Approximation algorithms for capacitated stochastic inventory systems with setup costs

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1 Approximaion algorihms for capaciaed sochasic invenory sysems wih seup coss The MIT Faculy has made his aricle openly available. Please share how his access benefis you. Your sory maers. Ciaion As Published Publisher Shi, Cong; Zhang, Huanan; Chao, Xiuli and Levi, Resef. Approximaion Algorihms for Capaciaed Sochasic Invenory Sysems wih Seup Coss. Naval Research Logisics (NRL) 61, no. 4 (April 2014): Wiley Periodicals, Inc. hp://dx.doi.org/ /nav Wiley Blackwell Version Auhor's final manuscrip Accessed Mon Aug 13 14:23:48 EDT 2018 Ciable Link hp://hdl.handle.ne/1721.1/ Terms of Use Creaive Commons Aribuion-Noncommercial-Share Alike Deailed Terms hp://creaivecommons.org/licenses/by-nc-sa/4.0/

2 Approximaion Algorihms for Capaciaed Sochasic Invenory Sysems wih Seup Coss Cong Shi, Huanan Zhang, Xiuli Chao, Resef Levi Indusrial and Operaions Engineering, Universiy of Michigan, Ann Arbor, MI shicong, zhanghn, Sloan School of Managemen, MIT, Cambridge, MA Absrac We develop he firs approximaion algorihm wih wors-case performance guaranee for capaciaed sochasic periodic-review invenory sysems wih seup coss. The srucure of he opimal conrol policy for such sysems is exremely complicaed, and indeed, only some parial characerizaion is available. Thus, finding provably near-opimal conrol policies has been an open challenge. In his paper we consruc compuaionally efficien approximae opimal policies for hese sysems whose demands can be nonsaionary and/or correlaed over ime, and show ha hese policies have a wors-case performance guaranee of 4. We demonsrae hrough exensive numerical sudies ha he policies empirically perform well, and hey are significanly beer han he heoreical wors-case guaranees. We also exend he analyses and resuls o he case wih bach ordering consrains, where he order size has o be an ineger muliple of a base load. Key words: invenory, seup cos, capaciy, approximaion algorihms, bounds, randomized cos-balancing, wors-case performance guaranees Hisory: Received Sepember 2013; revision received March 2014; acceped March Inroducion In his paper we sudy capaciaed sochasic periodic-review invenory sysems wih seup coss. The demand process may be nonsaionary (ime-dependen) and correlaed over ime, capuring demand seasonaliies and forecas updaes. These sysems are fundamenal bu nooriously hard o analyze in boh heory and compuaion. If he ordering capaciy in each period is infiniy, i is well-known ha sae-dependen (s, S) ype of policies are opimal for invenory sysems wih seup coss under independen demand processes. This srucure for opimal policies also holds rue for exogenous Markov-modulaed demands (e.g., Cheng and Sehi [7]) and models wih advance demand informaion (e.g., Gallego and Özer [9]). One migh expec ha some form of modified (s, S) policies is opimal for he capaciaed case, bu all sudies have rejeced he conjecure. In fac, even when he demands in differen periods are independen and idenically disribued, he srucure of he opimal conrol policies is very 1

3 complicaed and only some parial characerizaion is available in he lieraure. Thus, he design and compuaion of a provably near-opimal conrol policy have been an open challenge. I should be noed ha compuing he opimal conrol policy using dynamic programming may no be possible due o he curse-of-dimensionaliy, i.e., he need o keep rack of a sae variable of large dimension. For example, he demand process for our model may be nonsaionary, driven by he sae-of-economy or sae-of-he-world (e.g., he Markov modulaed demand process), or i may be a forecas-relaed demand process such as he Maringale Model of Forecas Evoluion (MMFE, see for example, Heah and Jackson [14]) in which he updaed forecas (as well as he realizaion of he supply capaciy in he nex period) is he original forecas plus a random error wih mean zero (see e.g. Lu e al. [22]). In hese scenarios, he demand srucure leads o a muli-dimensional sochasic dynamic program and compuing he opimal policies is usually inracable. 1.1 Main resuls and conribuions of his paper The major resuls and conribuions of his paper are summarized as follows. We also poin ou he major disincion of our proposed algorihms from previous work, in paricular, Levi e al. [20] and Levi and Shi [21]. Algorihms and heir wors-case analysis. We develop he firs approximaion algorihms for capaciaed sochasic periodic-review invenory sysems wih seup coss under a correlaed, nonsaionary and evolving sochasic demand srucure. The policy proposed will be referred o as a randomized 1/2-balancing policy (R/2). We show ha he proposed policies admi a consan wors-case performance guaranee of 4, regardless of any specific demand insance or inpu parameers. Noe ha his consan wors-case performance guaranee does no scale wih he sysem size or he lengh of he planning horizon or he inpu parameers. Since he srucure of opimal policies for hese sysems is no well undersood, he proposed invenory conrol policies provide valuable insighs ino how various cos componens should be balanced. As menioned in our lieraure review below, Levi and Shi [21] developed a 3-approximaion algorihm for he uncapaciaed model wih seup coss using an exac randomized balancing (i.e., exacly balance he marginal holding cos, he forced backlogging cos and he seup cos), and Levi e al. [20] proposed a concep of forced backlogging cos accouning for he capaciaed model wihou seup coss. However, exac balancing is no achievable in he presence of boh capaciy consrains and he seup cos. The main source of difficuly lies in he fac ha he policy may no be able o order a specific quaniy ha makes he marginal holding cos equal o he seup cos, since his paricular quaniy may exceed he ordering capaciy; in such cases, he policy has o runcae an order a he capaciy level. The approach employed in Levi and Shi [21] fails o work in his case. Insead of exac balancing, our proposed R/2 policy almos balances he marginal holding or forced backlogging cos wih half of he seup cos. We provide a unified and much simpler analysis of Levi and Shi [21] in he well-behaved cases, and a novel analysis in he ill-behaved cases. We also exend our resuls o capaciaed model wih seup cos under bach order consrains. Wih he bach order consrain, each order quaniy has o be an ineger muliple of a pre-specified base load, e.g., a ruck-load. We refer ineresed readers o Veino [25], Chao and Zhou [3], Chen 2

