Approximation Algorithms for Perishable Inventory Systems with Setup Costs

Transcription

1 Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss Huanan Zhang, Cong Shi, Xiuli Chao Indusrial and Operaions Engineering, Universiy of Michigan, MI 48105, {zhanghn, shicong, We develop he firs approximaion algorihm for periodic-review perishable invenory sysems wih seup coss. The ordering lead ime is zero. The model allows for correlaed demand processes which generalize he well-known approaches o model dynamic demand forecas updaes. The srucure of opimal policies for his fundamenal class of problems is no known in he lieraure. Thus, finding provably near-opimal conrol policies has been an open challenge. We develop a randomized proporional-balancing policy (RPB ha can be efficienly implemened in an online manner, and show ha i admis a wors-case performance guaranee beween 3 and 4. The main challenge in our analysis is o compare he seup coss beween RPB and he opimal policy in he presence of invenory perishabiliy, which depars significanly from he previous works of Chao e al. (2015b,a. The numerical resuls show ha he average performance of RPB is good (wihin 1% of opimaliy under i.i.d. demands and wihin 7% under correlaed demands. Key words : invenory, perishable producs, seup coss, randomized algorihms, wors-case analysis Received Ocober 2014; revisions received March 2015, Ocober 2015, December 2015; acceped January Inroducion We sudy he periodic-review perishable invenory sysems wih seup coss under a general class of associaed demand processes (see 2. Undoubedly, perishable producs are an indispensable par of our lives. For example, perishable producs such as mea, frui, vegeable, dairy producs, and frozen foods consiue he majoriy of supermarke sales. Moreover, virually all pharmaceuicals belong o he caegory of perishable producs. Blood bank provides anoher salien example where whole blood has finie lifeimes (see, e.g., Prasacos (1984. In general, he analysis of perishable producs is much harder han ha of non-perishables. The work on classical perishable invenory sysems daes back o Nahmias (1975 and Fries (1975 who characerized he srucure of he opimal ordering policy wih i.i.d. demands and no seup coss for backlogging and los-sales models, respecively. We refer ineresed readers o Karaesmen e al. (2011 and Chao e al. (2015b,a for a review of he field. Unforunaely, almos all he papers in he exising perishable invenory lieraure do no model posiive seup coss 1

2 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 2 (mainly due o is mahemaical inracabiliy, despie he fac ha fixed cos is ofen unavoidable in many pracical seings. As a resul, lile is known abou he srucure of he opimal policies for his class of problems wih seup coss. Thus, finding provably near-opimal conrol policies has been an open challenge. In his paper, we propose an efficien randomized proporional-balancing policy ha admis a wors-case performance guaranee beween 3 and 4. We also demonsrae via numerical experimens ha he proposed policy performs consisenly well. Relevan lieraure. Nahmias (1978 was he firs o analyze perishable invenory problems wih posiive seup coss. He showed ha he cos funcion is in general no K-convex and ha (s, S policy may no be opimal. Indeed, he opimal conrol policy for his class of problems is very complicaed, and here is no known srucural characerizaion for i. Neverheless, Nahmias (1978 demonsraed, compuaionally, ha (s, S ype policies perform well under i.i.d. demands. Lian and Liu (1999 also considered a periodic-review model wih posiive seup coss, and consruced a muli-dimensional Markov chain o model and analyze he invenory-level process. Lian e al. (2005 used he same coss and replenishmen assumpions as Lian and Liu (1999, and consruced a muli-dimensional Markov chain o model he invenory-level process o derive cos expressions. They numerically showed ha he variabiliy in he lifeime disribuion can have a significan effec on sysem performance. However, he aforemenioned heurisic policies can only be applied o i.i.d. demand processes, and furhermore hey lack any performance guaranees. Main resuls and conribuion. The conribuion of his noe is o presen he firs approximaion algorihm for periodic-review perishable invenory sysem wih seup coss ha admis a wors-case performance guaranee beween 3 and 4. The resul holds no only for independen demand processes, bu also for a general class of correlaed processes, which we call associaed demand processes. The proposed policy will be referred o as a randomized proporional-balancing policy (RPB. As seen from our lieraure review, his class of problems is fundamenal in perishable invenory lieraure ha has challenged researchers for decades; ye lile is known abou boh he srucure of opimal policies and he design of provably-good heurisic policies. Our approach builds on Chao e al. (2015b,a on perishable invenory sysems wihou seup coss. In paricular, we adop heir marginal cos accouning schemes of compuing marginal holding, backlogging and oudaing coss. The main idea underlying his marginal cos accouning approach is o decompose he oal cos in erms of he marginal coss of individual decisions (raher han he convenional per-period cos accouning. However, he nonlinear seup coss make he balancing of he various coss much more complicaed. Thus, we consruc a randomized algorihm o ackle his problem, sriking a righ balance beween differen cos componens. The wors-case performance analysis is ineviably much more sophisicaed, due o invenory perishabiliy. The key

3 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 3 echnical challenge is o amorize he seup coss of RPB agains he opimal policy. We noe ha Levi and Shi (2013 on sochasic lo-sizing problems had he same challenge bu heir pariion of periods is raher sraighforward wihou perishabiliy, while our problem needs o consider he invenory age informaion ha is essenial in he analysis of our sysem. (To his end, we also provide a deailed discussion in 4 on why he previous pariion in Levi and Shi (2013 fails o work in our perishable invenory sysems. This addiional informaion on invenory age enables us o amorize he seup cos incurred by RPB agains ha of he opimal policy. On he echnical level, we provide a novel sample-pah argumen ha shows ha he opimal policy has o place an order beween wo consecuive problemaic periods in which he relaionship beween he rimmed ending invenory levels of RPB and he opimal policy is unclear (due o randomized decisions. This is in sharp conras wih he averaging argumen used in Levi and Shi (2013. Moreover, Levi and Shi (2013 need no consider he age informaion in heir analysis. Our consrucion provides he righ and delicae framework o use he age informaion o analyze he sysem, which could be useful in analyzing oher more complex perishable invenory sysems (e.g., wih posiive lead imes and/or finie ordering capaciies. We also demonsrae hrough an exensive compuaional sudy ha our proposed algorihm performs quie well empirically (wih average error under 7% and maximum error of 11.28% under correlaed demands. Our algorihm has also comparable (if no, beer performance wih ha of Nahmias (1978 under i.i.d. demands, in which boh algorihms perform very close o opimal. Srucure and general noaion. The res of his paper is organized as follows. In 2, we formally describe he periodic-review perishable invenory sysems wih seup coss. In 3, we inroduce he randomized proporional-balancing policy (RPB. In 4, we carry ou a wors-case performance analysis of RPB. In 5, we demonsrae he empirical performance of RPB. Throughou his paper, we ofen disinguish beween a random variable and is realizaions using capial and lower-case leers, respecively. For any real numbers x and y, we denoe x + = max{x, 0}, x y = max{x, y}, and x y = min{x, y}. In addiion, for a sequence x 1, x 2,... and any inegers and s wih s, we denoe x [,s] = s j= x j and x [,s = s 1 j= x j. The indicaor funcion 1(A akes value 1 if A is rue and 0 oherwise, and sands for defined as. 2. Perishable Invenory Sysems wih Seup Coss We formally describe he sochasic periodic-review perishable invenory sysem wih seup coss over a planning horizon T (possibly infinie, indexed by = 1,..., T. The produc lifeime is m 2, i.e., iems perish afer saying in invenory for m periods if no consumed. The ordering lead ime is 0 (see, e.g., Karaesmen e al. (2011.

