, comaring he invenory osiions of he dual-balancing olicy and O T in eriod s does no seem o rovide sufficien informaion abou eriod s + L. For examle, consider a los-sales model wih L = 1, where in eriod he ieline vecor of olicy 1 is 1 = (3, 10) (i.e., on-hand invenory equal o 10 and an order of 3 unis laced in eriod ) and he ieline vecor of olicy 2 is 2 = (4, 1) (i.e., on-hand invenory equal o 1 and an order of 4 unis laced in eriod ). In eriod he invenory osiion of olicy 1 is y 1 = 13, higher han he invenory osiion of olicy 2, which is y 2 = 5. However, if he demand in eriod is greaer han 10, hen olicy 2 has greaer on-hand invenory in eriod + 1 (4 unis) han olicy 1, which is lef only wih 3 unis on-hand. Conversely, if he demand in eriod is no greaer han 9, hen olicy 1 has on-hand invenory in eriod + 1 no smaller han ha of olicy 2. The above examle suggess ha he eriod-by-eriod amorizaion scheme of he cos of he dualbalancing olicy agains he cos of O T, based on he invenory osiion as used in he backlogging analysis, does no seem o be useful when alied o he los-sales model. (In models wih backlogged demand if one olicy has a higher invenory osiion in eriod s, i will have higher on-hand invenory a lead ime ahead in eriod s + L.) To overcome his difficuly, he analysis resened below incororaes wo novel ideas. We use a global amorizaion of coss, ha is, we comare he overall cos of he dual-balancing olicy o ha of O T, where he comarison is no necessarily eriod-by-eriod. In addiion, we inroduce he new conce of runcaed invenory osiion, which is defined as follows. For each eriod s = 1,..., T, he runcaed invenory osiion wih resec o eriod (where [s L, s]), is defined o be he sum of he invenory on-hand in eriod s lus all ousanding orders laced by ime eriod. Le Y s denoe he runcaed invenory osiion in eriod s wih resec o eriod, ha is, Y s = I s + j=s+1 L Q j. (6) Observe ha he runcaed invenory osiion Y s refers o he sum of he on-hand invenory in eriod s lus all ousanding orders ha will arrive by ime eriod + L. Noe ha we consider a eriod earlier han s which imlies ha all he orders ha arrive by ime eriod + L are already known a ime eriod s. Secifically, Y ss = Y s is he radiional invenory osiion defined earlier in Secion 2, and Y s,s L = I s is he on-hand invenory a he beginning of eriod s. The runcaed invenory osiion is a generalizaion of he radiional invenory osiion conce commonly used in invenory heory (see Figure 1). Due o he comlex mahemaical srucure of los-sales models, he effec of he decision made in he curren eriod on fuure coss is very hard o quanify. The runcaed invenory osiion rovides a more racable way o analyze his effec; secifically, he effec of he curren ordering decision on he on-hand invenory a lead ime ahead. Moreover, i urns ou ha he conce of he runcaed invenory osiion rovides he righ framework for comaring beween he ieline vecors of any wo olicies; secifically, O T and he dual-balancing olicy. Thus, a cenral ar of he wors-case analysis resened below is based on his new conce. We believe ha i will have more alicaions in oher seings. The wors-case analysis in he model wih backlogged demand is based on comaring he (radiional) invenory osiion of he dual-balancing olicy and O T in each eriod, i.e., comaring Y B and Y O T. Insead, in he los-sales model, he analysis will be based on comaring he resecive runcaed invenory osiions Ys B and Ys O T in each eriod s [, +L]. Tha is, in each eriod s [, +L], we comare he resecive number of unis already ordered by he dual-balancing olicy and O T ha will be available by ime eriod + L (see Figure.2). Le T H be he se of all eriods T L such ha he runcaed invenory of he dual-balancing olicy wih resec o eriod is sricly smaller han he resecive runcaed invenory osiion of

