Asymptotic Optimality of Order-up-to Policies in Lost Sales Inventory Systems

Transcription

1 Asympoic Opimaliy of Order-up-o Policies in Los Sales Invenory Sysems Woonghee Tim Huh 1, Ganesh Janakiraman 2, John A. Mucksad 3, Paa Rusmevichienong 4 December 4, 2006 Subjec Classificaion: Invenory/Producion: Los Sales, Backorders, Finie and Infinie Horizon, Opimal Coss Absrac We sudy a single-produc single-locaion invenory sysem under periodic review, where excess demand is los and he replenishmen lead ime is posiive. The performance measure of ineres is he long run average holding cos and los sales penaly. For a large class of demand disribuions, we show ha when he los sales penaly becomes large compared o he holding cos, he relaive difference beween he cos of he opimal policy and he bes order-up-o policy converges o zero. For any given cos parameers, we esablish a bound on his relaive difference. Numerical experimens show ha he bes order-up-o policy performs well, yielding an average cos ha is wihin 1.5% of he opimal cos even when he raio beween he los sales penaly and he holding cos is jus Deparmen of Indusrial Engineering and Operaions Research, Columbia Universiy, New York, NY 10027, USA. huh@ieor.columbia.edu. 2 IOMS-OM Group, Sern School of Business, New York Universiy, 44 W. 4h Sree, Room 8-71, New York, NY gjanakir@sern.nyu.edu 3 School of Operaions Research and Indusrial Engineering, Cornell Universiy, Ihaca, NY 14853, USA. jack@orie.cornell.edu 4 School of Operaions Research and Indusrial Engineering, Cornell Universiy, Ihaca, NY 14853, USA. paarus@cornell.edu 1

2 1. Inroducion We sudy he opimal replenishmen policy in a periodic-review single-sage invenory sysem ha procures invenory from a source wih ample supply. There is a replenishmen lead ime of τ periods beween placing an order and is delivery. Demands in differen periods are independen and idenically disribued. In he even ha demand in a period exceeds he available on-hand invenory, excess demand is los and we incur a los sales penaly cos of $b per uni. We also charge holding coss on invenory on hand a he end of each period a he rae of $h per uni per period. We wish o find an ordering policy ha minimizes he long run average holding cos and los sales penaly. When demand is backordered insead of los, Karlin and Scarf (1958) show ha an order-up-o policy is opimal; ha is, here is an order-up-o-level o which we raise he invenory posiion defined as invenory available for immediae sales plus he amoun of invenory ha has been ordered and no ye delivered in each period. They also show ha his simple policy fails o be opimal in he los sales model. In many business environmens, los sales occur frequenly when cusomer demands are no me immediaely. Hence, finding he opimal replenishmen policy, characerizing is srucural properies, and developing heurisics ha work well in pracical seings are imporan. Moreover, in many imporan invenory sysems, we observe ha he los sales penaly b is generally much larger han he holding cos h, as shown in he following examples from reail and service pars environmens. In his paper, we propose simple invenory policies ha are guaraneed o perform well in such sysems. Reail: Consider a produc whose procuremen cos is $1 per uni o he reailer. Assume he reailer reviews he sysem and replenishes is invenory once a week, and sells he produc a $(1 + m) per uni, where m represens he mark-up. The los sales cos in his case is a leas $m per uni, no including any loss in cusomer goodwill due o unfulfilled demand. The holding cos per uni per period is simply he cos of holding $1 in invenory for a week. A a cos of capial of 15% per year, his is approximaely $ per uni per period. So, he raio beween he los sales penaly cos and he holding cos in his example is a leas 400m. A a 25% mark-up, which is quie common in many reail environmens, his raio is a leas

3 Service Pars Mainenance: Consider, for example, he business of mainaining service pars for personal compuers, phoocopiers, or elecommunicaion equipmens. Mos corporae cliens purchase service-level agreemens ha require he manufacurer, in he even of a failure, o bring he equipmen back o service wihin a specified ime window wihin sipulaed minimum probabiliies (for example, wihin 2 hours for 95% of failures and wihin 6 hours for 99% of failures). To mee hese agreemens, he equipmen manufacurers frequenly expedie service pars o cusomer locaions when he closes socking locaions do no have he necessary pars. Consider a $100 par ha has o be expedied a an addiional cos of $14. These sysems are ypically reviewed once a day. Assuming a cos of capial of 25% per year, he cos of holding his par in invenory for one day is abou $0.07. Here, he raio beween he los sales penaly cos in his case, he expediing premium of $14 and he holding cos is 200. Our main resul is ha, under mild assumpions on he demand disribuion, he class of orderup-o policies is asympoically opimal for hese sysems as he los sales penaly increases. In fac, we show asympoic opimaliy for a specific order-up-o policy ha is compued using he newsvendor formula wih appropriae parameers. For any given cos parameers, we also esablish an upper bound on he increase in he oal cos from using his specific order-up-o policy insead of he opimal policy. Finally, we presen several compuaional resuls o evaluae he performance of he opimal policy, he bes order-up-o policy, and he specific order-up-o policy menioned above, for a wide range of demand disribuions and cos parameers. 1.1 Noaion and Problem Descripion To faciliae he discussion of our main resuls, le us inroduce he noaion and he problem descripion. We will consider boh he los sales and he backorder sysems. In boh sysems, he lead ime beween he placemen of a replenishmen order and is delivery is denoed by τ. The index for ime periods is and D is he demand in period. We assume D 1, D 2,... are independen and idenically disribued random variables and we use D o denoe a generic random variable wih he same disribuion as D. Also, le D = τ+1 =1 D denoe he oal demand over τ + 1 periods, represening he oal demand over he lead ime including he period when we place he order. Le F denoe he disribuion funcion of D. 3