4 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 4 Cos srucure. The uni holding and shorage coss are denoed by h and b, respecively, and he uni oudaing cos is θ. Wihou loss of generaliy, we assume he uni purchasing cos c = 0 (see a deailed cos ransformaion in Chao e al. (2015b. In addiion, here is a seup cos K ha is incurred whenever an order is placed. The discoun facor is α [0, 1] (bu α is sricly less han 1 if T =, and he firm s objecive is o minimize he expeced oal discouned cos. (Our model and resuls can allow for non-saionary seup coss K as long as hey saisfy αk +1 K. Demand srucure. We adop he same demand srucure as ha of Chao e al. (2015a, where he demands D 1,..., D T are associaed (which is defined formally below. A he beginning of each period, he manager observes an informaion se denoed by F. The informaion se F conains he pas informaion accumulaed up o he beginning of period, including he realized demands d 1,..., d 1 in he firs 1 periods and possibly some oher exogenous informaion (e.g., sae of economy, weaher, ec. The informaion se {F [1, T ]} form he filraion over a probabiliy space (Ω, F, P. We assume ha he condiional expecaions of all relevan quaniies, given F, are well-defined. The formal definiion of associaed demand process is as follows. Definiion 1 (Chao e al. (2015a. A sochasic demand process {D ; = 1, 2,...} wih filraion {F, 1} is called associaed if, condiional on F, he random demand vecor D = (D, D +1,..., D T saisfies E[f(D g(d ] E[f(D ] E[g(D ] for all non-decreasing (or nonincreasing funcions f and g for which he expecaions E[f(D ], E[g(D ], E[f(D g(d ] exis. The class of associaed processes includes no only independen demand processes, bu also mos ime-series demand models such as auoregressive (AR and auoregressive moving average (ARMA demand models (Box e al. (2008, muliplicaive auo-regression model (Levi e al. (2008, demand forecas updaing models such as maringale models for forecas evoluion (MMFE (Heah and Jackson (1994, demand processes wih advance demand informaion (ADI (Gallego and Özer (2001, as well as economic-sae driven demand processes such as Markov modulaed demand processes wih sochasically monoone ransiion marix, among ohers. We refer ineresed readers o Chao e al. (2015a for more discussions. Sysem dynamics. In perishable invenory sysems, any invenory uni ha says in he sysem for m periods wihou meeing he demand expires and exis he sysem. Thus, we use a vecor o keep rack of he invenory age informaion, which resuls in a mulidimensional sae space. A he beginning of each period, = 1, 2,..., T, we are endowed wih he realized informaion f. (Here we abuse he noaion o use f o denoe he realized informaion up o ime, which should be disinguished wih he filraion F. The saring invenory a period is x = (x,1,..., x,m 1, where for i = 1,..., m 2, x,i is he invenory level of produc whose remaining lifeime is i periods, and x,m 1 is he number of produc whose remaining lifeime is m 1 minus he number

5 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 5 of backlogs. Afer receiving he order q in period, he random demand D will be realized (denoe is realizaion by d and saisfied o he maximum exen by FIFO issuing policy, i.e., he oldes invenory mees demand firs. Following he convenion by Nahmias (1975, we assume he invenory unis ha will perish a he end of his period also incur a holding cos. The discouned ( one-period cos is α 1 K 1(q > 0 + h(y d + + b(d y + + θe, where y = m 1 x i=1,i + q is he oal invenory level (afer ordering in period, and e = (x,1 d + is he oudaing invenory. Then, he sysem proceeds o period + 1 wih x +1 given by x +1,j = x,j+1 ( d + j x,i i=1 +, for 1 j m 2, (1 ( m 1 + x +1,m 1 = q d x,i. (2 i=1 For simpliciy we assume ha he sysem is iniially empy. Then he oal expeced discouned cos for any given FIFO issuance policy P can be wrien as [ C (P = E α (K ] 1 1(Q > 0 + h(y D + + b(d Y + + θe. (3 =1 Noe ha, he quaniies Q, Y, E and X depend on he policy P, and we shall make he dependency explici whenever necessary, i.e., wriing hem as Q P, Y P, E P and X P. 3. Randomized Proporional-Balancing Policy We presen he randomized proporional-balancing policy (denoed by RPB and is wors-case performance guaranee. We sar by reviewing he nesed marginal holding and oudaing cos accouning scheme used in Chao e al. (2015b,a. The main idea is o decompose he oal cos in erms of he marginal coss of individual decisions (firs used in Levi e al. (2007. Tha is, we associae he decision in period wih is affiliaed cos conribuions o he sysem. These coss are only affeced by fuure demands bu no by fuure decisions. These marginal coss may include coss (associaed wih he decision incurred in boh he curren and subsequen periods Review of he marginal cos accouning scheme Under FIFO issuing policy, afer a uni is ordered, he number of periods ha i says in he sysem is no affeced by fuure decisions bu only affeced by fuure demands. In oher words, he oal expeced holding and oudaing cos of any uni is deermined in he period in which ha uni is ordered.