4 [4] and Huh and Janakiraman [16] for deails concerning bach orders on he case wih infinie ordering capaciy. We propose a modified randomized 1/2-balancing policy and show ha he wors-case performance guaranee of he proposed policy is sill 4. Empirical performance. We show how hese policies can be parameerized o creae a broader class of policies. We demonsrae hrough exensive compuaional sudies ha he proposed algorihms perform well in an empirical sudy (around 5% 15% from he opimal cos), which is significanly beer han he heoreical wors-case performance guaranees. The proposed invenory conrol policies are compuaionally efficien wih a compuaional complexiy of O(T 2 ) where T is he lengh of he planning horizon, which is very efficien compared o he dynamic programming approach ha suffers from he well-known curse of dimensionaliy. 1.2 Lieraure review Sochasic periodic-review invenory sysems have araced he aenion of many researchers over he years. The dominan paradigm in he exising lieraure has been o formulae and analyze hese problems using dynamic programming. For many uncapaciaed invenory sysems wih seup coss, i can be shown ha some form of (s,s) policies are opimal (see, e.g., Scarf [24], Veino [27]). Cheng and Sehi [7] have exended he opimaliy proof o exogenous Markov-modulaed demands ha capure cycles and seasonaliy o some exen. Gallego and Özer [9] have esablished heir opimaliy for models under advance demand informaion, a demand model ha allows correlaion and forecas updaes. Myopic policies seem o perform well for some scenarios in uncapaciaed sysems and are even opimal in some specific seings (see Veino [26], Ignall and Veino [17] and Iida and Zipkin [18]). However, capaciaed problems are inherenly harder, srucurally and compuaionally, compared o heir uncapaciaed counerpars. The capaciy consrain makes fuure coss heavily dependen on curren decisions. Chen and Lambrech [6] demonsraed ha he opimal policy for capaciaed invenory sysems wih seup coss exhibis an X Y band srucure, wih X < Y. Tha is, if he invenory level is below X, order he full capaciy, and if he invenory level is over Y, order nohing; if he invenory level is beween X and Y, however, he ordering policy is complicaed and no known. Gallego and Scheller-Wolf [10] and Chen [5] provided some furher refinemens o his policy, bu again, he opimal conrol policy remains complicaed and can only be parially characerized when he invenory level a he beginning of a period is in he middle range. For example, in Gallego and Scheller-Wolf [10], i was shown ha he region beween he X Y bands can be furher divided ino wo subregions. In one of hem, i is opimal o eiher order nohing or o bring he invenory level o a leas some specified level, ha is, here exiss a lower bound for he opimal order up-o level in his range; in he oher subregion, he parameers of he soluion dicae which one of he wo cases hold: In he firs case i is opimal o order, again o a leas some specified level (hus only a lower bound is shown o exis), and in he second, he opimal policy is o eiher order he full capaciy or order nohing. Özer and Wei [23] sudied capaciaed invenory sysems wih advance demand informaion. They esablished he opimaliy of a sae-dependen modified base-sock policy for invenory sysems wih zero fixed ordering cos and for he sysems wih posiive fixed coss, hey resriced he ordering o he class of all-or-nohing policies and characerized he opimal policies wihin ha class. For models wih infinie ordering capaciies and independen and idenically disribued de- 3

5 mands, Federgruen and Zipkin [8] proposed an algorihm o compue he opimal (s, S) policy in an infinie horizon model. Bollapragada and Moron [2] proposed a simple, myopic heurisic for compuing he policies where he demands in differen periods are assumed o have he same form of disribuion funcion bu wih differen means, and he coefficien of variaion of he demands is assumed o be saionary. Gavirneni [11] designed a simple heurisic o compue (s, S) policies for nonsaionary and capaciaed model. Guan and Miller [13] proposed an exac and polynomial-ime algorihm for he uncapaciaed sochasic periodic-review invenory sysem wihou backlogging if he sochasic programming scenario ree is polynomially represenable. Guan and Miller [12] exended hese algorihms o allow for backlogging. Huang and Küçükyavuz [15] considered similar problems bu wih random lead imes. These models allow for sochasic and correlaed demands. The main limiaion comes from he fac ha he number of nodes in he sochasic programming scenario ree (he size of inpu) is likely o be exponenially large in he size of he planning horizon. Aali and Özer [1] proposed a close-o-opimal heurisic o manage a muli-iem wo-sage producion sysem subjec o Markov-modulaed demands and producion quaniy requiremens. All of he exising heurisics and algorihms, eiher lack any performance guaranees or can only be applied under resricive assumpions on he demand processes or he inpu size, and o he bes of our knowledge, no efficien compuaional policies have been repored for capaciaed models ha admi wors-case performance guaranees. Our work is closely relaed o he recen lieraure on approximaion algorihms in sochasic periodic-review invenory sysems, firs sared by Levi e al. [19]. Levi e al. [19] inroduced he concep of marginal cos accouning ha associaes he full planning horizon cos wih each decision a paricular policy makes. They proposed a dual-balancing policy ha admis a worscase performance guaranee of 2 for he uncapaciaed model wihou seup coss. Subsequenly, Levi e al. [20] inroduced he forced marginal backlogging cos-accouning scheme o analyze he capaciaed models wihou seup coss, and Levi and Shi [21] proposed he randomized cosbalancing policy o solve uncapaciaed sochasic lo-sizing problems wih seup coss. I is worhy o noe ha he sysems sudied in hese papers all have nice simple srucures for heir opimal conrol policies. However and as discussed above, he srucure of he opimal conrol policies for capaciaed sochasic invenory models wih seup coss is complicaed and has no been fully characerized; and designing an approximaion algorihm for he capaciaed sochasic invenory models wih seup coss remained a challenging ask. 1.3 Srucure of his paper The remainder of he paper is organized as follows. In Secion 2, we presen he mahemaical model for he capaciaed sochasic periodic-review invenory sysem wih seup cos. Secion 3 reviews he marginal cos accouning scheme proposed by [20]. More specifically, we presen he marginal holding cos accouning scheme in Secion 3.1 and he forced backlogging cos accouning scheme in Secion 3.2. In Secion 4, we propose a novel randomized 1/2-balancing policy and discuss he key ideas. Then we show ha he policy has a wors-case performance guaranee of 4 in Secion 5. In Secion 6 we exend our resuls o sysems wih bach order consrains. Finally, Secion 7 is devoed o he numerical sudies for our proposed policies. The parameerized policies are compuaionally efficien and perform well under a correlaed demand srucure wih advance demand informaion (see, e.g., Gallego and Özer [9] and Özer and Wei [23]). 4