4 X L A he beginning of period, he replenishmen order placed in period τ is received. Le [0, ) denoe he invenory on hand a his insan in he los sales sysem. For he backorder sysem, le X B (, ) denoe he ne-invenory in period, ha is, he invenory on hand minus backorders a he insan afer receiving he delivery due in period. Afer receiving deliveries, a new replenishmen order is placed afer which he demand D is observed. For any h 0 and b 0, we denoe by L(h, b) he los sales sysem and by B(h, b) he backorder sysem, which is idenical o L(h, b) excep ha excess demand is backordered. In boh he los sales L(h, b) and backorder B(h, b) sysems, we charge holding coss on invenory on hand a he end of each period a he rae of $h per uni per period. While we incur los sales penaly of $b per uni of unme demand in he los sales model L(h, b), he shorage coss in he backorder sysems B(h, b) are charged a he rae of $b per uni of backordered demand per period. Given he holding cos h and los sales penaly b, we denoe by C L,S (h, b) and C L (h, b) he long run average cos in he los sales sysem L(h, b) under an order-up-o-s policy and under an opimal policy, respecively. The corresponding quaniies C B,S (h, b), and C B (h, b) are defined similarly, wih he inerpreaion of b as he backorder cos per uni per period. We denoe by S L (h, b) and S B (h, b) he bes order-up-o levels in he los sales L(h, b) and he backorder B(h, b) sysems, respecively. We noe ha in he backorder sysem B(h, b), order-up-o policies are opimal, and he bes order-up-o level is given by he newsvendor formula under he disribuion funcion F of D. 1.2 Conribuions and Organizaion of he Paper Our analysis provides some imporan insighs abou boh los sales and backorder invenory sysems, in addiion o he main resul on he asympoic opimaliy of order-up-o policies in los sales sysems. We now describe he organizaion of he paper and discuss he main conribuions of he individual secions. Table 1 provides a summary of he main resuls. In Secion 2, we provide a brief lieraure review and indicae how our research conribues o he curren research on los sales sysems. In Secion 3, we describe he assumpion on he demand disribuion ha we will use hroughou he paper. We hen show in Theorem 4 ha our assumpion encompasses a broad class of demand disribuions ha commonly occur in many invenory seings. 4

5 Caegory Descripion Resuls Backorder Robusness of Opimal For any ν > 0, lim b C B (h,νb) C B (h,b) Sysem Cos and Newsvendor lim b C B,S νb(h,b) C B (h,b) = 1, and = lim b C B,S b(h,νb) C B (h,νb) = 1, (Secion 4) Soluion (Theorem 6) where S b = S B (h, b) and S νb = S B (h, νb) Asympoic Equivalence Connecions of he Opimal Coss lim b C B (h,b) C L (h,b) = 1 Beween (Theorem 8) Los Sales Bounds on he Cos For any S, and of Any Order-up-o C B,S (h, b/(τ + 1)) C L,S (h, b) C B,S (h, b + τh) Backorder Policy (Lemma 13) Sysems Bounds on he Bes (Secion 5) Order-up-o Levels S B (2h(τ + 1), b h(τ + 1)) S L (h, b) S B (h, b + τh) (Theorem 14) Asympoic Opimaliy Main Resuls of Order-up-o Policies lim b min S C L,S (h,b) C L (h,b) (Secion 6) in Los Sales Sysems where S b+τh = S B (h, b + τh). (Theorem 15) = lim b C L,S b+τh(h,b) C L (h,b) = 1, Table 1: A summary of he main resuls in he paper. All asympoic resuls assume ha he disribuion of he demand over lead ime saisfies Assumpion 1, which is discussed in deail in Secion 3. Robusness of he Opimal Cos and he Opimal Policy in Backorder Sysems (Secion 4): As our firs conribuion, we show ha he opimal cos in he backorder sysem is robus agains changes in he backorder cos parameer b for large b. More precisely, he increase in oal cos resuling from incorrecly esimaing b becomes negligible for large b; ha is, for any h 0 and ν > 0, C B,S νb(h, b) C B (h, νb) lim b C B = lim (h, b) b C B (h, b) = lim C B,S b(h, νb) b C B (h, νb) = 1, where S b = S B (h, b) and S νb = S B (h, νb) denoe he opimal order-up-o levels in he backorder sysems B(h, b) and B(h, νb), respecively. Esimaing he backorder cos can be difficul in many applicaions because we have o assess he long-erm impac of a sockou and accoun for losses in cusomer goodwill from delays in order 5

6 fulfillmen. Suppose we misakenly esimae he backorder parameer o be νb (insead of b) and use he order-up-o-s νb policy in he B(h, b) sysems. The above resul shows ha he relaive increase in he oal cos from using such a policy converges o zero as b increases. We hus make precise he folk heorem ha he cos funcion in a ypical invenory conrol problem is fla around he opimal soluion. Ineresingly, he above resul holds only for demand disribuions saisfying Assumpion 1 (see Secion 3 for more deails). In Secion 4.1, we provide a counerexample ha does no saisfy Assumpion 1 and where he above resul fails. Connecions Beween Los Sales and Backorder Sysems (Secion 5): As our second conribuion, we also esablish relaionships beween los sales and backorder sysems, in erms of he opimal cos, he cos of any order-up-o policy, and he bes order-up-o level. In Theorem 8 in Secion 5, we show he asympoic equivalence beween he opimal cos in he los sales and he backorder sysems as he parameer b increases; ha is, for any h 0, C B (h, b) lim b C L (h, b) = 1. When he parameer b is large, his resul enables us o use he (easily compued) opimal cos of he backorder sysem B(h, b) as an approximaion for he opimal cos in he corresponding los sales sysem. In addiion o asympoic equivalence of he opimal coss, he long run average cos of any order-up-o policy in he los sales L(h, b) sysem is bounded above and below by he cos of he same policy in he backorder sysems B(h, b + τh) and B(h, b/(τ + 1)), respecively. Lemma 13 shows ha for any order-up-o level S, C B,S (h, b/(τ + 1)) C L,S (h, b) C B,S (h, b + τh). We also develop bounds on he bes order-up-o level in he los sales sysem, as shown in Theorem 14, ha S B (2h(τ + 1), b h(τ + 1)) S L (h, b) S B (h, b + τh). The above bounds represen he firs such resuls ha relae he cos of any order-up-o policy and he bes order-up-o level in he los sales sysem wih he corresponding quaniies in he wellsudied backorder sysem. These bounds are easily compuable since hey are simply represened by newsvendor formulas. Main Resuls (Secion 6): The resuls from Secion 4 and 5 se he sage for he main resul of he paper (Theorem 15): order-up-o policies are asympoically opimal in he los sales sysem, 6