6 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 6 Marginal holding and oudaing coss. Suppose an order quaniy q P 0 is placed by a policy P in period. Then he oal expeced marginal holding and oudaing cos of hose q P unis can be compued via a recursive equaion. For s < + m, le B [,s (x P denoe he oal number of oudaed unis from period up o period s 1 when he saring invenory a he beginning of period is x P. Wih he convenion ha B [, (x P 0, we can wrie B [,s (x P recursively by { s } B [,s (x P = max x P,j D [,s, B [,s 1 (x P, < s + m 1. (4 j=1 Using B [,s (x P, he nesed marginal holding cos H P can be wrien as, for 1 T, H P (+m 1 T = H P (q P ; x P h s= α (q s 1 P ( m 1 D [,s] + B [,s (x P j=1 x P,j + +, (5 and he nesed marginal oudaing cos Θ P can be wrien as, for 1 T m + 1, Θ P ( m 1 +, = Θ P (q P ; x P α +m 2 θe +m 1 = α +m 2 θ q P + x P,j B [,+m 1 (x P D [,+m 1] (6 j=1 and Θ P 0 for = T m + 2,..., T as he ordered unis do no expire wihin he planning horizon. I is clear ha boh H P ( and Θ P ( are increasing in q P. Backlogging cos and seup cos. I is clear ha in period, no fuure marginal backlogging cos is caused by he curren order q P, since any under-ordering can be correced by subsequen orders. As a resul, he marginal backlogging cos in period is he same as he convenional backlogging cos in period, which can be wrien as Π P ( m 1 = Π P (q P ; x P α 1 b D x P,i q P i=1 +. (7 Moreover, he seup cos in any period is also only affeced by q P, which is α 1 K 1(q P > 0. Sysem oal cos. Since he sysem is iniially empy, he expeced oal sysem cos C (P of policy P can be obained by summing (5, (6, (7 and he seup coss over from 1 o T, and hen aking expecaions. Thus, (3 can be rewrien as C (P = E [ =1 ( H P (q P ; x P + Π P (q P ; x P + Θ P (q P ; x P + α 1 K 1(q P > 0 ], (8 and he sysem dynamics are governed by (1 and (2.

7 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss RPB policy To describe he randomized proporional-balancing (RPB policy, we modify he definiion of he informaion se f o also include he implemened decisions of he randomized policy up o period 1. To deermine wheher or no o place an order, and how much o order, in period, RPB compues he following quaniies: (a Compue ˆq ha balances he condiional expeced marginal holding and oudaing coss of hese unis agains he condiional expeced backlogging cos in period. Tha is, ˆq solves mh+θ E [ H RP B 2(m 1h+θ ] [ ] (ˆq + Θ RP B (ˆq f = E Π RP B (ˆq f θ, (9 where ˆq is referred o as he proporional balancing quaniy, and θ he proporional balancing cos. The soluion o (9 is unique and can be compued efficienly via bisecion search. (b Compue he (proporional balancing-k-quaniy q ha solves mh+θ E [ H RP B 2(m 1h+θ ( q + Θ RP B ( q f ] = K. Tha is, ordering q ha balances a proporion of he marginal holding and oudaing coss wih seup cos K. (c Compue E[Π RP B ( q f ], i.e., he condiional expeced backlogging cos in period if one orders he balancing-k-quaniy q in period. (d Compue E[Π RP B (0 f ], i.e., he condiional expeced backlogging cos in period resuling from no ordering in period. Descripion of RPB policy. Le P denoe he probabiliy ha RPB policy places an order in period which is a-priori random, and le p = (P f. Given f, he RPB policy deermines he ordering decisions according o wo cases below: Case (I. If he proporional balancing cos exceeds K, i.e., θ K, hen he RPB policy orders he balancing quaniy q RP B = ˆq in period wih probabiliy p = 1. Case (II. If he proporional balancing cos is less han K, i.e., θ < K, hen he RPB policy orders he balancing-k-quaniy (i.e., q RP B = q in period wih probabiliy p and orders nohing wih probabiliy 1 p, where he probabiliy p is compued by solving equaion p K = p E[Π RP B ( q f ] + (1 p E[Π RP B (0 f ]. (10 This complees he descripion of he RPB policy. We offer some inuiive explanaions as follows. In Case (I, ha is when θ K, he fixed ordering cos K is no dominan compared o he oher

8 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 8 cos componens, hence he policy only srives o achieve (he righ balance beween marginal holding, oudaing, and shorage coss. In Case (II, he underlying reasoning behind he choice of he paricular randomizaion in (10 is ha, he policy aemps o balance beween he hree cos componens, namely, holding and oudaing cos, backlogging cos and seup cos associaed wih period. In paricular, since we order he balancing-k-quaniy wih probabiliy p and do no order anyhing wih probabiliy 1 p, he condiional expeced proporional holding and oudaing cos for his case is mh+θ 2(m 1h+θ = mh+θ 2(m 1h+θ = p K. E[H RP B (q RP B { p E[H RP B + Θ RP B (q RP B f ] (11 ( q + Θ RP B ( q f ] + (1 p E[H RP B (0 + Θ RP B (0 f ] } By he selecion of p in (10, he condiional expeced backlogging cos is E[Π RP B (q RP B f ] = p E[Π RP B ( q f ] + (1 p E[Π RP B (0 f ] = p K. (12 Since p is he ordering probabiliy in Case (II, he expeced fixed ordering cos is also p K. I can be shown he p ha solves (10 is 0 p = E[Π RP B (0 f ] < 1, (13 K E[Π RP B ( q f ] + E[Π RP B (0 f ] where he above inequaliies follow from he fac ha θ < K and q > ˆq, which implies ha E[Π RP B ( q f ] < E[Π RP B (ˆq f ] = θ < K. We remark ha when he seup cos K = 0, he RPB policy reduces exacly o he PB policy proposed in Chao e al. (2015b, i.e., RPB is a generalizaion of PB in he presence of seup coss. The following is he main heoreical resul of his paper. Theorem 1. When he demand process is associaed, he RPB policy for perishable invenory sysem wih seup cos and m 2 periods of produc lifeime has a wors-case performance guaranee ( of 3 + (m 2h, i.e., for any insance of he problem, he expeced cos of he RPB policy is a mh+θ ( mos 3 + (m 2h imes he expeced cos of an opimal policy. mh+θ The balancing raio on he lef hand side of (9 is chosen such ha he resuling RPB policy admis he ighes wors-case performance guaranee. Suppose an arbirary balancing raio β (0, 1] is used o consruc RPB. Then one can show ha i admis a wors-case performance guaranee of (2β + 1/ min{β, β 0 }, where β 0 = mh+θ 2(m 1h+θ (which is obained from (22 in our worscase analysis. This wors-case performance guaranee is minimized when β = β 0.