6 2 Capaciaed Periodic-Review Invenory Sysem wih Seup Coss In his secion, we provide he mahemaical formulaion of he capaciaed periodic-review invenory sysem wih seup cos. Our model allows for nonsaionary and generally correlaed demand srucure. The ordering capaciy in each period is denoed by u. The planning horizon is T periods which can be eiher finie or infiniy, and we index he period by = 1,..., T. Demand srucure. The demands D 1,..., D T over he planning horizon T are random. A he beginning of each period s, we are given wha we call an informaion se denoed by f s. The informaion se f s conains all of he informaion ha is available a he beginning of ime period s. More specifically, he informaion se f s consiss of he realized demands d 1,..., d s 1 over he inerval [1, s), and possibly some exogenous informaion denoed by (w 1,..., w s ). The informaion se f s in period s is one specific realizaion in he se of all possible realizaions of he random vecor F s = (D 1,..., D s 1, W 1,..., W s ). The se of all possible realizaions is denoed by F s. Wih he informaion se f s, he condiional join disribuion of he fuure demands (D s,..., D T ) is known. The only assumpion on he demands is ha for each s = 1,..., T, and each f s F s, he condiional expecaion E[D f s ] is well-defined and finie for each period s. In paricular, we allow for non-saionariy and correlaion beween he demands in differen periods. Cos srucure. In each period, = 1,..., T, four ypes of coss are incurred, a per-uni ordering cos c for ordering any number of unis a he beginning of period, a per-uni holding cos h for holding excess invenory from period o + 1, a per-uni backlogging penaly b ha is incurred for each unsaisfied uni of demand a he end of period, and a seup cos K ha is incurred in each period wih sricly posiive ordering quaniy. Unsaisfied unis of demand are usually called backorders. Each uni of unsaisfied demand incurs a per-uni backlogging penaly cos b in each period unil i is saisfied. In addiion, we consider a model wih a lead ime of L 0 periods beween he ime an order is placed and he ime a which i acually arrives. We remark ha he analysis and resuls remain rue when he seup cos K depends on period as long as K K +1 is saisfied for all. We assume wihou loss of generaliy ha he discoun facor α = 1, and ha c = 0 and h, b 0, for each (see he discussion in [19]). Sysem dynamics. The goal is o coordinae a sequence of orders ha minimizes he overall expeced seup cos, holding cos and backlogging cos. More specifically, in each period, = 1,..., T, we place an order of Q [0, u] unis. Given a feasible policy P L, he dynamics of he sysem are described using he following noaion. Le NI denoe he ne invenory a he end of period. Thus, NI + and NI are invenory on hand and backlog quaniies in period, respecively, where for any real number x, we le x + = maxx, 0}. Since here is a lead ime of L periods, one also considers he invenory posiion of he sysem, which is he sum of all ousanding orders plus he curren ne invenory. Le X be he invenory posiion a he beginning of period before he order in period is placed, i.e., X = NI j= L Q j, and Q j [0, u] denoes he number of unis ordered in period j. Similarly, le Y be he invenory posiion afer he order in period is placed, i.e., Y = X + Q. Noe ha for every possible policy P L, once he informaion se f F is known and order Q is placed, he values ni 1, x and y are known, where hese are he realizaions of NI 1, X and Y, respecively. A he end of each period, he coss incurred are holding cos h NI + and backlogging cos b NI. In addiion, if he order quaniy Q > 0, hen he fixed ordering cos K is incurred. Thus, he oal cos of a feasible policy P L is 5

7 C (P L) = ( h NI +,P L =1 + b NI,P L ) + K 1(Q P L > 0), (1) where 1(A) is he indicaor funcion aking value 1 if saemen A is rue and 0 oherwise. The objecive is o find he opimal ordering decisions Q P L, based on informaion f, = 1,..., T, ha minimizes he oal cos (1). 3 Marginal Cos Accouning Scheme The cos accouning scheme described in (1) above decomposes he cos by periods. Following Levi e al. [19] and [20], we nex describe an alernaive cos accouning scheme ha is called marginal cos accouning scheme. The main idea underlying his approach is o decompose he cos by decisions. Tha is, he decision in period is associaed wih all coss ha, afer ha decision is made, become unaffeced by any fuure decisions, and are only affeced by fuure demands. This may include coss in all subsequen periods. 3.1 Marginal holding cos accouning Le D [s,] denoe he cumulaive demand over he inerval [s, ], i.e., D [s,] = j=s D j. We firs focus on he holding coss and assume, wihou loss of generaliy, ha unis in invenory are consumed on a firs-ordered firs-consumed basis. This implies ha he overall holding cos of he q s unis ordered in period s (i.e., he holding cos hey incur over he enire horizon [s, T ]) is a funcion only of fuure demands, and is unaffeced by any fuure decision. Specifically, based on he assumpion ha invenory is consumed on a firs-ordered firs-consumed basis, he q s unis on order will be used o saisfy demand only when he x s unis presenly in he sysem have been compleely consumed. Among hese q s unis, he number of hose sill remaining in invenory a he end of period j (where j s + L) is precisely ( q s (D [s,j] x s ) +) +. Thus, he oal marginal holding cos associaed wih he decision o order q s unis in period s is, recall ha he discoun facor α = 1, defined o be T j=s+l h ( j qs (D [s,j] x s ) +) +. Noe ha a he ime he order qs is placed, he invenory posiion x s is already known and indeed he marginal holding cos is jus a funcion of fuure demands. In addiion, once he order ( in period s is deermined, he backlogging cos a lead ime ahead in period s + L, i.e., b s+l D[s,s+L] (x s + q s ) ) +, is also affeced only by he fuure demands. This leads o a marginal cos accouning scheme. For each feasible policy P L, le Hs P L be he holding cos incurred by he Q P s L period s, for s = 1,..., T L, over he inerval [s, T ]. Then, H P L s = Hs P L (Q P s L ) = j=s+l unis ordered in h j ( Q P L s (D [s,j] X s ) +) +. (2) I is readily verified ha when we sum up he marginal holding coss of all uni ordered, we would obain he oal holding cos for all he periods. Tha is, =1 L h NI +P L = H (,0] + H P L, (3) 6 =1

8 where H (,0] denoes he oal holding cos incurred by unis ordered before period 1, which is independen of he ordering decisions during he planning horizon [1, T ]. 3.2 Forced backlogging cos accouning In capaciaed models, i is no longer rue ha a misake of ordering oo lile in he curren period can always be fixed by decisions made in he fuure periods. Levi e al. [20] proposed a new backlogging cos accouning ha associaes wih decision of how many unis o order in period wha is called forced backlogging cos resuling from his decision in fuure periods. Consider some period s. Suppose ha x s is he invenory posiion a he beginning of period s and ha he number of unis ordered in period is q s u. Le q s be he resuling unused slack capaciy in period, i.e., q s = u q s 0. Focus now on some fuure period s + L when his order arrives and becomes available. Suppose ha for some realizaion of he demands, we have ha d [s,] (x s + q s + ( s L)u) > 0. (4) This implies ha here exiss a shorage in period, and moreover, even if in each period afer period s and unil period L he orders placed were up o he maximum available capaciy, his par of he shorage in period would sill exis and incur he corresponding backlogging cos. The acual shorage may be even higher han (4) and is equal o d [s,] (x s + q s + q j ) > 0, j (s, L] (recall ha q j u for each period j). In oher words, given our decision in period s, his par of he shorage could no be avoided by any decision made over he inerval (s, L] (clearly, any order placed afer period L will no be available by ime ). We conclude ha, if more unis had been ordered in period s, hen a leas some of he shorage in period could have been avoided. More precisely, he maximum number of unis of shorage ha could have been avoided by ordering more unis in period s is equal o min q s, [ ] } + d [s,] (x s + q s + ( s L)u). The inuiion is ha by ordering more unis in period s, we could have avered par of he shorage in period, bu clearly no more han he unused slack capaciy q s, since we could no have ordered in period s more han addiional q s unis. In his case, we would say ha his par of he backlogging cos in period was forced by he decision in period s. Denoe W s, as he backlogging cos in period associaed wih decision made in period s. Then we can wrie (D[s,] W s = b min (X s + Q s + ( s L)u) ) } +, (u Qs ). This is significanly differen from he radiional backlogging cos accouning, in which his cos would be associaed wih period L. Since he decision a period s could affec all succeeding period s backlogging cos, hen he forced backlogging coss ha are incurred by any feasible policy P L in a period s is given by Π P L s = =s+l 7 W P L s. (5)