7 or min S C L,S (h, b) C L,Sb+τh(h, b) lim b C L = lim (h, b) b C L = 1, (h, b) where S b+τh = S B (h, b + τh). The above resul shows ha, in fac, he opimal order-up-o level for he backorder sysem B(h, b + τh) is asympoically opimal for he los sales sysem L(h, b). Theorem 15 also provides an explici and compuable bound on he rae of convergence for any finie value of b. Compuaional Invesigaion (Secion 7): In addiion o esablishing asympoic opimaliy of base sock policies in los sales invenory sysems, we also discuss exensive compuaional invesigaion. In our experimen, we indicae he performance of base sock policies under differen problem parameers. We show how order-up-o policies perform agains oher replenishmen heurisics (Secion 7.2), deermine he impac of increasing demand on oal cos (Secion 7.3), and sudy how well order-up-o policies would do when he demand exhibis high variance-o-mean raios (Secion 7.4). Our compuaional resuls show ha he cos of he bes order-up-o policy is wihin 1.5% of he opimal cos even when he raio beween he los sales penaly and he holding cos is jus 100. Moreover, our order-up-o policy coninues o perform well even for demands wih large means or high variance. This resul suggess ha such a policy should perform well in many pracical invenory applicaions. 2. Brief Lieraure Review There are hree main sreams of research on los sales invenory sysems: he analysis of he opimal invenory policy, he analysis of hese sysems under an arbirary policy or under policies of specific kinds, and he compuaional invesigaion of he performance of easily implemenable heurisics. We now briefly review hese hree research sreams in ha order. Karlin and Scarf (1958) firs sudy he los sales invenory sysem wih a lead ime of one period. They demonsrae ha order-up-o policies are no opimal for hese sysems; he opimal order quaniy is a decreasing funcion of he amoun of invenory on hand, wih he rae of decrease being smaller han one. For he general lead ime case, hey analyze he sysem under order-upo policies and exponenially disribued demands, and derive an expression for he seady sae disribuion of on-hand invenory level. Moron (1969) exends Karlin and Scarf s resuls on he opimal ordering policy o he general lead ime case. He also derives upper and lower bounds on he opimal order quaniy in a period as funcions of he sae vecor. Recenly, Zipkin (2006b) 7

8 presens an elegan derivaion of Moron s srucural resuls and exends he resuls o more general los sales invenory sysems (for example, allowing capaciy resricions). Levi e al. (2005) develop a heurisic based on he dual balancing echnique inroduced originally for backorder sysems by Levi e al. (2004). They show ha his heurisic aains an expeced cos per period ha is a mos wice ha achieved by an opimal policy for a large class of demand models. Janakiraman e al. (2005) show analyically ha he opimal cos of managing a los sales invenory sysem is smaller han ha of managing a backorder sysem when he backorder cos parameer in he laer sysem has he same magniude as he los sales cos parameer in he former sysem. Under varying assumpions, Karush (1957), Downs e al. (2001) and Janakiraman and Roundy (2004) all consider los sales invenory sysems under order-up-o policies and show he convexiy of he expeced cos per period wih respec o he order-up-o level. Reiman (2004) sudies he class of order-up-o policies and he class of consan order policies (policies ha order a consan quaniy every period regardless of he sae of he sysem). He derives expressions for he order-up-o level and for he consan order-quaniy ha are asympoically opimal wihin he respecive classes of policies as he penaly cos becomes large. He also invesigaes he comparaive performance of he wo policies as he lead ime grows. Moron (1971) compuaionally invesigaes he performance of he myopic policy as a heurisic for problems wih a lead ime of one or wo periods. For los sales problems wih addiional feaures (for example, a se-up cos), Nahmias (1979) derives an inuiively appealing heurisic and invesigaes is performance for he cases of one and wo period lead imes. Recenly, for problems wih lead imes ranging from one o four periods, Zipkin (2006a) invesigaes he performance of he opimal order-up-o policies, he myopic policy, a modified myopic policy ha is based on he coss incurred in wo periods, he dual balancing policy, a generalizaion of base-sock policies suggesed by Moron (1971), and he opimal consan-order policy. To reduce he compuaional effor involved in evaluaing each of hese policies, he presens elegan analyical bounds on he size of he effecive sae space under any given policy. Our paper has elemens of all hree research sreams. Our asympoic opimaliy resuls conribue o an undersanding of he srucure of he opimal policy by esablishing condiions under which he opimal cos is asympoically equal o he cos of he bes base sock policy. Our bounds on he performance of a specific order-up-o policy and he analysis leading o such bounds illuminae he srucural properies of base sock policies and esablish connecions beween los sales 8