9 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 9 Remark on discree demand and order quaniies: If he demand and order quaniies are discree, hen we can always wrie ˆq = ˆλ ˆq 1 + (1 ˆλ ˆq 2, where ˆq 1 and ˆq 2 = ˆq are consecuive inegers wih ˆq 1 ˆq < ˆq 2 and 0 < ˆλ 1. Similarly, we wrie q = λ q 1 + (1 λ q 2, where q 1 and q 2 = q are consecuive inegers wih q 1 q < q 2 and 0 < λ 1. In Case (I, RPB orders eiher ˆq 1 unis (wih probabiliy ˆλ or ˆq 2 unis (wih probabiliy 1 ˆλ. In Case (II, RPB orders eiher q 1 unis (wih probabiliy p λ, or q 2 unis (wih probabiliy p (1 λ, or zero uni (wih probabiliy 1 p. This randomized procedure will no affec our wors-case performance guaranee, and we refer readers o he deailed discussions in 4.3 of Levi e al. (2007 and also 6 of Shi e al. ( Wors-Case Performance Analysis We carry ou a wors-case performance analysis of RPB policy. For simpliciy, we provide he proofs wih a discoun facor α = 1 (albei he analysis holds under general α [0, 1]. We firs describe a concep called rimmed invenory level (see Chao e al. (2015a. The rimmed invenory level, denoed by Y,s for any s 1, is defined as he invenory a he beginning of period s for he producs ha are ordered in period or earlier. Equivalenly, Y,s is he invenory level Y s (afer ordering a he beginning of period s less he oal order quaniy in periods + 1,..., s, i.e., Y,s = Y s s s =+1 Q s. By definiion, Y, = Y and we also have Y,s = Y,s D [s,s E [s,s, s < s + m 1, (14 Y,s = Y,+m 1 D [+m 1,s, s > + m 1. (15 Noe ha Y,s can be negaive when s is large enough. These rimmed invenory levels serve as a generalizaion of he radiional invenory levels, as hey provide criical parial informaion on he age of he on-hand producs. Due o he naure of perishable sysems, i is impossible o quanify he effec of he decision made in he curren period on fuure coss only hrough he radiional oal invenory level Y. The rimmed invenory levels provide a racable way o analyze his effec, and also provide he righ framework for coupling he marginal holding, oudaing and seup coss in differen sysems. Furhermore, for any realizaion f, if p < 1, hen we denoe RPB as he policy ha orders exacly he same as RPB before period, bu orders q a (insead of using a randomized decision beween 0 and q, and orders 0 aferwards. We now compare he coss of RPB policy agains ha of an opimal policy, denoed by OPT. For each realizaion f T, we pariion all he periods ino he following ses: T 1H = { : p = 1 and y OP T T 1Π = { : p = 1 and y OP T } > y RP B ; (16 } y RP B ; (17

10 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 10 T 2Π = { : p < 1 and x RP B { T 2H = : p < 1 and x RP B T 2M = } y OP T ; (18 < y OP T { : p < 1 and x RP B < y OP T and s [, ( + m 1 T ], y OP T,s and s [, ( + m 1 T ], y OP T,s } > y RP B,s y RP B,s ; (19 }. (20 Lemma 1 shows ha he backlogging cos of OPT can cover ha of RPB wihin se T 1Π T2Π, which is formally saed below. We relegae is deailed proof o he elecronic companion. Lemma 1. For each sample pah f T, we have =1 Π OP T [ =1 ] 1 ( T 1Π T2Π. (21 Π RP B Lemma 2 shows how o amorize he sum of marginal holding and oudaing coss agains ha of OPT, which follows idenical argumens used in Chao e al. (2015a; hence is proof is omied. Lemma 2. For each sample pah f T, we have ( =1 H OP T + Θ OP T mh + θ 2(m 1h + θ [ =1 (H RP B ] + Θ RP B 1 ( T 1H T2H. (22 The nex resul is a key lemma for his paper. I suggess ha he oal seup cos incurred by OPT can cover he oal seup cos incurred by RPB for periods wihin he se T 2M. Noe ha T 2M is he mos problemaic se of periods because we are unsure wheher he rimmed ending invenory levels of RPB are higher or lower han hose of OPT due o randomized decisions. Lemma 3. For each sample pah f T, he following inequaliy holds for he OPT and RPB policies, Proof. =1 1(q OP T > 0 =1 1(q RP B > 0 and T 2M. (23 For a fixed sample pah f T, each period deerminisically belongs o one of he ses in T 1H, T 1Π, T 2Π, T 2H and T 2M. Denoe all he periods in T 2M wih posiive ordering quaniies by { T2M : q RP B > 0 } = { 1, 2,..., n }. See Figure 1 for an illusraion of all such periods. If his se is empy, i.e., n = 0, hen (23 rivially holds. Oherwise, i follows from he definiion of { 1, 2,..., n } ha he righ hand side of (23 is equal o n. Thus, i suffices o prove ha OPT places a leas n orders. We firs show ha, 1 =1 1(q OP T > 0 1. (24