9 I is, again, readily verified ha he summaion of forced backlogging cos in all periods is equal o he oal backlogging cos. Tha is, =1 b NI P L = Π L (,0] + =1 Π P L, (6) where Π (,0] denoes all he forced backlogging coss of he ordering decisions made before period 1, which is independen of he policy used. 3.3 Toal cos of any feasible policy Le C (P L) be he oal cos incurred by using he conrol policy P L. By (3) and (6), we can rewrie C (P L) as C (P L) = = ( h NI +P L =1 T L ( =1 K 1(Q P L + b NI P L > 0) + H P L ) + K 1(Q P L > 0) + Π ) P L + H (,0] + Π (,0]. Since H (,0] and Π (,0] are consans ha are no affeced by he policy used, we will ignore hem in he subsequen analysis and wrie he effecive cos of a policy P L as L( C(P L) = K 1(Q P L =1 > 0) + H P L + Π ) P L. Clearly, o compare he performances of differen policies, i suffices o compare heir corresponding effecive coss. 4 The Randomized 1/2-Balancing (R/2) Policy In his secion, we propose a policy called randomized 1/2-balancing policy (R/2, or R-half policy) which aims o srike a balance beween hree ypes of coss, namely, he marginal holding cos, he forced backlogging cos, and he seup cos. There are wo sources of difficulies in designing cos-balancing algorihms for capaciaed sochasic periodic review invenory sysems wih seup coss. The firs one is ha we are unable o perfecly balance he hree ypes of coss menioned above. For insance, we may no be able o order he quaniy ha brings he marginal holding cos up o he seup cos K, since he paricular quaniy can exceed he capaciy consrain u. In hese cases, he balancing policy has o place a runcaed order a he full capaciy. This creaes difficulies in analyzing he performance bounds of he policy since i is no so clear which cos componen of he opimal policy can pay for a consan fracion of he cos incurred by he balancing policy. The second source of difficuly is he need o balance he nonlinear seup cos agains he forced backlogging cos ha may have large spikes because of he variabiliy of he demands. Thus, he balancing policy needs o employ 8

10 a randomized decision rules o make he expeced seup cos incurred in each period a coninuous funcion, raher han an indicaor funcion K if an order is placed and 0 oherwise. However, he randomized decision rules also inroduce uncerainies in he relaionships beween he ending invenory posiion of he opimal policy and he balancing policy. In some periods, i is no a-priori clear how o use he cos of he opimal policy o pay for ha of a balancing policy. To describe he new policy, we modify he definiion of he informaion se f o also include he randomized decisions of he randomized balancing policy up o period 1. Thus, given he informaion se f, he invenory posiion x a he beginning of period is known. However, he order quaniy in period is sill unknown because he policy randomizes among various order quaniies. 4.1 Compuing auxiliary balancing quaniies and coss A he beginning of each period wih he realized informaion se f s, we can efficienly compue he following auxiliary ordering quaniies and coss, since he marginal holding cos H( ) and he forced backlogging cos Π( ) are given in (2) and (5) in closed forms. Firs, compue he balancing quaniy ˆq and he balancing cos θ such ha θ E[H R/2 (ˆq ) f ] = E[ Π R/2 (ˆq ) f ]. The balancing quaniy perfecly balances he condiional expeced marginal holding cos agains R/2 he condiional expeced forced backlogging cos associaed wih he order ˆq. Since Π (u) = 0, i follows ha ˆq u. Noe ha H ( ) is convex and increasing on [0, ) and Π( ) is convex and decreasing o 0. Thus, ˆq always exiss and can be compued efficienly via bi-secion search. Then, compue he holding-cos-k/2 quaniy q ha solves E[H R/2 ( q ) f ] = K 2. The holding-cos-k/2 quaniy makes he condiional expeced marginal holding cos equal o K/2, and i is well-defined since H ( ) is convex and increasing on [0, ). A cavea is ha ordering q may no be feasible due o he capaciy consrain u in each period. More specifically, if E[H R/2 (u) f ] < K/2, hen he quaniy q exceeds he capaciy u and herefore canno be ordered in full amoun. I is naural o consider he order quaniy min q, u} which runcaes he holdingcos-k/2 quaniy a u. Thirdly, we compue he condiional expeced forced backlogging cos φ if one orders he minimum of q and he capaciy u in period. Tha is, φ E[ Π R/2 (min q, u}) f ]. And finally, we compue he condiional expeced forced backlogging cos ψ resuling from no ordering anyhing in period. Tha is, ψ E[ Π R/2 (0) f ]. 4.2 Descripion of he R/2 policy Using he quaniies compued above, we propose he following procedure for a randomized policy for period. (i) If he balancing cos exceeds K/2, i.e., θ K/2, hen he R/2 policy orders he balancing quaniy ˆq wih probabiliy p = 1; 9

11 (ii) if he balancing cos is less han K/2, i.e., θ < K/2, hen he R/2 policy orders he runcaed holding-cos-k/2 quaniy min q, u} wih probabiliy p and order nohing wih probabiliy 1 p. The probabiliy p is compued by he following equaion, I follows from (7) ha p K 2 = p φ + (1 p )ψ. (7) p = ψ K/2 φ + ψ. We argue ha 0 p < 1. This is because, in (ii) i holds ha E[H R/2 (ˆq ) f ] = θ < K/2 = E[H R/2 ( q ) f ], hence we mus have q > ˆq. In addiion, u ˆq by he consrucion of ˆq. Thus, ˆq min q, u}, which implies ha φ θ < K/2. orders In summary, we denoe he order quaniy of he R/2 policy by q R/2 q R/2 = ˆq, wih probabiliy p = 1 in case (i), min q, u}, wih probabiliy p in case (ii), 0, wih probabiliy 1 p in case (ii),. Then he R/2 policy where p in case (ii) is given by (7). The R/2 policy is depiced in Figure 1. This concludes he descripion of he R/2 policy. Noe ha p is a-priori random and is realized wih he informaion se f F. Following he convenion we use P o denoe his a-priori random probabiliy. Similarly, we use Q R/2 o represen he random a-priori ordering quaniy in period. 4.3 Key ideas of he R/2 policy In he nex secion, we shall show ha he R/2 policy described above has an expeced wors-case performance guaranee of 4. Here we firs provide he inuiion and keys ideas underlying his policy. When he balancing cos θ exceeds K/2, we have K 2θ, implying ha he seup cos K is smaller han he he oal expeced marginal holding and forced backlogging coss in period. The seup cos in his case is a less dominan facor. Moreover, if he R/2 policy does no place an order, he condiional expeced forced backlogging cos is poenially very large. Thus, i is worhwhile o order he balancing quaniy q R/2 = ˆq wih probabiliy 1. When he balancing cos θ is below K/2, he seup cos K becomes more dominan, and herefore i is no advisable o order wih probabiliy 1. I is naural o aemp o perfecly balance he hree ypes of he coss, namely, marginal holding, forced backlogging and seup coss. Due o he ordering capaciy consrain u, he opimal balancing raio is no longer 1 : 1 : 1 for each ype of he coss. Inuiively, we wan o increase our frequencies of ordering, keeping he sum of he marginal holding and forced backlogging equal o he seup coss. In paricular, since we order he runcaed holding-cos-k/2 10