9 and backorder invenory sysems. Finally, we complemen Zipkin (2006a) by invesigaing he compuaional performance of order-up-o policies over a larger class of problem insances, especially when he los sales penaly is significanly higher han he holding cos. We show ha when he raio b/h is large, order-up-o policies perform exremely well, wih an average cos ha is wihin 1.5% of he opimal. 3. Assumpion on he Demand Disribuion Recall ha D τ+1 =1 D denoes he oal demand over τ + 1 periods, represening he oal demand over he lead ime including he period when we place he order. Le F denoe he disribuion funcion of D. For any 0, we define he mean residual life m D () as follows: E [ D D > ], if < sup {x : F (x) < 1}, m D () = 0, oherwise. Through ou his paper, we make he following assumpion on he disribuion of D. Assumpion 1. lim m D ()/ = 0. The above assumpion implies ha he expeced mean residual life of D a does no grow faser han. Before proceeding wih examples of demand disribuions saisfying Assumpion 1, le us recall he following definiion. Definiion 2. A coninuous (resp. discree) random variable Y wih a disribuion funcion F and a densiy funcion f (resp. probabiliy mass funcion p) has an increasing failure rae (IFR) propery if f(x)/(1 F (x)) (resp. p(x)/(1 F (x))) is nondecreasing in x. The following resul provides an equivalen characerizaion of an IF R random variable. The proof appears in Secion 1.B.1 of Shaked and Shanhikumar (1994). Lemma 3. A random variable Y is IFR if and only if for any 0 1 < 2, he residual life of Y a 2 is sochasically smaller han he residual life of Y a 1 ; ha is, for any s 0, P { Y 2 > s } { Y > 2 P Y 1 > s } Y > 1. The following heorem, whose proof appears in Appendix A, shows ha Assumpion 1 encompasses a large class of discree and coninuous demand disribuions ha commonly occur in many invenory sysems. 9

10 Theorem 4. If any of he following condiions holds, hen D saisfies Assumpion 1. (a) The demand D in each period (eiher discree or coninuous) is bounded; ha is, here exiss M such ha P {D M} = 1. (b) The demand D in each period (eiher discree or coninuous) has an increasing failure rae (IFR) disribuion. (c) D has a finie variance and he disribuion F of D has a densiy funcion f and a failure rae funcion r() of F ha does no decrease o zero faser han 1/; ha is, where for any 0, r() = f()/(1 F ()). lim r() =, The above heorem shows ha Assumpion 1 encompasses a very large of demand disribuions used in many supply chain models, including any bounded demand random variables. For unbounded demand, par (b) of Theorem 4 shows ha many commonly used disribuions also saisfy Assumpion 1. Examples include geomeric disribuions, Poisson disribuions (see Corollary 5.2 in Ross e al. (2005)), negaive binomial disribuions wih parameer r > 0 and 0 < p < 1, exponenial disribuions, and Gaussian disribuions. When he demand disribuion does no exhibi an IFR propery, par (c) of he above heorem shows ha Assumpion 1 remains saisfied as long as he failure rae does no decrease o zero oo quickly. The following example shows a disribuion ha is no IFR, ye sill saisfies par (c) of Theorem 4. Example 1. Suppose ha D follows a Weibull disribuion wih scale parameer λ > 0 and shape parameer 0 < k < 1. Then, D has he following disribuion, densiy, and failure rae funcions: for any x > 0, f(x) = kxk 1 e (x/λ)k λ k, F (x) = 1 e (x/λ)k, and r(x) = k ( x ) k 1, λ λ and he firs wo momens of D are ( E [D] = λγ ) k and E [ D 2] ( = λ 2 Γ ), k where Γ( ) denoes he Gamma funcion. Since 0 < k < 1, i is easy o verify ha D is no IFR, bu he failure rae funcion r sill saisfies par (c) of Theorem 4. 10

11 4. Asympoic Properies for Backordered Sysems We sar our analysis by showing asympoic properies of he opimal policy in he backorder sysem as he backordere cos parameer b becomes large. To faciliae our discussion, le us inroduce he following noaion. For any y 0, le ψ(y; h, b) denoe he raio beween he expeced backorder and holding coss given he invenory posiion y in he B(h, b) sysem, ha is, ψ(y; h, b) = be [ (D y) +] he [ (y D) +]. The following lemma esablishes upper and lower bounds on he increase in he oal cos from using sub-opimal policies. Lemma 5. For any h 0, b 0, and ν > 0, le S b = S B (h, b) and S νb = S B (h, νb) denoe he opimal policies in he B(h, b) and B(h, νb) backorder sysems, respecively. Then, he relaive difference beween he opimal coss of B(h, b) and B(h, νb) backorder sysems can be bounded as follows: 1 + ψ (S νb ; h, νb) 1 + (1/ν)ψ (S νb ; h, νb) = CB (h, νb) C B,S CB (h, νb) νb (h, b) C B (h, b) CB,Sb(h, νb) C B = 1 + νψ (S b; h, b) (h, b) 1 + ψ (S b ; h, b) Proof. The firs and second inequaliies follows from he fac ha C B (h, b) C B,S νb(h, b) and C B (h, νb) C B,S b(h, νb), respecively. To esablish he firs equaliy, noe ha C B (h, νb) C B,S = νbe [D S νb] + + he [S νb D] + νb (h, b) be [D S νb ] + + he [S νb D] + = 1 + ψ (S νb; h, νb) 1 + (1/ν)ψ (S νb ; h, νb), where he las inequaliy follows from dividing he numeraor and denominaor by he [S νb D] +. The proof of he second equaliy of he lemma is similar. The bounds in Lemma 5 lead direcly o he main asympoic resul of his secion, which is saed in he following heorem. Theorem 6. Under Assumpion 1, he following resuls hold for any h 0. (a) The raio beween he expeced backorder cos per period o he expeced holding cos per period under he opimal policy converges o zero as he backordere cos b increases, ha is, lim ψ ( S B (h, b); h, b ) = 0. b 11