11 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 11 Figure 1 Illusraion of boh Inequaliy (24 and Inequaliy (25 for sample pah f T. This is rue because boh OPT and RPB have he same saring invenory and are facing he same demand, if OPT has no ye placed any order before or in period 1, hen we canno have y OP T 1 > x RP B 1. Hence, OPT places a leas one order as shown in Figure 1. If n = 1, hen (23 is he same as (24 which has already been proved. Now suppose n 2. We wan o show ha i+1 = i +1 1(q OP T > 0 1, for all i = 2, 3,..., n. (25 Tha is, OPT places a leas one order in each and every inerval { i + 1,..., i+1 }. (Figure 1 provides an illusraion of (25. Fix an i {2, 3,..., n}. Since i T 2M and he RPB policy orders in i, i follows from he definiion of T 2M ha for all s { i, i + 1,..., i + m 1}, i holds ha y OP T i,s y OP T i,s y RP B i,s for all s { i, i + 1,..., T } since by (15, he difference of y OP T i,s when s > i + m 1. This shows y RP B i,s. This implies and y RP B i,s is fixed y OP T i+1, i y RP B i+1, i. (26 We prove (25 by conradicion. Assume ha, on he conrary, OPT does no place any order from i + 1 o i+1. By he definiion of rimmed on-hand invenory, we can see ha, for any policy P, y P i+1, i is always a lower bound of x P i+1 and y P i+1. Thus, since we have assumed ha OPT makes no order from i o i+1, we have y OP T i+1 (26, we have y OP T i+1 = y OP T i+1, i. Similarly, we have y RP B i+1, i x RP B i+1. Togeher wih = y OP T i+1, i y RP B i+1, i x RP B i+1, (27 which conradics o he fac ha i+1 T 2M all i = 2, 3,..., n. wih x RP B i+1 < y OP T i+1. This proves ha (25 holds for Finally, summing up (24 and (25 for all i = 2, 3,..., n proves (23. Q.E.D. Remark 1. A key sep in he proof of Lemma 3 is he consrucion of se T 2M defined in (20. This definiion is very differen from he original se T LS 2M defined in Levi and Shi (2013 for nonperishable invenory sysems, which only requires he relaionship beween he aggregae invenory posiions y OP T y RP B. This is because for non-perishable invenory sysems, he condiion on aggregae invenory levels is sufficien o esablish a similar resul as (23. Bu for perishable

12 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss D =2 O OPT =1 _ RPB O =3 ordered a -1 ordered a Figure 2 OPT _ RPB OPT _ RPB A counerexample wih m=2 o show he imporance of age disribuion in se decomposiion OPT _ RPB invenory sysems where he age informaion becomes criical, y OP T y RP B is no sufficien o guaranee he same ype of resul. For example, consider he following simples nonrivial seing wih m = 2, he se T 2M defined in (20 can be wrien as { T 2M = : p < 1 and x RP B < y OP T, y OP T } y RP B, and y OP T,+1 y RP B,+1. However, he se T LS 2M used in Levi and Shi (2013 is given by { T LS 2M = : p < 1 and x RP B < y OP T, y OP T } y RP B. Now consider he following scenario as depiced in Figure 2. In period, OPT has 3 invenory unis of age 1 and 3 new invenory unis of age 0; RPB has 5 invenory unis of age 1 and 2 new invenory unis of age 0. Suppose he demand is 2 in period and is saisfied using FIFO issuance. A he end of period, OPT has 1 oudaing uni and RPB has 3 oudaing unis. In period + 1, OPT (having 3 unis on-hand overakes RPB (having only 2 unis on-hand. Noe ha in his case, period saisfies he condiions in T LS 2M bu does no saisfy he las condiion in T 2M. Thus, hough he invenory level of RPB is higher in period, afer one period we would have y OP T,+1 > y RP B,+1 (because RPB has more oudaing invenory unis in period. The difficuly arises because OPT needs no o make any order in period + 1 o overake he invenory level of RPB. Therefore, i canno be argued ha OPT has placed more orders and he resul in Lemma 3 canno be esablished for he se T LS 2M. By he definiion of T 2H, he indicaor funcion 1( T 2H (i.e., wheher a period belongs o he se T 2H canno be deermined only by F (he informaion up o ime. Hence in order o compare he expeced coss beween RPB and OPT, we require he demand process o be associaed, a concep inroduced in Chao e al. (2015a ha embodies a noion of posiive correlaion beween fuure demands. Lemma 4 esablishes ha under ha condiion, in each period, he expeced

13 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 13 marginal holding and oudaing coss wih known informaion 1( T 1H T2H is in fac greaer han or equal o hose wihou knowing he informaion. Lemma 4. If he demand process D 1,..., D T is associaed, hen for each period = 1,..., T, [ E 1 [ E 1 ( T 1H T2H ( T 1H T2H (H RP B ] [ F E H RP B + Θ RP B ] + Θ RP B F F ]. (28 Proof. Condiioning on F, we have he following four cases (a p = 1, (b p < 1 and x RP B y OP T, (c p < 1, x RP B < y OP T and y RP B < y OP T, and (d p < 1, x RP B < y OP T y RP B. Firs, we noice ha P( T 1H T2H F akes value 0 or 1 if and only if eiher case (a or (b or (c happens; in such cases, i is sraighforward o verify ha (28 holds. Thus, in he remainder of his proof, we only focus our aenion on he non-rivial case (d, which implies ha 0 < P( T 1H T2H F = P( T 2H F < 1. According o he definiion of marginal holding and oudaing coss, i is clear ha condiioning on F, H RP B + Θ RP B is decreasing in fuure demands (D,..., D +m. If we can show ha 1 ( T 2H F is also decreasing in fuure demands (D,..., D +m, hen (28 follows since he demand process is associaed (see Essary e al. (1967. To show ha 1 ( T 2H F is decreasing in fuure demands (D,..., D +m, we firs define wha we call swiching evens, for s < ( + m 1 T, { A,s [Y RP B,s } Y OP T,s ] [Y RP B,s+1 < Y OP T,s+1 ]. Given case (d, by he definiion of T 2H, he even { T 2H } happens if and only if he swiching even A,s occurs for some s [, ( + m 1 T. We claim ha he occurrence of he swiching even A,s implies ha E RP B s > 0, i.e., RPB mus have some oudaing unis in period s. (To see his, A,s happens if OPT overakes he rimmed invenory level of RPB in period s + 1, which is impossible if no invenory unis in RPB expires in period as boh sysems face he same demands. The fac ha E RP B s > 0 implies ha no invenory unis ordered afer period s m + 1 are consumed by demand a he beginning of period s + 1. Hence, we mus have Y,s+1 RP B = q RP B [s m+2,] wih probabiliy 1. By he same argumen, we can see ha if E RP B s > 0 and Y RP B,s+1 = q RP B [s m+2,] wih probabiliy 1, hen Y RP B,s+1 < Y OP T,s+1. Thus, he swiching even A,s can be rewrien as { A,s = [Y RP B,s } Y OP T,s ] [E RP B s > 0] [q RP B [s m+2,] < Y OP T,s+1 ]. (29