12 Cos Cos Cos forced backlogging cos marginal holding cos ψ forced backlogging cos marginal holding cos K 2 ψ forced backlogging cos marginal holding cos K 2 θ K 2 0 balancing quaniy capaciy (u) θ 0 balancing quaniy φ capaciy (u) K holding-cos- 2 quaniy θ 0 balancing quaniy φ capaciy (u) Figure 1: A graphical depicion of how he R/2 policy orders in he following hree scenarios: (1) when he balancing cos exceeds K/2, he policy orders he balancing quaniy; (2) when he balancing cos is below K/2 and he holding-cos-k/2 quaniy is below he full capaciy, order he holding-cos-k/2 quaniy wih probabiliy p and nohing wih probabiliy 1 p ; (3) when he balancing cos is below K/2 and he holding-cos-k/2 quaniy exceeds he full capaciy, order he full capaciy wih probabiliy p and nohing wih probabiliy 1 p. Noe ha p is compued from equaion (7). quaniy min q, u} wih probabiliy p and nohing wih probabiliy 1 p, he condiional expeced marginal holding cos in his case is E[H R/2 (q R/2 ) f ] = p E[H R/2 (min q, u}) f ] + (1 p ) E[H R/2 (0) f ] p E[H R/2 ( q ) f ] + (1 p ) E[H R/2 (0) f ] = p K/2. By he consrucion of he ordering probabiliy p in (7), he condiional expeced forced backlogging cos is E[ Π R/2 (q R/2 ) f ] = p E[ Π R/2 (min q, u}) f ] + (1 p ) E[ Π R/2 (0) f ] = p φ + (1 p )ψ = p K/2. Finally, since p is he ordering probabiliy, he expeced seup cos is p K, which is wice of p K/2. I follows ha his randomized decision rule almos balances in a parameerized way, up o he capaciy consrain, he hree ypes of coss associaed wih he period. The balancing raio is 1 : 1 : 2 for he marginal holding, he forced backlogging and he seup coss. Remark In a way, he balancing randomized R/2 policy we employed for marginal holding, he forced backlogging and seup coss is opimal in erms of achieving he bes wors-case bound. Indeed, we could show ha, if he balancing raio is a : b : c, hen he wors case bound reaches is minimum a a : b : c = 1 : 1 : 2. For example, if our balancing raio is 1 : 1 : 1, hen we would obain a wors-case bound of 6. 11

13 5 Wors-Case Analysis of he R/2 Policy In his secion, we provide a wors-case analysis of he randomized 1/2-balancing policy (R/2) and show ha he R/2 policy has a provable wors-case performance guaranee of 4. In Secion 7, we demonsrae hrough exensive numerical sudies ha he R/2 policy empirically performs well, and i is significanly beer han he provable wors-case performance guaranees. We formally sae Theorem 1, which is he main resul of his paper. Theorem 1. For each insance of he capaciaed periodic-review sochasic invenory sysem wih seup cos, he expeced cos of he randomized 1/2-balancing policy (R/2) is a mos 4 imes he expeced cos of an opimal policy OP T, i.e., E[C (R/2)] 4 E[C (OP T )]. The proof of Theorem 1 is divided ino a sequence of lemmas. Firs, le Z R/2 be a random variable defined as Z R/2 Θ, if Θ K/2; K P 2, oherwise, (8) where Θ E[H R/2 (Q R/2 ) F ] = E[ Π R/2 (Q R/2 ) F ] is he balancing cos and P is he ordering probabiliy in period. Noe ha Z R/2 and P are random variables ha are realized wih he informaion se f F in period. In he following lemma we show ha he expeced cos of he R/2 policy can be upper bounded using he Z R/2 variables defined in (8). Lemma 1. Le C (R/2) be he oal cos incurred by he R/2 policy. Then we have, E[C (R/2)] 4 L =1 E[Z R/2 ]. Proof. We firs show ha Z R/2 E[H R/2 (Q R/2 ) F ], Z R/2 = E[ Π R/2 (Q R/2 ) F ] and Z R/2 P K/2 wih probabiliy 1. Given any informaion se f, we know he invenory level x and all he quaniies θ, ψ, φ, p defined above are also known deerminisically. We spli he analysis ino wo cases. Firs, if θ K/2, hen q R/2 = ˆq wih probabiliy P = p = 1 implying z R/2 = θ K/2. In addiion, we have z R/2 = E[H R/2 (ˆq ) f ] = E[ Π R/2 (ˆq ) f ], and he claim follows. Second, if θ < K/2, hen q R/2 = min q, u} wih probabiliy p and q R/2 = 0 wih 1 p. Thus, by he consrucion of he probabiliy p, we have z R/2 = p K/2 = E[ Π R/2 (q R/2 ) f ] and hence he claim again follows. z R/2 = p K/2 = E[H R/2 ( q ) f ] E[H R/2 (q R/2 ) f ], 12