12 (b) For large values of b, he opimal cos and he opimal policy are robus agains changes in he backorder cos; ha is, for any ν > 0, C B,S νb(h, b) C B (h, νb) lim b C B = lim (h, b) b C B (h, b) = lim C B,S b(h, νb) b C B (h, νb) = 1, where S b = S B (h, b) and S νb = S B (h, νb). Proof. To esablish he resul in par (a), noe ha he opimal order-up-o level S B (h, b) is given by he newsvendor formula: S B (h, b) = inf { y : P {D y} b }, b + h which implies ha P { D S B (h, b) } b/(b + h) and P { D > S B (h, b) } h/(b + h) by he lef coninuiy of he disribuion funcion. Therefore, ψ ( S B (h, b); h, b ) = b P { D > S B (h, b) } E [ D S B (h, b) D > S B (h, b) ] h P {D S B (h, b)} E [ S B (h, b) D D S B (h, b) ] E [ D S B (h, b) D > S B (h, b) ] E [ S B (h, b) D D S B (h, b) ] = ( ( md S B (h, b) ) ) ( S B (h, b) S B (h, b) S B (h, b) E [ D D S B (h, b) ] Since E [ D D S B (h, b) ] E [D], he desired resul follows from Assumpion 1 and he definiion of S B (h, b). To prove par (b), noe ha i follows from par (a) and he bounds in Lemma 5 ha lim b C B (h, νb) C B (h, νb) C B,S = lim νb (h, b) b C B (h, b) = lim C B,S b(h, νb) b C B = 1, (h, b) and he desired resul follows from he fac ha ). C B,S νb(h, b) C B (h, b) = CB (h, νb)/c B (h, b) C B (h, νb)/c B,S νb (h, b) and CB,Sb(h, νb) C B (h, νb) = CB,Sb(h, νb)/c B (h, b) C B (h, νb)/c B (h, b). Theorem 6 shows ha, for disribuions saisfying Assumpion 1, he newsvendor soluion and he opimal cos are robus agains inaccurae esimaion of he backorder parameer when he backorder parameer b is large. However, when Assumpion 1 fails, he resul of Theorem 6 many no longer hold, as we now demonsrae. 12

13 4.1 Pareo Disribuions: An Example Where Theorem 6 Fails For any θ > 1, le he densiy funcion f θ be defined by: for any x 0, f θ (x) = θ (1 + x) 1+θ, and le F θ (x) = x 0 f θ(u)du denoe he corresponding disribuion funcion. The following proposiion shows ha F θ does no saisfy Assumpion 1 and he opimal cos is sensiive o he backorder parameer, even for large values of b. Proposiion 7. For any θ > 1, b 0, h 0, and ν > 0, if D has a disribuion funcion F θ, hen Proof. Please see Appendix B. m D () lim = 1 θ 1 and C B (h, νb) lim b C B (h, b) = ν1/θ. 5. Connecions beween Los Sales and Backorder Invenory Sysems The following resul provides an inuiive basis for conjecuring ha he opimal policy in he backorder sysem is also asympoically opimal in he los sales sysem. Theorem 8. Under Assumpion 1, as b increases, he opimal cos in he los sales L(h, b) sysem converges o he opimal cos in he backorder sysem B(h, b); ha is, for any h 0, C L (h, b) lim b C B (h, b) = 1. Proof. Janakiraman e al. (2005) esablished he following bounds on he opimal cos in he los sales sysem: C B (h, b/(τ + 1)) C L (h, b) C B (h, b). From Theorem 6(b), lim b C B (h, b)/c B (h, b/(τ + 1)) = 1, which gives he desired resul. Nex, we esablish connecions beween he dynamics in boh he los sales and he backorder sysems under he same order-up-o policy. Le X L,S and X B,S denoe he on-hand invenory in he los sales sysem and he ne invenory in he backorder sysem, respecively, a he beginning of period under an order-up-o-s policy. Similarly, we use and BACK B,S o denoe he los sales incurred in period and he backorders exised a he end of period, respecively, under he order-up-o-s policy. By definiion, X L,S, and BACK B,S are non-negaive random variables. The following lemma esablishes he relaionship among hese random variables. 13

14 Lemma 9. Assume boh he los sales and he backorder sysems sar a he same sae in period 1 and he invenory posiion in his sae is S or less. Then, for every demand sample pah and for every τ + 1, X B,S X L,S and BACK B,S 1 i= τ i BACK B,S. Proof. I follows from he dynamics of los sales sysems under he order-up-o-s policy (see Janakiraman and Roundy (2004)) ha X L,S = S 1 i= τ D i + 1 i= τ i = X B,S + 1 i= τ i, for any τ + 1, which proves he firs par of he lemma. Since x + y (x y) + for any x R and y 0, we have BACK B,S 1 i= τ i = = ( ( D X B,S D X L,S ) + 1 i= τ ) + = LOST L,S where he las inequaliy follows from he fac ha X B,S i ( X L,S. ( D X B,S D X B,S 1 i= τ ) + = BACK B,S, i ) + The resul of Lemma 9 relaes he ne invenory in he backorder sysem wih he on-hand invenory in he los sales sysem for any finie ime period. To use his resul for sudying he long run average cos in L(h, b), his resul should be exended o he seady-sae on-hand invenory, which we will denoe by X L,S. This is our nex sep. Before proving properies of X L,S well known (and rivial o verify) ha X B,S and X B,S, i is imporan o esablish heir exisence. I is exiss and τ D. X B,S d S However, in general, for any given saring sae and order-up-o level S, i is no rue ha he disribuion of X L,S =1 converges o a saionary disribuion. In fac, Huh e al. (2006) give such an example. Ineresingly, we are able o show wo resuls ha help us resolve his difficuly. Lemma 10. For every S and any saring sae in period 1, he sequence of he expeced cos per period over he inerval [1, T ] given by T =1 E[h (XL,S D ) + + b (D X L,S ) + ] T converges o a limi ha is independen of he saring sae. 14