14 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 14 Thus, we can rewrie he even as Since E RP B s { T 2H } = { (+m 1 T 1 s= [E RP B s > 0] [ q RP B [s m+2,] < Y OP T,s+1 ] }. (30 are he oudaing unis in period s (which were ordered in period s m + 1, i is clear ha given F, E RP B s is decreasing in he demand process afer, i.e., 1(E RP B s > 0 F is decreasing in (D, D +1,..., D T. Because he rimmed invenory level Y OP T,s+1 is decreasing in (D, D +1,..., D T, and q RP B [s m+2,] is already known deerminisically a ime, i follows ( ha 1 q RP B < Y OP T [s m+2,],s+1 F is also decreasing in (D, D +1,... D T. Given case (d, by (30, we conclude ha 1 ( T 2H F is indeed decreasing in fuure demands (D,..., D +m. Q.E.D. Remark 2. The main idea behind he proof of Lemma 4 is o apply he concep of associaed random variables. I can be shown ha 1( T 1H T2H is a decreasing funcion of fuure demands. Togeher wih he fac ha H RP B + Θ RP B is also a decreasing funcion of fuure demands, he resul hen follows from he properies of associaed sochasic processes. To (inuiively see why 1( T 1H T2H is a decreasing funcion of fuure demands, we consider he same example as shown in Figure 2. If d is equal o 4, hen we have y OP T,+1 = y RP B,+1 = 2 and in his case T 2M. We can also see ha if d < 4, hen falls in he se T 2H while if d 4, hen i falls in he se T 2M. This implies ha i is more likely o be in he se T 1H T2H when he demand is lower. Le Z RP B be a random variable defined by Z RP B { Λ, if p = 1; p K, oherwise, (31 where Λ mh+θ RP B E[H 2(m 1h+θ + Θ RP B F ] = E[Π RP B F ], and p is he ordering probabiliy. Lemma 5 below follows from he sandard argumens of condiional expecaion, he consrucion of RPB, and (31. We relegae is deailed proof o he elecronic companion. Lemma 5. Le C (RP B be he expeced oal cos incurred by he RPB policy. Then we have, [ C (RP B 3 + ] (m 2h E[Z RP B ]. mh + θ =1 Combining Lemmas 2, 3, 4, and 5, we are able o prove Theorem 1, and we again relegae is deailed proof o he elecronic companion.

15 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss Compuaional Experimens An imporan quesion is how well RPB performs empirically. In his secion we repor on numerical resuls. All compuaions were done using Malab R2014a on a deskop compuer wih an Inel(R Xeon(R CPU 3.20 Ghz. The purpose of our compuaional sudy is wo-fold. (a Under i.i.d. demands, we would like o es he empirical performance of RPB agains he heurisic policy proposed in Nahmias (1978 (denoed by NA. To he bes of our knowledge, NA is he only benchmark heurisic algorihm (wihou any wors-case performance guaranees available for perishable invenory conrol problems wih seup coss. This approximae (s, S-ype policy is repored o perform very close o opimaliy under i.i.d. demands, wih average error under 1% and maximum error of 3.7%. The quesion is wheher our RPB policy can achieve, if no beer, a leas a comparable performance o NA under i.i.d. demands. (b Under correlaed demands, since here are no benchmark heurisic algorihms repored in he exising lieraure, we can only compare he performance of RPB agains an opimal policy (denoed by OPT. Thus in our experimenal seing under correlaed demands we generae a sufficienly rich se of small problem samples, where he opimal policy can be evaluaed (using brue-force dynamic programming wih reasonable compuaional effor o provide a baseline for comparison. Following Levi and Shi (2013, he proposed RPB policies can be paramerized o obain a general class of policies, and he wors-case analysis discussed above can be viewed as choosing one parameer value ha achieves a wors-case performance guaranee for any problem insance (see he discussion a he end of 3.2. Alernaively, one can ry differen parameer values for a given problem insance, and idenify a parameer ha empirically performs he bes (in erms of average and/or wors-case performance for ha paricular insance. This gives rise o policies ha have a leas he same performance guaranees, bu are likely o perform beer empirically, as we refined he parameers according o he specific insance being solved Design of Experimens and Performance Merics We have conduced our experimens under boh an i.i.d. demand seing and a correlaed demand seing, using similar examples as in Chao e al. (2015b. (a Under he i.i.d. demand seing, following Nahmias (1976, 1977, we es wo demand disribuions, i.e., exponenial disribuion and Erlang-2 disribuion, boh wih mean 10. (b Under he correlaed demand seing, we adop he Markov modulaed demand process (MMDP wih hree saes of he economy. The MMDP is governed by he sae of he economy: poor (1, fair (2, and good (3. If he sae of he economy in period is i (i = 1, 2, 3, hen he demand in period is id, where D has mean 10 and follows