14 Applying he above resuls, we obain E[C (R/2)] = = L =1 L =1 L =1 E[H R/2 (Q R/2 R/2 ) + Π (Q R/2 ) + K 1(Q R/2 > 0)] [ ] E E[H R/2 (Q R/2 R/2 ) + Π (Q R/2 ) + K 1(Q R/2 > 0) F ] E[2Z R/2 L + P K] 4 =1 E[Z R/2 ]. This complees he proof of he lemma. To complee he wors-case analysis, we need o show ha he expeced cos of an opimal policy denoed by OP T is a leas T L =1 E[ZR/2 ]. This will be done by amorizing he cos of OP T agains he cos of he R/2 policy. In he subsequen analysis, we decompose he se of periods 1, 2,... T L} ino he following random pariion of six ses: T 1H = : Θ K } OP T and Y Y R/2 ; (9) 2 T 1Π = : Θ K } OP T and Y < Y R/2 ; (10) 2 T 2H = : Θ < K } OP T and Y X R/2 + Q R/2 and Q R/2 = 2 Q u ; (11) T 2S = : Θ < K } OP T and Y X R/2 + Q R/2 and Q R/2 = u < 2 Q ; (12) T 2Π = : Θ < K } and XR/2 Y OP T ; (13) 2 T 2M = : Θ < K } and XR/2 < Y OP T < X R/2 + Q R/2 and Q R/2 = min 2 Q, u}. (14) Noe ha he ses (9) (14) are disjoin and heir union is he complee se of periods. I is also sraighforward o check ha condiioning on f, i is already known which par of he pariion period belongs. We firs analyze he ses T 1H, T 2H, T 1Π and T 2Π since we can idenify he cos componens of he opimal policy larger han hose in he R/2 policy. This gives rise o Lemma 2 below. Lemma 2. The oal holding and backlogging coss incurred by OP T, denoed by H OP T and Π OP T respecively, saisfy [ ] E[H OP T ] E Z R/2 1( T 1H T2H ), (15) E[Π OP T ] E [ Z R/2 1( T 1Π T2Π ) ]. (16) 13

15 Proof. Noe ha in each period (T 1H T2H ), we have Y OP T Y R/2 wih probabiliy one. By he argumen of Lemma 4.2 in [19], since he invenory level of he opimal policy is higher han ha of he R/2 policy, he opimal policy mus have ordered Q R/2 no laer han he R/2 policy. Thus, he oal holding cos associaed wih Q R/2 in he opimal policy mus exceed ha of he R/2 policy. I remains o check ha E[H R/2 (Q R/2 ) F ] = Z R/2 in he wo ses T 1H and T 2H. For T 1H, since he R/2 policy orders he balancing quaniy, i.e., Q R/2 = ˆQ, E[H R/2 (Q R/2 ) F ] = E[H R/2 ( ˆQ ) F ] = Θ = Z R/2. Now for T 2H, since Q R/2 = Q u by he consrucion of T 2H, E[H R/2 (Q R/2 ) F ] = E[H R/2 (min Q, u)} F ] = E[H R/2 ( Q K ) F ] = P 2 = ZR/2 Noe ha Q is he holding-cos-k/2 quaniy; in he se T 2H, he R/2 policy can order his quaniy in full amoun since i is below he capaciy u. Thus, we conclude ha [ ] E[H OP T ] E H R/2 (Q R/2 ) 1( T 1H T2H ) = E = E [ E [ [ H R/2 (Q R/2 ) 1( T 1H T2H ) F ]] Z R/2 1( T 1H T2H ) Noe ha in each period T 1Π T2Π, we have Y OP T [ < Y R/2 wih probabiliy one. By he argumen of Lemma 2 in [20], we have E[Π OP T R/2 ] ] E Π 1( T 1Π T2Π ). Since E[ Π R/2 (Q R/2 ) F ] = Z R/2 is auomaic by he consrucion of he R/2 policy, we have [ ] E[Π OP T ] E Π R/2 (Q R/2 ) 1( T 1Π T2Π ) = E = E [ E [ This complees he proof of Lemma 2. [ Π R/2 (Q R/2 ) 1( T 1Π T2Π ) F ]] Z R/2 1( T 1Π T2Π ) ] ]... In each period T 2S, he R/2 policy can no longer order he holding-cos-k/2 quaniy in full amoun due o he capaciy consrain u, i.e., Q R/2 = u < Q. Then we have E[H R/2 (Q R/2 ) F ] = E[H R/2 (u) F ] < E[H R/2 ( Q K ) F ] = P 2 = ZR/2 14.

16 Thus, we can no longer argue ha he holding cos of he opimal policy is greaer han [ ] E Z R/2 1( T 2S ), even hough he ending invenory of he opimal policy is higher han ha of he R/2 policy. However, we show[ in Lemma 3 ha half of he oal seup coss incurred by he opimal policy can ] be used o pay E ZR/2 1( T 2S ) incurred by he R/2 policy. Lemma 3. Half of he oal seup coss incurred by he opimal policy is lower bounded by [ ] [ ] 1 2 E K 1(Q OP T > 0) E Z R/2 1( T 2S ). (17) Proof. In each period T 2S, we have Q R/2 = u < Q, he R/2 policy will order he capaciy u wih probabiliy p incurring sricly less han p K/2 expeced marginal holding cos. Since = P K/2 in each period T 2S, we shall show ha half of he seup coss incurred by he opimal policy is greaer han E [ P K 2 1( T 2S) ]. Z R/2 Fix a period T 2S. Firs we claim ha he number of orders placed by he opimal policy over he inerval [1, ] is a leas he number of orders in which he R/2 policy orders he full capaciy u over [1, ]. We prove he claim by conradicion. Suppose oherwise, he number of orders placed by he opimal policy over he inerval [1, ] is m and he number of orders in which he R/2 policy orders exacly he capaciy u over [1, ] is n and m < n. The maximum invenory posiion of he opimal policy in period is mu X OP T s=1 D s Y OP T, whereas he minimum invenory posiion of he R/2 policy in period is nu D s Y R/2 = X R/2 + u. X R/2 s=1 Since m < n and boh policies sar wih he same invenory posiion in period 1, i.e., X1 OP T = X R/2 1, his implies ha Y OP T < X R/2 + u which conradics o he fac ha T 2S. The claim hus holds rue. Thus, by leing A be he even ha he R/2 policy orders exacly he capaciy u, we have [ ] [ ] 2 E Z R/2 1( T 2S ) = E P K 1( T 2S ) [ ] = E K 1(A T 2S ) 15 E [ K 1(Q OP T > 0) ].