15 Proof. Please see Appendix C. Since he long run average cos is he quaniy of ineres o us in his paper, Lemma 10 implies ha we can limi our analysis o any specific saring sae. In he nex lemma, we show ha for a specific saring sae we choose, he saionary disribuion of he on hand invenory, {X L,S }, exiss. Lemma 11. Assume he saring sae (in period 1) is such ha here are S/(τ + 1) unis on hand and S/(τ + 1) unis due o be delivered in each of he periods 2,..., τ. Then, he sequence of he disribuions of he random variables {X L,S } converges. Proof. Please see Appendix D. We will use X L,S o denoe a random variable whose disribuion is he limiing disribuion from Lemma 11. We can now define C L,S (h, b) mahemaically as follows: [ (X C L,S L,S (h, b) = he D ) ] + + be [ (D ) X L,S + ], where he random variable D denoes he demand in a single period. Corollary 12. The random variable X B,S and he random variable i.e. for any z 0, P { X B,S is sochasically smaller han he random variable X L,S is sochasically smaller han he random variable BACK B,S, > z } P { X L,S > z } and P { LOST L,S > z } P { BACK B,S > z }. Proof. Noice ha we assumed ha X L,S represens he limiing disribuion of X L,S when he saring sae vecor has S/(τ + 1) unis in each componen. So, his saring sae saisfies he assumpion of Lemma 9. The resul follows direcly from his lemma. The nex resul esablishes upper and lower bounds on he cos of any order-up-o policy in he los sales sysem in erms of he coss in he backorder sysem. Lemma 13. The long run average cos of he order-up-o-s policy in he los sales sysem L(h, b) is bounded above (resp. below) by he cos of he same policy in he backorder sysem B(h, b + τh) (resp. B(h, b/(τ + 1))), i.e. C B,S (h, b/(τ + 1)) C L,S (h, b) C B,S (h, b + τh). 15

16 Proof. Recall ha he random variable D denoes he oal demand over τ + 1 periods and X L,S = S 1 i= τ D i + 1 i= τ [ ( ) ] + he X L,S D + be [ = he X L,S = he = he [ [ D ] + he S S ( = he S 1 i= τ i= τ i= τ D i + L,S LOSTi. Then, for any, we have [ ( ) ] + D X L,S [ ( ) ] + D X L,S + be ] 1 i= τ D i ] + h = he [ (S D) +] [ he Since he sochasic process i i= τ [ E D i i= τ [ ( ) ] + D X L,S [ + he ] [ + be ) + ( ) + D i he D i S + h BACK B,S ] + h { } X L,S : 1 i= τ [ E C L,S (h, b) = he [ (S D) +] he [ BACK B,S i ] [ + be ] i= τ [ E ] [ + be converges o X L,S, we have i ] ] [ + be ]. ] [ ] + (b + (τ + 1)h)E LOST L,S he [ (S D) +] + (b + τh)e [ BACK B,S ] = C B,S (h, b + τh), [ where he inequaliy follows from Corollary 12 which implies ha E This esablishes he upper bound in he saemen of he lemma. Nex, we esablish he lower bound. Since BACK B,S 1 L,S i= τ LOSTi ] [ E BACK B,S probabiliy one by Lemma 9, aking he expecaion on boh sides and aking he limi as increases [ ] [ ] / o infiniy, i follows ha E E (τ + 1). Therefore, i follows from he above expression for C L,S (h, b) ha BACK B,S C L,S (h, b) he [ (S D) +] ( ) b + (τ + 1)h + h E [ BACK B,S ] = C B,S (h, b/(τ + 1)). τ + 1 ] ]. wih We now relae he bes order-up-o level in he los sales and backorder sysems. Theorem 14. For any h 0 and b 0, he bes order-up-o level in he los sales sysem L(h, b) is 16

17 (a) bounded above by he bes order-up-o level in he backorder sysem B(h, b + τh) wih a backorder penaly cos parameer of b + τh, ha is, S L (h, b) S B (h, b + τh) ; (b) bounded below by he bes order-up-o level in he backorder sysem B(2h(τ + 1), b h(τ + 1)) wih a holding cos parameer of 2h(τ+1) and a backorder penaly cos parameer of b h(τ+1), ha is, S L (h, b) S B (2h(τ + 1), b h(τ + 1)). Proof. Please see Appendix E and Appendix F. 6. Performance Bounds and Asympoic Resuls for Los Sales Sysems We will now esablish he asympoic opimaliy of order-up-o policies in he los sales sysem. We will, in fac, show he asympoic opimaliy of he order-up-o-s B (h, b+τh) policy, corresponding o he opimal policy in he backorder sysem B(h, b + τh). Noice ha his is he upper bound we derived for he bes order-up-o level for he los sales sysem (Theorem 14(a)). For any finie b, we also derive an upper bound on he loss in performance from using his policy relaive o he opimal policy. Recall ha for any y 0, ψ(y; h, b) = be [ (D y) +] he [ (y D) +]. The main resul of his secion is saed in he following heorem. Theorem 15. For any h 0 and b 0, le S b+τh = S B (h, b+τh) and S b/(τ+1) = S B (h, b/(τ +1)) denoe he opimal order-up-o policies in he backorder sysems B(h, b + τ h) and B(h, b/(τ + 1)), respecively. Then, (a) The raio beween he cos of he order-up-o-s b+τh and he opimal policies in he los sales sysem L(h, b) can be bounded as follows: C L,S b+τh(h, b) C L (h, b) 1 + ( (b+τh)(τ+1) b ) ψ ( S b/(τ+1) ; h, b/(τ + 1) ) 1 + ψ ( S b/(τ+1) ; h, b/(τ + 1) ). (b) Under Assumpion 1, he order-up-o-s b+τh policy is asympoically opimal in he los sales sysem L(h, b), i.e. min S 0 C L,S (h, b) lim b C L (h, b) = lim b C L,Sb+τh(h, b) C L (h, b) = 1. 17