16 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 16 one of he following disribuions: exponenial, and Erlang-2. The sae of he economy follows a Markov chain wih ransiion probabiliies p 11 = 0.6, p 12 = 0.3, p 13 = 0.1, p 21 = 0.4, p 22 = 0.2, p 23 = 0.4, p 31 = 0.1, p 32 = 0.3, and p 33 = 0.6. This Markov chain is sochasically monoone (and hence he demand process is associaed. The parameers of our compuaional model are chosen as follows. The holding cos for all esing insances is normalized o h = 1, and he discoun facor is α = We se he planning horizon T = 20, he per-uni backlogging cos b {5, 10, 15}, he per-uni oudaing cos θ {5, 10, 15} and he seup cos K {5, 10, 15}. We also se he produc lifeime m = 2. (We noe ha Nahmias (1978 focused on m = 2 only and commened on he expensive compuaional overhead for m = 3. The sysem is iniially empy. We define he performance raio of our RPB policy agains a benchmark ( C (RP B policy P as r = 1 100%, where C (RP B is he cos of RPB and C (P is he cos C (P of policy P. The benchmark policy P is NA (proposed by Nahmias (1978 under i.i.d. demands whereas P is OPT (solved using brue-force dynamic program under correlaed demands Numerical Resuls Table EC.1 from he elecronic companion summarizes our numerical resuls for he i.i.d. demand seing. The performance raio is very small for all es insances, which suggess ha he performance of RPB is comparable (in fac slighly beer bu no saisically significan o ha of NA under i.i.d. demands. This also suggess boh RPB and NA perform exremely close o opimaliy (in ha NA s error is on average under 1% of he opimal soluion. We also noice ha boh algorihms perform beer when he seup cos is relaively small. Table EC.2 provides he oal cos breakdown for he i.i.d. demand seing (varying he values of K while fixing b = θ = 5. We observe ha he seup cos componen accouns for a significan porion of he oal coss, and he percenage of seup coss increases in K. The ordering frequency is around 70% o 80% wih m = 2, and decreases in K. Table EC.3 summarizes our numerical resuls for he correlaed demand seing. The performance raio is small for all es insances, wih average error under 7% and maximum error of 11.28%. This indicaes ha he RPB policy performs well, and i is significanly beer han he heoreical wors-case performance guaranee. Under boh he i.i.d. and correlaed demand seings, our numerical resuls sugges ha he performance of RPB is raher insensiive o b or θ, bu improves as K ges smaller. We also es he sabiliy of RPB he coefficien of variaion of he oal coss due o randomizaion. We consider he case wih exponenial demand wih mean 10, and evaluae he coss of RPB (wih randomized

17 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 17 decisions for he same cos parameers and demand realizaions. We find ha he coefficien of variaion of coss (sandard deviaion divided by mean cos is a mos 1.57%, suggesing ha he adverse effec due o randomizaion is raher small and RPB is quie sable. Supplemenal Maerial An elecronic companion o his paper is available a hp://or.journal.informs.org/. Acknowledgmens The auhors hank he area edior Chung-Piaw Teo, he anonymous associae edior, and wo referees for heir consrucive commens and suggesions, which helped improve boh he conen and he exposiion of his paper significanly. The research of Huanan Zhang and Cong Shi is parially suppored in par by NSF grans CMMI and CMMI The research of Xiuli Chao is parially suppored by NSF grans CMMI and CMMI References Box, G. E. P., G. M. Jenkins, G. C. Reinsel Time Series Analysis: Forecasing and Conrol. Wiley, Fourh Ediion. Chao, X., X. Gong, C. Shi, C. Yang, H. Zhang, S. X. Zhou. 2015a. invenory sysems wih posiive lead imes. // Approximaion algorihms for capaciaed perishable Working paper, Universiy of Michigan, Ann Arbor. Available a hp: Chao, X., X. Gong, C. Shi, H. Zhang. 2015b. Approximaion algorihms for perishable invenory sysems. Operaions Research 63( Essary, J. D., F. Proschan, D. W. Walkup Associaion of random variables, wih applicaions. The Annals of Mahemaical Saisics 38( Fries, B Opimal ordering policy for a perishable commodiy wih fixed lifeime. Operaional Research 23( Gallego, G., Ö. Özer Inegraing replenishmen decisions wih advance demand informaion. Managemen Science 47( Heah, D. C., P. L. Jackson Modeling he evoluion of demand forecass wih applicaion o safey sock analysis in producion/disribuion sysem. IIE Transacions 26( Karaesmen, I. Z., A. Scheller-Wolf, B. Deniz Managing perishable and aging invenories: Review and fuure research direcions. G. Karl Kempf, Pınar Keskinocak, Reha Uzsoy, eds., Planning Producion and Invenories in he Exended Enerprise: A Sae of he Ar Handbook, Volume 1. Inernaional Series in Operaions Research and Managemen Science, Vol. 151 (Springer, New York, Levi, R., G. Janakiraman, M. Nagarajan los-sales. Mahemaics of Operaions Research 33( Levi, R., M. Pál, R. O. Roundy, D. B. Shmoys Mahemaics of Operaions Research 32( A 2-approximaion algorihm for sochasic invenory conrol models wih Approximaion algorihms for sochasic invenory conrol models. Levi, R., C. Shi Approximaion algorihms for he sochasic lo-sizing problem wih order lead imes. Operaions Research 61( Lian, Z., L. Liu A discree-ime model for perishable invenory sysems. Annals of Operaions Research 87( Lian, Z., L. Liu, M. F. Neus A discree-ime model for common lifeime invenory sysems. Mahemaics of Operaions Research 30(

18 Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss 18 Nahmias, S Opimal ordering policies for perishable invenory-ii. Operaional Research 23( Nahmias, S Myopic approximaions for he perishable invenory problem. Managemen Science 22( Nahmias, S Higher order approximaions for he perishable invenory problem. Operaions Research 25( Nahmias, S The fixed charge perishable invenory problem. Operaions Research 26( Prasacos, G. P Blood invenory managemen: an overview of heory and pracice. Managemen Science Shi, C., H. Zhang, X. Chao, R. Levi Approximaion algorihms for capaciaed sochasic invenory sysems wih seup coss. Naval Research Logisics (NRL 61( Brief Bio: Huanan Zhang is a Ph.D. candidae in he Deparmen of Indusrial and Operaions Engineering a he Universiy of Michigan. His primary research ineres lies in sochasic opimizaion and online learning problems wih applicaions o invenory and supply chain managemen, and revenue managemen. Cong Shi is an assisan professor in he Deparmen of Indusrial and Operaions Engineering a he Universiy of Michigan. His research lies in sochasic opimizaion and online learning heory wih applicaions o invenory and supply chain managemen, and revenue managemen. He won firs prize in he 2009 George Nicholson Suden Paper Compeiion. Xiuli Chao is a professor in he Deparmen of Indusrial and Operaions Engineering a he Universiy of Michigan. His recen research ineress include sochasic modeling and analysis, invenory conrol, game applicaions in supply chains, and daa-driven opimizaion.