9 : 9 q s q s 1 s, s y = y s s,s 2 y = i s y s,s 1 Figure 1: The runcaed invenory osiion in eriod s (L=2). O T, for each eriod s [, + L]. Tha is, T H = { T L : s [, + L], Y B s < Y O T s }. (7) Le T Π be he comlemen of T H, i.e., he se of eriods T L for which here exiss some s [, + L] where he runcaed invenory osiion of he dual-balancing olicy wih resec o eriod + L is no smaller han he resecive runcaed invenory osiion of O T. Tha is, T Π = { T L : s [, + L] wih Y B s Y O T s }. (8) Recall ha in he los-sales model, having a higher invenory osiion in eriod does no guaranee higher on-hand invenory in eriod + L. Moreover, for cerain realizaions of he demands over he inerval [, + L), he runcaed invenory osiion of he dual-balancing olicy wih resec o eriod migh be higher han he resecive runcaed invenory osiion of O T in some eriods and lower in ohers. In fac, i is ossible o observe an alernaing behavior, where he relaion beween he runcaed invenory osiion of he dual-balancing olicy and ha of O T may change several imes over he inerval. More recisely, for some eriod and s [, + L), we will say ha he resecive runcaed invenory osiion of he dual-balancing olicy wih resec o eriod and ha of O T alernae in eriod s if one of he following evens occur or [Ys B < Ys O T ] [Ys+1, B Ys+1, O T ], [Ys B Ys O T ] [Ys+1, B < Ys+1, O T ]. Tha is, in he wo consecuive eriods s and s + 1, he inequaliies relaing he runcaed invenory osiions of he dual-balancing olicy and ha of O T alernae. For each T H, we know ha in each eriod over he inerval [, +L], O T had (sricly) more unis available by ime eriod + L. In aricular, here is no alernaion in he resecive relaion beween he runcaed invenory osiion of he dual-balancing olicy wih resec o eriod and ha of O T over he inerval [, + L). On he oher hand, for each T Π, here was a leas one eriod over ha inerval when he dual-balancing olicy had unis available by ime eriod + L a leas as many as O T had. Noe ha his does no necessarily imly alernaions (e.g., when he runcaed invenory osiion of he dual-balancing olicy wih resec o eriod is higher in eriod and hroughou (, + L]), nor does i exclude more han one alernaion (i.e., i is ossible ha he resecive runcaed invenory osiion of O T and ha of he dual-balancing olicy will alernae several imes over [, + L)). Nex we sae and rove wo key lemmas ha will show how o amorize he cos of he dual-balancing olicy agains he cos of O T. The corresonding wo lemmas hold wih robabiliy 1, i.e., for each samle ah of he demands D 1,..., D T or equivalenly, for each f T F T. (In he saemens and roofs

10 10 : Key Order laced in his eriod Order laced 2 eriods earlier Order laced 1 eriod earlier Invenory on hand q + 3 q + 1 q + 2 q + 2 q q + 1 q + 1 q q 1 y, q 2 q 1 y + 1, i i + 1 q y + 2, i + 3 y + 3, i + 2 d = 2 +1 d + 1 = 8 +2 d + 2 = 0 +3 Figure 2: Evoluion of he runcaed invenory osiion wih resec o eriod over [, + L] (L = 3)

11 : 11 of hese lemmas we shall omi he exression wih robabiliy 1.) In he firs of hese lemmas we will show ha he overall holding cos incurred by O T, denoed by H O T is greaer han he holding coss incurred by unis ordered by he dual-balancing olicy in eriods T H. Lemma 3.2 The holding cos incurred by O T is greaer han he holding cos incurred in he dualbalancing olicy by unis ordered in eriods T H, i.e., H O T T H H B. roof. Recall ha by definiion Y+L, = I +L. However, his imlies ha, for each T H, we have I+L B < IO +L T, i.e., he on-hand invenory of O T in eriod +L is higher han ha of he dual-balancing olicy. We have already seen ha he on-hand invenory a he beginning of eriod + L, jus before he unis ordered in eriod have arrived, is equal o (I+L 1 D +L 1) +. In aricular, I B +L = Q B + (I B +L 1 D +L 1 ) + < I O T +L. Wihou loss of generaliy, we assume ha suly unis are consumed on a firs-ordered-firs-consumed basis. We can hen associae an index o each uni of suly currenly on-hand according o he number of unis on-hand o be consumed rior o ha uni (where unis ordered in he same eriod are sored arbirarily). Noe ha since we allow fracional orders, he suly unis are defined infiniesimally. In aricular, he Q B unis ordered by he dual-balancing olicy in eriod are indexed in eriod + L in he range ((I B +L 1 D +L 1 ) +, (I B +L 1 D +L 1 ) + + Q B s ]. (9) Since T H and he on-hand invenory of O T in eriod + L is higher, we conclude ha in eriod + L here exis suly unis on-hand in O T wih he same range of indices as in (9). We now mach airs of unis of suly wih he same resecive index (in eriod + L) in he dual-balancing olicy and O T, resecively. In aricular, in eriod +L we mach he suly unis