17 The las inequaliy holds rue because of our previous claim ha he number of orders placed by he opimal policy is no smaller han he oal number of full capaciy orders placed by he R/2 policy wihin [1, ] where T 2S. This complees he proof of Lemma 3. In each period T 2M, he R/2 policy orders he runcaed holding-cos-k/2 quaniy Q R/2 = min Q, u} wih probabiliy p and nohing wih probabiliy 1 p. The randomized decision rule inroduces uncerainies in he relaion beween he invenory posiions afer ordering of he R/2 policy and he opimal policy. Thus, we canno [ argue ha he holding cos or he backlogging ] cos of he opimal policy is greaer han E ZR/2 1( T 2M ). We resor o he seup coss incurred by he opimal policy again, and show in Lemma [ 4 ha half of he oal seup coss ] incurred by he opimal policy is sufficien o pay E ZR/2 1( T 2S ) incurred by he R/2 policy. Lemma 4. Half of he oal seup cos incurred by he opimal policy is lower bounded by [ ] [ ] 1 2 E K 1(Q OP T > 0) E Z R/2 1( T 2M ). (18) Proof. Consider an arbirary sample pah wih f T F T. We denoe he period in which he opimal policy makes he nh order by n. Then we can pariion he planning horizon 1,..., T } = [0, 1 ) [ 1, 2 ) [ N 1, N ) [ N, N+1 ), where N+1 = T + 1 and N is he oal number of orders ha he opimal policy have placed hrough T. Firs we claim ha here does no exis a period s [0, 1 ) such ha s T 2M. Since he R/2 policy and opimal policy have he same iniial invenory X1 OP T = X R/2 1 and face he same demands, if he opimal policy has no placed any orders, we mus have Xs R/2 which implies ha s does no belong o he se T 2M. Therefore he claim is proved. X OP T s = Y OP T s, Nex we claim ha he R/2 policy will a mos make one order in each se of periods T 2M [ i, i+1 ) where 1 i N. In each period T 2M [ i, i+1 ), we have X R/2 < Y OP T < X R/2 +Q R/2. By he consrucion of he R/2 policy, we will order Q R/2 wih probabiliy p and nohing oherwise. Now le A be he even in which he R/2 policy places an order and define he sopping ime k = inf m i : A m T 2M }. If k i+1, he claim holds since he R/2 policy does no place any orders wihin T 2M [ i, i+1 ). Now suppose ha k < i+1. I suffices o show ha T 2M [k + 1, i+1 ) =. Since k T 2M and he R/2 policy places an order, hen we mus have Yk OP T < X R/2 k + Q R/2 k = Y R/2 k. In addiion, we know ha Ym OP T = Xm OP T for all m ( i, i+1 ) since he opimal policy does no place any orders in he se ( i, i+1 ). Then for each period j [k + 1, i+1 ), by he dynamics of he model, we have Y OP T j = X OP T j = Y OP T k j 1 m=k D m < Y R/2 k j 1 m=k D m Y R/2 j 1 k + m=k Q m j 1 m=k D m = X R/2 j, which implies ha j does no belong o he se T 2M. Thus he second claim is also proved. 16

18 Then we can conclude wih probabiliy 1, i holds ha 2 E L =1 Thus, by he above inequaliy, we have [ L ] =1 Z R/2 1( T 2M ) = E This complees he proof of Lemma 4. K 1(A T 2M ) NK. [ T L =1 E [NK] = E P K 1( T 2M ) [ ] = E [ T L =1 K 1(Q OP T > 0) Summing up inequaliies (15), (16), (18) and (17), we hen obain E[C (OP T )] L =1 ]. K 1(A T 2M ) E[Z R/2 ]. (19) Hence, by (19) and Lemma 1, we have esablished Theorem 1, i.e., he R/2 policy has an expeced wors-case performance guaranee of 4. Before closing his secion, we provide some inuiions why our proposed policy has a wors case performance guaranee of 4 bu no 3 (in which Levi and Shi [21] were able o prove for he uncapaciaed sochasic problems). I can be readily observed ha he se T 2S defined in (12) can be merged ino he se T 2H defined in (11) in models wih infinie ordering capaciies. This follows from he fac ha he policy can always order up o holding-cos-k quaniy Q. Following he argumens in Lemma 2, we can show ha he holding cos incurred by OP T can cover our balancing cos in T 2H T2S. Togeher wih he analysis of he remaining pariions, his leads o he 3-approximaion algorihm in Levi and Shi [21] for he uncapaciaed lo-sizing problem. Wih ordering capaciy consrains, he holding cos incurred by OP T can no longer cover our balancing cos in T 2S. Insead, we have shown ha he seup coss incurred by OP T can be used o cover his gap. Since analyzing he problemaic se T 2M requires he use of seup coss incurred by OP T once again, we have in fac used he seup coss incurred by OP T wice. If our balancing raio is 1 : 1 : 1 (marginal holding, he forced backlogging and seup coss), hen we would obain a wors-case bound of 6, which is no opimal in erms of achieving he ighes wors-case bound. As we discussed earlier, he wors case bound reaches is minimum a 1 : 1 : 2, which yields a 4-approximaion algorihm. ] 6 Exensions o Bach Ordering Sysem In his secion we exend our resuls o sochasic periodic-review invenory models wih seup cos under bach ordering consrains. The bach consrain specifies ha, every order quaniy has o be an ineger muliple of a pre-specified base bach size, say q 0, which can be, for example, a box, a palle, a ruckload, ec. The case of bach ordering wih infinie order capaciy has been sudied in 17

19 he lieraure, see, for example Veino [25], and Chen [4], among ohers. Wihou loss of generaliy, we can assume he capaciy is also an ineger muliple of bach size, since he excess quaniy which is less han a base bach size q 0 can never be used under bach order consrains. More specifically, le u = mq 0 where m is a given posiive ineger, and a feasible policy can only order quaniies of iq 0 for some ineger i aking value from 0, 1,..., m}. 6.1 Modified randomized 1/2-balancing (M R/2) policy Since he ordering quaniy can only be an ineger muliple of he base quaniy q 0, he marginal holding cos funcion H (Q ) and he forced backlogging cos funcion Π (Q ) are defined only a Q = iq 0 where i = 0, 1,..., m. For oher non-negaive ineger value Q, we can exend he wo funcions H (Q ) and Π (Q ) by inerpolaing piecewise linear exensions of hese bach quaniies. More specifically, for any ineger value Q, here exiss a scalar λ [0, 1) such ha where Q = (1 λ )Q lower + λ Q upper, Q lower = Q /q 0 q 0, Q upper = Q /q q 0, and he floor funcion a is he larges ineger less han or equal o a. The corresponding marginal holding cos and forced backlogging cos are defined, using linear inerpolaion, as E[H (Q ) F ] = (1 λ ) E[H (Q lower ) F ] + λ EH (Q upper E[ Π (Q ) F ] = (1 λ ) E[ Π (Q lower ) F ], ) F ] + λ E Π (Q upper ) F ]. I is clear ha hese exended cos funcions H (Q ) and Π (Q ) preserve he properies of convexiy and monooniciy. Cos marginal holding cos forced backlogging cos θ 0 bach size balancing quaniy capaciy (u) Figure 2: A graphical depicion of he exended cos funcions by linear inerpolaion. We now propose a modified randomized 1/2-balancing policy (M R/2). A he beginning of each period wih he realized informaion se f, we compue he auxiliary order quaniies and cos funcions discussed in Secion 4.1. Noe ha hese auxiliary funcions are defined only on ineger 18

