Analysis of Tailored Base-Surge Policies in Dual Sourcing Inventory Systems

Transcription

1 Analysis of Tailored Base-Surge Policies in Dual Sourcing Invenory Sysems Ganesh Janakiraman, 1 Sridhar Seshadri, 2, Anshul Sheopuri. 3 Absrac We sudy a model of a firm managing is invenory of a single produc by sourcing supplies from wo supply sources, a regular supplier who offers a lower uni cos and a longer lead ime han a second, emergency, supplier. A pracically implemenable policy for such a firm is a Tailored Base-Surge (TBS) Policy (Allon and Van Mieghem, 2010) o manage is invenory: Under his policy, he firm procures a consan quaniy from he regular supplier in every period and dynamically makes procuremen decisions for he emergency supplier. Allon and Van Mieghem describe his pracice as using he regular supplier o mee a base level of demand and he emergency supplier o manage demand surges, and hey conjecure ha his pracice is mos effecive when he lead ime difference beween he wo suppliers is large. We confirm hese saemens in wo ways. Firs, we show he following analyical resul: When demand is composed of a base demand random componen plus a surge demand random componen, which occurs wih a cerain small probabiliy, he bes TBS Policy is close o opimal (over all policies) in a well defined sense. Second, we also numerically invesigae he cos-effeciveness of he bes TBS policy on a es bed of problem insances. The emphasis of his invesigaion is he sudy of he effec of he lead ime difference beween he wo suppliers. Our sudy reveals ha he cos difference beween he bes TBS policy and he opimal policy decreases dramaically as he lead ime of he regular supplier increases. On our es bed, his cos difference decreases from an average (over he es bed) of 21 % when he lead ime from he regular supplier is wo periods (he emergency supplier offers insan delivery) o 3.5 % when ha lead ime is seven periods. Keywords: Invenory/Producion Sysems, Dual Sourcing, Muliple Suppliers, Opimal Policies 1. Inroducion We sudy a periodically reviewed, single locaion invenory sysem experiencing sochasic demand and having access o wo supply sources, one wih a lower uni cos (R, for Regular) and he oher wih a shorer lead ime (E, for Express or Emergency). We focus aenion on Tailored Base-Surge Allocaion Policies (we will refer o his class of policies as TBS) which work as follows: Source a consan quaniy from supplier R in every period and use a base-sock (order-up-o) 1 Naveen Jindal School of Managemen, The Universiy of Texas a Dallas 2 Indian School of Business 3 IBM Research 1

2 policy for supplier E. TBS policies have pracical appeal as discussed in Allon and Van Mieghem (2010) [hereon, referred o as AVM] who also menion ha similar policies called sanding order policies have been proposed earlier by Rosenshine and Obee (1976) and Janssen and de Kok (1999). TBS policies also have he same spiri as an indusry pracice called base-commimen conracs 4 (Simchi-Levi e al., 2008). Our work is mos closely relaed o AVM (who also coined he name TBS). The focus of heir sudy was he opimizaion wihin TBS policies, i.e., he opimal choice of he consan quaniy o be sourced from R and he order-up-o level o be used for E. As far as moivaion for hese policies is concerned, AVM argue ha hese policies are easy o manage and ha hey are inuiive R is used o handle a consan, base level of demand while he use of E is ailored o mee any surge in demand above he base. They also conjecure ha such a policy is effecive when he lead ime difference beween he wo suppliers is high. We validae hese saemens in he following wo ways: (1) Analyical Resuls: When demand comes from a wo poin disribuion and when he probabiliy mass a he smaller demand level is sufficienly large, he bes TBS policy is opimal. (Secion 5.1) Moreover, when demand is composed of a sum of wo random variables, he firs of which represens base demand and he second of which represens surge demand which only occurs wih a sufficienly small probabiliy, TBS policies work well. More precisely, he relaive difference beween he cos of he bes TBS policy and he opimal cos is smaller han 1.2 imes he raio of he sandard deviaion of base demand o he expeced surge demand. In oher words, when he surge probabiliy is small bu he expeced surge, given ha here is a surge, is large, he bes TBS policy is close o opimal. (Secion 5.2) (2) Numerical Resuls: We conduc a numerical invesigaion of he cos-effeciveness of he bes TBS policy (Secion 6). On our es bed of problem insances, his effeciveness increases dramaically as he lead ime of he regular supplier increases from wo o seven periods (while he lead ime of he emergency supplier is zero). More specifically, he relaive difference beween he cos of he bes TBS policy and he opimal cos over all policies decreases from 21 % o 3.5 %. The remainder of he paper is organized as follows. We describe our model in deail and presen our noaion in Secion 2. A review of he relaed lieraure is given in Secion 3. Secion 4 4 See, for e.g., he discussion on page 138 of he supply chain ex by Simchi-Levi e al. (2008). 2