ha are indexed in he above range in O T o he Q B unis ordered by he dual-balancing olicy in eriod (see also Figure 3). Observe ha unil he I+L B unis on-hand a he beginning of eriod + L will be consumed, neiher he dual-balancing olicy nor O T incur los-sales coss. Moreover, since he demands over [ + L, T ] are he same for O T and he dual-balancing olicy, i is clear ha each air of resecive mached suly unis of O T and he dual-balancing olicy will incur he same holding cos over [ + L, T ], for each samle ah of demands D +L,..., D T. Since each air of unis are consumed a he same ime eriod, i is readily verified ha each suly uni of O T can be mached o a mos one suly uni of he dual-balancing olicy. This concludes he roof. Noe ha he above roof sill holds for ime-deenden holding cos arameers and osiive nonincreasing er-uni ordering cos arameers, where he er-uni ordering cos is incororaed ino he marginal execed holding cos and is balanced agains he marginal execed los-sales enaly cos. In he second lemma, we amorize he los-sales enaly coss of he dual-balancing olicy which are associaed wih eriods T Π. In he roof of his lemma, we use a global amorizaion raher han a eriod-by-eriod one. For each T Π, we know ha here exiss some eriod s [, + L] such ha he runcaed invenory osiion of he dual-balancing olicy wih resec o eriod is no smaller han he one of O T, i.e., Ys B Ys O T. However, as we have already observed, his does no guaranee ha in eriod + L he invenory on-hand of he dual-balancing olicy is no smaller han he one of O T. Tha is, i is sill ossible o have I+L B < IO +L T, which imlies ha we can no amorize he los-sales enaly cos incurred by he dual-balancing olicy in eriod + L agains he resecive cos of O T in his eriod. The nex lemma shows ha in his case, eriod + L belongs o an inerval of eriods over which he los-sales enaly coss incurred by O T are higher han he resecive los-sales enaly coss incurred by he dual-balancing olicy. This leads o a global amorizaion of he cos of he dual-balancing olicy wih he cos of O T. Lemma 3.3 The los-sales enaly incurred by O T, denoed by Π O T, is greaer han he los-sales enaly coss of he dual-balancing olicy which are associaed wih eriods T Π, i.e., Π O T T Π Π B.

12 12 : Q OT ( I OT 1 ) 1 D + + L + L Mached Unis Q B ( I B 1 ) 1 D + + L + L I OT I B + L + L Figure 3: Mached suly unis in eriod + L where T H. roof. Consider he following random ariion of he eriods L + 1,..., T. For each realizaion of demands d 1,..., d T, consider he resuling realizaion of he se T Π, and ariion he eriods in he following way. Sar in eriod T and look for he laes eriod T Π wih he roery ha eriod + L is no marked (iniially all eriods are unmarked) and i B +L < io +L T (we abuse he noaion and use T Π o denoe he deerminisic se of eriods resuling from he realized demands). If no such exiss hen we erminae. If such a eriod exiss, le be ha eriod and le w be he earlies eriod in [, + L] for which he runcaed invenory osiion of he dual-balancing olicy wih resec o is no smaller han he resecive runcaed invenory osiion of O T. Tha is, w = min{j [, + L] : yj B yo j T } (observe ha w is he realizaion of a random variable, denoed by W, which is defined for each eriod T Π ). By our assumion does belong o T Π, hence w is indeed well-defined. We now mark all he eriods in [w, + L]. Nex we coninue recursively over he eriods 1,..., w 1. Tha is, we look for he laes w 1 such ha T Π and wih he roery ha + L is unmarked and i B +L < io +L T and reea he above. The above rocedure induces a random ariion of he eriods L+1,..., T ino marked and unmarked eriods, resecively. Le M be he (random) se of all marked eriods. In aricular, his random ariion induces a ariion of he se T Π ino eriods s T Π such ha s + L M, i.e., s + L is marked and eriods s T Π such ha s + L is no marked. Firs consider he laer se. For each eriod s T Π such ha s + L / M, we know ha Is+L B IO s+l T, for if no s + L would have been marked. This imlies ha for all eriods {s T Π : s + L / M}, we have Π B s Π O s T. Now consider all he eriods {s T Π : s + L M}. Since all marked inervals are disjoin, i is sufficien o show ha, for each marked inerval of he ye [W, + L], he overall los-sales enaly coss incurred by O T over ha inerval are higher han he resecive los-sales enaly coss incurred by he dual-balancing olicy over ha inerval. In aricular, his will imly ha he los-sales coss of he dual-balancing olicy associaed wih eriods in he se {s T Π : s + L / M} are lower han he