18 Proof. We know from Janakiraman e al. (2005) ha C B (h, b/(τ + 1)) C L (h, b) and i follows from Lemma 13 ha C L,S b+τh(h, b) C L (h, b) where ν b = (b + τh)(τ + 1)/b. CB,S b+τh(h, b + τh) C B (h, b/(τ + 1)) = CB (h, b + τh) C B (h, b/(τ + 1)) 1 + ν ( bψ Sb/(τ+1) ; h, b/(τ + 1) ) 1 + ψ ( S b/(τ+1) ; h, b/(τ + 1) ), Noe ha he equaliy follows from he definiion of S b+τh and he las inequaliy follows from Lemma 5. This proves par (a). Since lim b ν b = τ + 1 and by Theorem 6(a) i follows ha which is he desired resul. C L,S b+τh(h, b). lim ψ ( S b/(τ+1) ; h, b/(τ + 1) ) = 0, b C L,S b+τh(h, b) lim b C L = 1, (h, b) To complee he proof, noe ha C L (h, b) min S C L,S (h, b) 7. Compuaional Invesigaion In his secion, we compare he oal cos of he opimal policy, he bes base sock policy, and he base sock policy suggesed in Theorem 15. In Secion 7.1, we describe he mehodologies for our experimens. In Secion 7.2, we consider he performance of our proposed order-up-o policies on problem insances considered by Zipkin (2006b), enabling us o benchmark our performance agains oher replenishmen policies. Then, in Secion 7.3, we sudy he performance of base sock policies as he expeced demand increases, focusing on he commonly used Poisson demand models. Finally, in Secion 7.4, by considering negaive binomial disribuions, we explore he impac of increasing variance-o-mean raios on he performance of base sock policies. 7.1 Mehodologies To compue he long run average cos of he opimal replenishmen policy for a given discree demand disribuion, we consider he average cos dynamic programming formulaion. The sae space S for our dynamic program consiss of τ-dimensional vecors given by S = {(z 0, z 1,..., z τ 1 ) : z i Z + {0}}, where z 0 denoes he on-hand invenory afer receiving he replenishmen order and for 1 i τ 1, z i denoes he replenishmen quaniies ha will arrive i periods from now, corresponding o he 18

19 order placed τ i periods in he pas. In our experimen, he demand D in each period has he propery ha P {D = 0} > 0. I follows ha from Proposiion 2.6 in Bersekas (1995) ha for any h 0 and b 0, he opimal cos C L (h, b) is he unique soluion of he following average cos dynamic program: for any x S, C L { (h, b) + g(x) = min he [z0 D] + + be [D z 0 ] + + E [ g ( (z 0 D) + + z 1, z 2,..., z τ 1, u )]}, u 0 where g( ) denoes he differenial cos vecor and u represens he ordering quaniy. Since we will consider unbounded demand in our experimen, we apply he elegan sae-space reducion echnique inroduced by Zipkin (2006b), enabling us o consider a dynamic program wih only a finie number of saes (alhough he size of he sae space sill increases exponenially wih he lead ime). Then, o deermine he opimal replenishmen policy, we hen apply he relaive value ieraion mehod for 1000 ieraions or unil he change beween ieraions is less han (see Bersekas (1995); Zipkin (2006b) for more deails). We noe ha for any S 0, a similar dynamic programming formulaion can be used o deermine he long run average cos of he order-up-o-s policy. In his case, for any x S, insead of minimizing over all possible ordering quaniies as in opimal dynamic program above, [ he ordering quaniy is given by u = S τ 1 +. i=0 i] z We can hen apply he same relaive value ieraion mehod o deermine he long run average cos C L,S (h, b). To deermine he bes order-up-o level, we use he fac ha he oal cos C L,S (h, b) is convex in S (Downs e al. (2001) and Janakiraman and Roundy (2004)) and he bes order-up-o level is bounded above by S B (h, b + τh) (Theorem 14(a)) and below by S B (2h(τ + 1), b h(τ + 1)) (Theorem 14(b)). 7.2 Represenaive Problems In his secion, we repor he compuaional resuls for represenaive problems considered in Zipkin (2006b), enabling us o compare he cos of our base sock policies wih oher replenishmen heurisics. We consider Poisson and Geomeric demand disribuions, boh wih mean 5. The lead ime ranges from 1 o 4 periods. Assuming a holding cos of $1, we consider he los sales penaly ranging from $1 o $199. We compare he cos of he opimal policy, he bes base sock policy, and he order-up-o-s B (h, b + τh) policy, which is shown o be asympoically opimal (Theorem 15). Table 2 and 3 show he coss of hese polices for Poisson and geomeric disribuions, respecively. (We remark ha Zipkin (2006b) repors he cases where he los sales penaly is $4, $9 or $19.) 19

20 From Table 2 and 3, we observe ha as he los sales penaly increases, he cos of he bes base sock and he order-up-o-s B (h, b + τh) policies converge o he opimal cos as prediced by Theorem 15. For b = 199, he coss of boh base sock policies differ from he opimal by a mos 5%. However, for a specific cos parameer, he performance of our base sock policies ends o degrade as he lead ime increases. When comparing wih he performance of oher heurisics on he same problem insances (as repored in Zipkin (2006b)), he performance of our base sock policies are comparable wih oher heurisics. 7.3 Impac of Varying Mean Demand In his secion, we explore he performance of base sock policies as he mean demand changes. To faciliae our discussion, we assume a lead ime of wo periods (τ = 2) and consider a Poisson demand disribuion whose mean varies from 1 o 10. Table 4 shows a comparison among he opimal cos, he cos of he bes base sock policy, and he order-up-o-s B (h, b + τh) policy. From Table 4, we observe ha, for a specific los sales penaly, he relaive difference beween he opimal cos and he cos of he order-up-o-s B (h, b + τh) policy remains prey small even when he mean demand increases. We observe a similar paern for differen values of lead imes as well. This observaion has an imporan pracical implicaion. When he mean demand is small, compuing he bes order-up-o level is relaively easy since he range of base sock levels o consider is small; in fac, he opimal policy iself migh be compuaionally feasible. On he oher hand, for large means, compuing he opimal policy (or even he bes orderup-o level) is compuaionally more difficul because he search space is larger. Noneheless, our experimenal resuls indicae ha he simple and easily compuable base sock policy (wih he order-up-o level of S B (h, b + τh)) coninues o perform well even his seing, yielding oal cos ha is wihin 2% of he opimal (for b = 199). This resul suggess a pracical and effecive replenishmen heurisic: compue he opimal order-up-o policy exacly for small demand, and use he base sock level S B (h, b + τh) as an approximaion for larger demand. 7.4 Impac of Increasing Variance-o-Mean Raio In his secion, we explore he impac of he variance-o-mean raio on he performance of base sock policies. As in he previous secion, we assume he lead ime is 2 (τ = 2) and he demand D in each period follows a negaive binomial demand disribuion wih parameer (r, p) where r {1, 2} and 20