20 muliples of he base bach size q 0. Thus, o properly balance he cos funcions, we need o define heir corresponding lower and upper quaniies. Firs, compue he balancing quaniy ˆq and he balancing cos θ such ha θ E[ Π (ˆq ) f ] = E[ Π (ˆq ) f ]. Then, here exiss a scalar ˆλ [0, 1) such ha where ˆq lower ˆq = (1 ˆλ )ˆq lower = ˆq /q 0 q 0 and ˆq upper = ˆq /q q 0. + ˆλ ˆq upper Nex, compue he holding-cos-k/2 quaniy q such ha E[H ( q ) f ] = K/2. There exiss anoher scalar λ [0, 1) such ha where q lower q = (1 λ ) q lower = q /q 0 q 0 and q upper = q /q q 0. + λ q upper, Third, compue he resuling forced backlogging cos φ E[ Π (minˆq, u}) f ] if one orders he minimum of he holding-cos-k/2 quaniy and he capaciy u. Finally, compue he forced backlogging cos ψ E[ Π (0) f ] if one orders nohing. The modified randomized 1/2-balancing (MR/2) order policy we propose for he case wih bach ordering consrain is described as follows: (i) If he balancing cos θ K/2, hen he MR/2 policy orders ˆq lower and ˆq upper wih probabiliy ˆλ. wih probabiliy 1 ˆλ (ii) If he balancing cos θ < K/2, hen compue he ordering probabiliy p from p K/2 = p φ + (1 p )ψ similar o (7). The MR/2 policy orders min q lower, u} wih probabiliy p (1 λ ), order min q upper, u} wih probabiliy p λ, and order nohing wih probabiliy 1 p. by To summarize, if we denoe he order quaniy of he MR/2 policy by q MR/2, hen i is given ˆq lower, wih probabiliy 1 ˆλ in case (i); q MR/2 = ˆq upper, wih probabiliy ˆλ in case (ii); min q lower, u}, wih probabiliy p (1 λ ) in case (ii); min q upper, u}, wih probabiliy p λ in case (ii); 0, wih probabiliy 1 p in case (ii). I is clear ha he modified randomized 1/2-balancing policy balances he hree ypes of coss in a similar manner as he original randomized 1/2-balancing policy wihou he bach order consrains. 19

21 6.2 Wors-case analysis To conduc he performance analysis, we define all he ses similarly o (9) o (14) as follows. T 1H = : Θ K } OP T and Y X MR/2 + ˆq upper, (20) 2 T 1Π = : Θ K } OP T and Y X MR/2 + ˆq lower and ˆq lower < ˆq upper, (21) 2 T 2H = : Θ < K } OP T and Y X MR/2 + min q upper, u} and q upper u, (22) 2 T 2S = : Θ < K } OP T and Y X MR/2 + min q lower, u} and u q lower < q upper, (23) 2 T 2Π = : Θ < K } and XMR/2 Y OP T, (24) 2 T 2M = : Θ < K } and XMR/2 < Y OP T X MR/2 + min q lower, u}. (25) 2 To ensure ha he union of he ses from (20) o (25) is he complee se of all periods, we show by Lemma 5 ha, condiional on he same demand realizaion, he base bach load q 0 mus divide he absolue difference beween invenory levels of he opimal policy and he M R/2 policy. Thus, i is impossible o have Y OP T (X MR/2 + min q lower, u}, X MR/2 + min q upper, u}) when he balancing cos Θ K/2. Similarly, since u is an ineger muliple of q 0, i is also impossible for his o happen when he balancing cos Θ < K/2. Lemma 5. For any realizaion, he base bach size q 0 mus divide Y OP T Y MR/2, he difference beween he invenory posiions of he opimal policy and he MR/2 policy for each period = 1,..., T L. Proof. Consider an arbirary period s and suppose he opimal policy ordered m OP s T q 0 while he MR/2 policy ordered m MR/2 s q 0, where m OP s T and m R/2 s are nonnegaive inegers. Suppose ha he saring invenory posiions a he beginning of period 1 are he same for boh policies, i.e., X1 OP T = X MR/2 1. Then for an arbirary period = 1,..., T L, we have = = Y OP T ( Y MR/2 ( X OP T 1 + ( s=1 (m OP T s s=1 m OP T s m MR/2 s ) ) q 0 D [1,) ) ) q 0. ( X MR/2 1 + ( s=1 m MR/2 s ) q 0 D [1,) ) Thus, he base bach load q 0 mus divide Y OP T Y MR/2. 20

22 By Lemma 5, we have consruced he disjoin ses (20) - (25) and heir union is a complee se. I can be readily verified ha he Lemmas 1, 2, 3 and 4 coninue o hold if we replace all he ses (9) (14) wih (20) (25). We formally sae he resul for capaciaed sochasic invenory problem wih seup cos under bach order consrains. Theorem 2. For each insance of he capaciaed sochasic periodic-review invenory problem wih seup cos under bach ordering consrains, he expeced cos of he modified randomized 1/2- balancing policy (MR/2) is a mos four imes he expeced cos of an opimal policy OP T, i.e., E[C (MR/2)] 4E[C (OP T )]. Remark Consider a special case of he bach ordering sysem wih he base bach order size equal o he capaciy, i.e., we resric ourselves o all-or-nohing ordering policies. (Özer and Wei [23] esablished he opimaliy of a sae-dependen hreshold policy wih his class of policies.) In his special case, we can convenienly ransform he original uni ordering cos c q plus he seup cos K 1(q = u) ino an equivalen modified uni ordering cos c q where c = c + K/u, since q can only ake values 0 or u. Then he model will be reduced o he one sudied in Levi e al. [20] where a dual-balancing policy yields a 2-approximaion. I should be noed ha his simple ransformaion fails o work for any more-han-wo poin ordering policies, since he seup cos exhibis a concave ordering cos srucure. 7 Numerical Experimens In his secion, we conduc a numerical sudy on he performance of he R/2 policy developed in Secion 5. As noed by Levi and Shi [21], he randomized cos-balancing policy can be parameerized o obain general classes of policies, respecively, and he wors-case analysis discussed above can hen be viewed as choosing parameer values ha perform well agains any possible insance. In conras, one can ry o find he bes parameer values, for each given insance. This gives rise o policies ha have a leas he same wors-case performance guaranees, bu are likely o work beer empirically, since we refined he parameers according o he specific insance being solved. Using simulaion based opimizaion, we have implemened his approach and esed he empirical performance of he resuling policies. The policies were esed using he demand model of advance demand informaion proposed by Gallego and Özer [9], and Özer and Wei [23]. To he bes of our knowledge, hese are he only papers ha repored opimal compuaional coss (by brue force dynamic programming) for he capaciaed sochasic periodic-review invenory sysem wih seup cos and dependen demand srucures. Parameerized policies. We describe a class of parameerized policies involving parameers β, γ and η, where β conrols he holding-cos-βk/2 quaniy, γ conrols he raio of he marginal holding cos o he forced backlogging cos and η conrols he level of he forced backlogging cos resuling from no ordering. Specifically, he parameerized policy firs compues several quaniies. 1) The balancing quaniy ˆq ha solves E[H R/2 (ˆq ) f ] = γe[ Π R/2 (ˆq ) f ] := θ. 2) The holding-cos-β K 2 quaniy q ha solves E[H R/2 ( q ) f ] = β K 2. 21