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20 ec2 e-companion o Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss Elecronic Companion o Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss by Zhang, Shi and Chao EC.1. Compuaional Complexiy of RPB We show ha he RPB policy is compuaionally very efficien wih compuaional complexiy O ( C m T for some posiive consan C, where m is produc lifeime and T is he lengh of he planning horizon. We also sress ha he RPB policy can be efficienly implemened in an online manner, i.e., he decision a any ime is compued based only on he curren observed sae of he sysem and does no depend on fuure decisions. This is a desired propery if one wishes o avoid he prohibiive compuaional burden of solving large dynamic programs. Nex we prove ha RPB has a compuaional complexiy O ( C m T. For each period = 1,..., T, he complexiy for evaluaing he marginal holding and oudaing coss is O ( C m since here are m layers of inegraion involved in he exac compuaion. For pracical purposes, evaluaing he expeced marginal coss using Mone Carlo simulaions (e.g., generaing sample pahs according o he join condiional demand disribuion can cu down he compuaional ime dramaically. This compuaional overhead is unavoidable as he complexiy is he same as compuing he single-period oudaing coss. This suggess ha even myopic policies ha only minimize he curren-single-period cos have o incur his compuaional overhead. Since he complexiy for carrying ou bisecion search is O ( log U (where U is an upper bound on he balancing quaniies, he algorihm runs in ime O ( C m T log U O ( C m T. In conras, compuing he exac opimal policy using dynamic programming is exponenial in he lengh of he planning horizon T. EC.2. Omied Proofs of Lemmas and Theorems Proof of Lemma 1. By he definiion of Π and he consrucion of ses T 1Π and T 2Π, we have Π RP B T 1Π T2Π = b (d y RP B + b T 1Π T2Π where he firs inequaliy holds since y OP T Proof of Lemma 5. Using he sandard argumens of condiional expecaions and by he consrucion of RPB, we have (d y OP T T 1Π T2Π + =1 Π OP T, y RP B when T 1Π T2Π. Q.E.D. C (RP B = =1 E[H RP B + Θ RP B + Π RP B + K 1(q RP B > 0]

21 e-companion o Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss ec3 = = = =1 =1 + E ( 3 + [ + E [ 3 + E [ E[H RP B + Θ RP B + Π RP B + K 1(q RP B > 0 F ] ] { [ [ (E[H RP B E E 1 ( T 1H T1Π + Θ RP B + Π RP B + K 1(q RP B > 0 ]] F ] [ [ E 1 ( T 2H T2M T2Π (E[H RP B + Θ RP B + Π RP B + K 1(q RP B > 0 ]]} F ] { [ [ ]] E E 1 ( T 1H T1Π Λ F ] (m 2h mh + θ =1 [ ]]} E 1 ( T 2H T2M T2Π p K F ] (m 2h E[Z RP B ], mh + θ =1 where he second equaliy holds because condiioning on F, 1( T 1H T1Π and 1( T 2H T2M T2Π are known; he inequaliy is valid due o he consrucion of RPB; and he las equaliy follows from he definiion of Z RP B. Q.E.D. Proof of Theorem 1. C (OP T = E [ H OP T Denoe he expeced oal cos of OPT by C (OP T, hen we have + Θ OP T [ mh + θ E 2(m 1h + θ =1 +1 ( T 1Π T2Π [ [ mh + θ E E + Π OP T + [ 1 Π RP B =1 = 2(m 1h + θ =1 +1 ( T 1Π T2Π Π RP B [ mh + θ E 2(m 1h + θ P( T 1H =1 + P ( T 1Π T2Π F = =1 E[Z RP B ] (m 2h mh+θ K 1 ( q OP T > 0 ] ] ( T 1H T2H (H RP B + Θ RP B ] + K [1( T 2M 1(q RP B > 0] [ ] 1 ( T 1H T2H (H RP B + Θ RP B + K [1( T 2M 1(q RP B > 0] F ]] E [ Π RP B T2H F E [ H RP B C (RP B. + Θ RP B F ] F ] + P ( T2M F E [ K 1(q RP B > 0 F ] ] The firs inequaliy follows from Lemmas 2 and 3; and he las inequaliy follows from Lemma 5. Now i remains o check he validiy of he second inequaliy. Lemma 4 implies ha [ ] [ E 1 ( T 1H T2H (H RP B + Θ RP B F P ( T 1H T2H F E H RP B + Θ RP B F ], which gives us he desired inequaliy for he holding and oudaing cos par.

22 ec4 e-companion o Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss By condiioning on he informaion se F, he indicaor funcion 1( T 1Π T2Π becomes known. Hence we can wrie [ ] [ E 1 ( T 1Π T2Π Π RP B F = P ( T 1Π T2Π F E Π RP B F ]. Moreover, given he informaion se F, he seup cos K 1(q RP B he RPB policy. Thus, we have > 0 is fully deermined by E [ K [1( T 2M 1(q RP B > 0] F ] = P ( T2M F E [ K 1(q RP B > 0 F ]. Combining he above argumens, we have shown ha he second inequaliy holds as well. Q.E.D. EC.3. Numerical Resuls K=5 K=10 K=15 K=5 K=10 K=15 θ b C (RPB C (NA r C (RPB C (NA r C (RPB C (NA r Erlang-2 Demand Disribuion % % % % % % % % % % % % % % % % % % % % % % % % % % % Exponenial Demand Disribuion % % % % % % % % % % % % % % % % % % % % % % % % % % % Table EC.1 Compuaional Performance of RPB and NA under IID Demands (h = 1, α = 0.95

23 e-companion o Zhang, Shi and Chao: Approximaion Algorihms for Perishable Invenory Sysems wih Seup Coss ec5 Erlang-2 Demand Disribuion Exponenial Demand Disribuion K E(H E(Π E(Θ Seup Coss % 45.7% 13.1% 20.9% % 43.4% 9.9% 30.9% % 41.9% 9.6% 32.6% % 51.8% 16.0% 15.7% % 45.1% 14.4% 25.3% % 43.0% 12.7% 31.4% Table EC.2 Toal Cos Breakdown under IID Demands (h = 1, α = 0.95, b = θ = 5 K=5 K=10 K=15 K=5 K=10 K=15 θ b C (RPB C (OPT r C (RPB C (OPT r C (RPB C (OPT r Erlang-2 Demand Disribuion % % % % % % % % % % % % % % % % % % % % % % % % % % % Exponenial Demand Disribuion % % % % % % % % % % % % % % % % % % % % % % % % % % % Table EC.3 Compuaional Performance of RPB and OPT under MMDP Demands (h = 1, α = 0.95