13 : 13 los-sales enaly coss incurred by O T in eriods which do no belong o M, and ha he los-sales enaly coss of he dual-balancing olicy associaed wih eriods in he se {s T Π : s + L M} are smaller han he los-sales coss incurred by O T in eriods ha belong o M (see Figure 3.4). The roof of he lemma will hen follow. ' ' T B OT i '' + L i '' + L Unmarked eriod ' T T y < y B s, OT s, y y B w, OT w, B OT i + L < i + L 1 '' ' '' +L ' +L s w +L T Marked inerval Marked inerval corresonding o corresonding o eriod ' eriod Figure 4: The ariion ino marked and unmarked eriods. Dashed arrow denoes he lead ime inerval saring a a cerain eriod. Solid arrow denoe a marked inerval. For each T Π wih + L unmarked (e.g., eriod ), we have i B +L io +L T. For each such inerval [W, + L], we know ha Y B W, +L Y O T W, +L, and IB +L < IO T +L. Moreover, he difference beween he los-sales enaly cos incurred by he dual-balancing olicy in eriod + L and he resecive los-sales enaly cos incurred by O T in ha eriod, is bounded by imes he difference beween he resecive on-hand invenory levels. Tha is, Π B ΠO T (IO T +L I B +L). (10) Nex we use he following ideniy ha is valid for every feasible olicy. For each, s such ha s + L, I +L = Y s D [s,+l) + Π [s L,), (11) where Π [s L,) is he cumulaive los-sales enaly coss over he inerval [s, + L), i.e., Π [s L,) = 1 j=s L Π j. Equaion (11) describes he dynamics of a model wih los sales. Secifically, he on-hand invenory in eriod + L is equal o he runcaed invenory osiion in ime eriod s wih resec o eriod minus he cumulaive demand over he inerval [s, + L) lus he cumulaive los sales over ha inerval. Observe ha D [s,+l) Π [s L,) is he number of suly unis consumed by he demand over he inerval [s, + L). Now consider Equaion (11) for eriods and W alied o O T and he dual-balancing olicy, resecively, and subsiue ino Equaion (10). We ge ha Π B ΠO T ΠO [W T L, ) ΠB O T [W, ) + (YW, Y B W, ) (12) Π O T [W L, ) ΠB [W L, ).

14 14 : The las inequaliy follows from he fac ha YW B, Π B = Π B + T Π : T Π, +L M This concludes he roof of he lemma. : T Π, +L M Π O T + Y W O T, : T Π, +L / M. We now ge ha : T Π, +L/ M Π B (13) Π O T Π O T. We noe ha Lemma 3.3 holds also in he case where here are ime-deenden los-sales enaly arameers 1,..., T, as long as hey are non-increasing. The roof is almos idenical, bu now w is defined o be he laes eriod j [, + L], such ha he cumulaive los sales of he dual-balancing over [j, + L] is no higher han he corresonding los sales of O T over ha inerval. (The roof of Lemma 3.3 imlies ha he newly defined w does exis.) This enables us o amorize he los sales incurred by he dual-balancing olicy, in each eriod such ha L T Π, wih los sales incurred by O T in eriods earlier han. (Secifically, for each eriod s [w, + L], we amorize he los sales of he dual-balancing in eriod s wih los sales of O T incurred in eriods [w, s].) In aricular, he lemma is valid in models wih discouned coss. Lemmas 3.2 and 3.3 imly ha H O T + Π O T H B + Π B. T H T Π Taking execaion we ge ha E[C(O T )] E[ ( 1( T H ) H B + 1( T Π ) Π B )]. (14) However, as we have already seen, in he los-sales model he runcaed invenory osiions of he dualbalancing olicy and O T wih resec o eriod can alernae over he inerval [, + L) from higher o lower. Thus, unlike he analysis of model wih backlogged demand [5], condiioning on some f F does no necessarily realize he indicaors 1( T H ) and 1( T Π ) above. Tha is, i is ossible ha in eriod we sill do no know wheher T H or T Π. Insead, we will condiion on he evens [ T H ] and [ T Π ], resecively, and ge ha E[C(O T )] = E[E[ 1( T H ) H B F ] + E[ 1( T Π ) Π B F ]] (15) E[ r( T H F ) E[H B (F, T H )] + r( T Π F ) E[Π B (F, T Π )]]. However, by condiioning on [ T H ] and [ T Π ], resecively, we consider informaion ha suersedes he original informaion se f F based on which he dual-balancing olicy has made he ordering decision a