21 Lead Los Opimal Bes Base Sock S B (h, b + τh) Time Sales Cos Level Cos % Diff From Level Cos % Diff From Penaly Opimal Cos Opimal Cos % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Table 2: Performance of base sock policies for he Poisson disribuion wih mean 5. 21

22 Lead Los Opimal Bes Base Sock S B (h, b + τh) Time Sales Cos Level Cos % Diff From Level Cos % Diff From Penaly Opimal Cos Opimal Cos % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Table 3: Performance of base sock policies for he Geomeric disribuion wih mean 5. 22

23 Los Mean Opimal Bes Base Sock S B (h, b + τh) Sales Demand Cos Level Cos % diff from Level Cos % diff from Penaly opimal opimal % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Table 4: Performance of base sock policies for differen Poisson disribuions when lead ime is 2. 23

24 0.1 p 0.5. Thus, E [D] = r(1 p)/p and V ar[d] = r(1 p)/p 2, leading o a variance-o-mean raio of 1/p. Table 5 compares he coss of differen policies for differen variance-o-mean raios. We observe from he able ha our base sock policies are quie robus. The relaive difference beween he opimal cos and he cos of order-up-o policies seems o be independen of he variance-o-mean raio. We observe similar resuls even for larger lead imes (no included in he paper due o space consrain), suggesing ha our policies should perform well in many pracical seings where he demand exhibis significan variance. A. Proof of Theorem 4 Proof. Since D = τ+1 =1 D, if he demand in each period has a bounded suppor, so is D. Par (a) hen follows immediaely from he definiion of m D (). To prove par (b), if he demand in each period has an IFR disribuion, i follows from Corollary 1.B.20 on page 23 in Shaked and Shanhikumar (1994) ha D also has an IFR disribuion. Then, i follows from Lemma 3 ha for any s 0, P { D 2 > s D > 2 } P { D 1 > s D > 1 }, which implies ha he mean residual life m D () is a decreasing funcion in, giving us he desired resul. To prove par (c), we can assume wihou loss of generaliy ha F (x) < 1 for all x 0. For any x, le F (x) = 1 F (x). I hen follows from he definiion of m D () ha m D () = F (u)du F. () Since E [ D 2] <, i follows ha E [D] = 0 F (u)du <. Therefore, lim F (u) = 0. Moreover, we have ha E [ D 2] = 0 2u F (u)du. Since D has a finie second momen, i follows ha lim F () = 0, implying ha boh he numeraor and he denominaor in he expression for m D ()/ converge o zero a increases o infiniy. Since D is assumed o be a coninuous random variable, we can apply 24

25 Los Negaive Binomial Variance- Opimal Bes Base Sock S B (h, b + τh) Sales Parameer o-mean Cos Level Cos % diff from Level Cos % diff from Penaly r p Raio opimal opimal % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Table 5: Performance of base sock policies for negaive binomial disribuions wih differen variance-o-mean raio when he lead ime is 2. 25

26 L Hospial s Rule o conclude ha which is he desired resul. m D () lim = lim F () F () f() = lim 1 r() 1 = 0, B. Proof of Proposiion 7 Proof. I is easy o verify ha 1 F θ (x) = 1/(1 + x) θ. I follows ha E [D] = 0 1 F θ (x)dx = 1/(θ 1). Then, using he fac ha m D () = P {D > z} dz/p {D > }, we can also show ha m D () = (1 + )/(θ 1), which proves he firs par of Proposiion 7. To esablish he second par, noe ha by definiion S B (h, b) = F 1 θ (b/(b + h)), which implies ha S B (h, b) = ( ) b+h 1/θ h 1. Then, we have ha [ (D E S B (h, b) ) ] + = P { D > S B (h, b) } [ ] E D S B (h, b) D > S B (h, b) h = b + h m ( D S B (h, b) ) = h ( 1 + S B (h, b) ) (b + h)(θ 1), where he las equaliy follows from he formula for m D ( ). Thus, and herefore, Thus, [ (S E B (h, b) D ) ] + = E [ S B (h, b) D ] [ (D + E S B (h, b) ) ] + = S B (h, b) 1 θ 1 + h ( 1 + S B (h, b) ) (θ 1)(b + h), C B (h, b) = [ (S he B (h, b) D ) ] [ + (D + be S B (h, b) ) ] + = ( h S B (h, b) 1 ) + h ( 1 + S B (h, b) ) = hθsb (h, b). θ 1 θ 1 θ 1 which is he desired resul. C B (h, νb) lim b C B (h, b) = lim S B (h, νb) b S B (h, b) = lim b ( νb+h h ( b+h h ) 1/θ 1 ) 1/θ = ν 1/θ, 1 C. Proof of Lemma 10 Proof. Le M = sup {x : P (D x) = 0} denoe he lowes possible single period demand. Huh e al. (2006) show he convergence of he sochasic process {X L,S } for all S > M (τ + 1). This 26