he beginning of eriod. Tha is, E[H B (F, T H )] and E[Π B (F, T Π )] migh no be equal o E[H B F ] = E[Π B F ] = Z. In aricular, he roblem arises for informaion ses f F for and he condiional robabiliies (condiioning on f ) ha [ T H ] and [ T Π ] are boh osiive (If his is no he case, hen we know wheher T H or T Π already a he beginning of eriod while observing f.) which y B < y O T Nex we will show ha if he demands in differen eriods are indeenden of each oher, hen he wo inequaliies and r( T H F ) E[H B (F, T H )] r( T H F ) E[H B F ] (16) r( T Π F ) E[Π B (F, T Π ))] r( T Π F ) E[Π B F ] (17) hold wih robabiliy 1, and his ogeher wih Equaion (15) and he fac ha Z = E[H B F ] = E[Π B F ] imly ha he dual-balancing has a wors-case erformance guaranee of 2. Equaions (16) and (17) imly ha condiioning also on he evens [ T H ] and [ T Π ], resecively, imlies ha

15 : 15 he resecive execed coss are even higher han wha was execed a eriod condiioning only on f. Noe ha his is he only ar of he analysis ha requires addiional assumions on he demand disribuions (beyond having finie mean). In Secion 5, we shall generalize he analysis, and show ha Inequaliies (16) and (17) hold under several oher demand srucures ha incororae correlaion beween demands in differen eriods. Inuiively, we require ha he demands do no have a cerain bad roery. Tha is, we would like o exclude a siuaion where high demands over a cerain inerval of eriods, say (j, j), imly low demands over he res of he horizon [j, T ]. Indeed, if he demands are indeenden, his bad siuaion is excluded. For each eriod = 1,..., T L and s [, + L), le A s be he even ha a he beginning of eriod s he runcaed invenory osiion of O T wih resec o eriod is higher han he one of he dual-balancing olicy, while a he beginning of eriod s + 1 he runcaed invenory osiion of he dual-balancing olicy wih resec o eriod is no smaller han he one of O T. Tha is, A s = [Ys B < Ys O T ] [Ys+1, B Ys+1, O T ]. Observe ha condiioning on an informaion se f such ha y B < y O T, hen T Π only if he even A j occurs for some j [, + L). In he nex lemma we characerize some of he roeries of he even A s defined above. Lemma 3.4 For each eriod = 1,..., T L and s [, + L), le A s be as defined above. Suose ha he even A s occurred. Then, (i) The cumulaive amoun of orders laced by he dual-balancing olicy over he inerval [s + 1 L, ] is higher han he corresonding amoun of orders of O T over ha inerval, i.e., j=s+1 L QB j j=s+1 L QO j T. (ii) The invenory on-hand of O T a he beginning of eriod s exceeds ha of he dual-balancing olicy by more han Q s = j=s+1 L QB j j=s+1 L QO j T, i.e., Is O T > Is B + Q s. (iii) The even A s can be exressed as [Ys B < Ys O T ] [ Q s 0] [D s > Is O T Q s ]. (iv) The dual-balancing olicy has incurred osiive los sales in eriod s, and hence is on-hand invenory a he beginning of eriod s+1 is equal o he size of he order laced in eriod s+1 L, denoed by Q B s+1 L. Tha is, A s [Π B s L > 0] [IB s+1 = Q B s+1 L ]. roof. Recall Equaion (6) ha, for each olicy, we have Ys = Is + j=s+1 L Q j. Assume ha (i) does no hold, i.e., ha A s has occurred and ha j=s+1 L QB j < j=s+1 L QO j T. Since Ys B < Ys O T, we conclude ha Is B Is O T < j=s+1 L QO j T j=s+1 L QB j. However, i is readily verified ha his imlies ha he invenory on-hand of he dual-balancing olicy a he end of eriod s, (I s B D s ) +, does no exceed he resecive invenory on-hand of O T, (Is O T D s ) +, by more han j=s+1 L QO j T j=s+1 L QB j. Tha is, (I B s D s ) + (I O T s D s ) + < j=s+1 L Q O T j j=s+1 L which imlies ha Y B s+1, < Y O T s+1, and leads o a conradicion. The roof of (i) hen follows. The roof of (ii) follows from (i) and he fac ha Y B s < Y O T s. I is now clear ha given (i) and (ii) above, he even A s is equivalen o he even [D s > Is O T Q s ], which imlies (iii). Finally, (ii) and (iii) imly (iv). In he nex wo lemmas we show ha if he demands are indeenden of each oher, hen he Inequaliies (16) and (17) do hold. (We again omi he saemen wih robabiliy 1 as long as he conex is clear.) Lemma 3.5 Assume ha D 1,..., D T are indeenden of each oher. Then for each eriod = 1,..., T L, we have r( T H F ) E[H B F ] r( T H F ) E[H B (F, T H )]. Q B j,