3 conains preliminary resuls which are used in our echnical analysis. The main analyical resuls are presened in Secion 5 and he numerical resuls are presened in Secion Model and Noaion Le D refer o he dual sourcing invenory sysem. The lead ime from R is l R periods and ha from E is l E periods. We use δ o denoe he lead ime difference, i.e. δ = l R l E. Le θ denoe any feasible ordering policy. The sequence of evens in every period under his policy is he following: (1) The order placed in period l R from R for q θ,r l R unis is delivered and, if l E > 0, he order placed in period l E from E for q θ,e l E unis is delivered. The ne-invenory, defined as he amoun of invenory on hand - amoun of backordered demand, a his insan is x θ. (2) Ordering decisions are made: q θ,r hese q θ,e 0 unis are ordered from R and q θ,e 0 unis are ordered from E. If l E = 0, 0 unis are immediaely delivered. (3) Demand, d, is realized. (4) The cos c θ for his period is charged according o he following expressions: c θ = h (x θ d ) + + b (d x θ ) + + c q θ,e if l E > 0 and c θ = h (x θ + q θ,e d ) + + b (d x θ q θ,e ) + + c q θ,e if l E = 0. Here, h and b represen he per-uni holding and backorder coss, respecively, and c is he unicos premium charged by E over R. (We do no include a uni procuremen cos for R because he difference in he uni coss of E and R, and no hese uni coss hemselves, exclusively deermines he opimal policy. 5 ) We assume ha c < b (l R l E ), failing which he problem is rivial in he sense ha single-sourcing from R would hen be opimal. (This is because he maximum benefi of ordering a uni from E insead of R is he reducion in he backorder cos associaed wih a uni of demand, which is bounded above by b (l R l E ).) Demands in differen periods are assumed o be i.i.d, i.e., independenly and idenically disribued. Le D denoe he random demand in any period we use F o denoe is disribuion funcion. We denoe he mean and sandard deviaion of D by µ and σ, respecively, boh of which are assumed o be finie. Throughou he paper, we assume ha demand is no deerminisic, i.e. σ > 0. 5 A more deailed explanaion follows. Consider a cos model in which he cos incurred in period under policy θ is c θ, where c θ is idenical o c θ excep ha i includes a cos c R for procuring a uni from R and ha he cos for procuring a uni from E is c E. Any policy θ wih a finie long run average cos will be such ha he long run average quaniy procured from E and R combined per period is exacly equal o he mean demand per period, µ. Using his fac, i can be verified ha he difference in he long run average coss of wo policies θ and θ under he cos funcion c θ is he same as he difference under c θ when c = c E c R. Thus, he opimal policy under boh cos models is he same. Moreover, under he cos funcion c θ, he quaniy c R µ can be viewed as a sunk cos ha every policy has o incur per period and disregarding ha quaniy (which is done under he cos model, c θ ) while comparing he cos of a TBS policy wih he opimal cos only describes he relaive performance of he TBS policy conservaively. 3

4 The performance measure we use in his paper is C θ, he long run average cos of policy θ, defined below: C θ = lim sup T The opimal long run average cos is given by T =1 E[cθ ] T. C D, = inf θ Cθ. In several places in he paper, i will be necessary o show he dependence of he opimal cos on he problem parameers; we will hen use he noaion C D, (h, b, c, l E, l R, F ). 3. Lieraure Review Our paper is sharply focused on he analysis of he effeciveness of he simple TBS policies in dual sourcing invenory sysems relaive o he opimal policies which have a complicaed srucure. Consequenly, we confine ourselves o specifically discussing hose resuls in he lieraure which are criical o undersanding he value of our work. We refer he reader o AVM for a deailed review of earlier papers which suggesed he use of TBS policies. For an exensive review of he lieraure on muliple-supplier invenory sysems, please see Minner (2003). Oher recen papers in which hese sysems have been sudied (Sheopuri e al., 2010; Song and Zipkin, 2009; Veeraraghavan and Scheller-Wolf, 2008) also provide and discuss several relaed references. Several oher policies have been proposed and esed in hese papers. We limi our aenion in his discussion o explaining wha is known in he lieraure abou opimal policies in dual sourcing sysems, why simple policies like TBS policies are valuable and wha we know abou TBS policies. Sheopuri e al. (2010) show ha he problem of finding an opimal policy in a dual sourcing sysem is a generalizaion of he problem of finding an opimal policy in a single supplier sysem wih a posiive lead ime when excess demand is los. Their argumen is based on considering he following special case of he dual sourcing problem: Assume ha he backorder cos is prohibiively high in a dual sourcing sysem. Assume also ha he lead ime from E is -1; ha is, we are allowed o place an order on he emergency supplier afer observing demand in a period and his order is delivered insanly and he resuling invenory is available o saisfy he demand ha arose in ha period. Under hese assumpions, i becomes opimal o use he emergency supplier exclusively o clear any demand in a period ha we are unable o mee from invenory a he beginning of ha period. Thus, he quaniy sourced from he emergency supplier can be hough of as he 4

5 los sales incurred by he regular supply sysem and he emergency procuremen cos is he los sales penaly cos. This los sales invenory problem wih lead imes is considered a challenging problem in he sense ha he opimal policy does no admi any nea srucure like ha of orderup-o policies (Karlin and Scarf, 1958). (This has spurred an acive line of research sudying he effeciveness of heurisic policies (Zipkin, 2008a; Huh e al., 2009).) Thus, opimal policies in dual sourcing sysems, in general, are no simple policies like base-sock policies. In fac, he opimal ordering quaniies from he wo suppliers are funcions of a sae vecor (of invenories) whose lengh is he difference in he lead imes of he wo suppliers. Thus, he problem of compuing he opimal policy by solving he dynamic program suffers from he curse of dimensionaliy. Equally imporanly, he likelihood of pracicing managers implemening ha policy is diminished by he complexiy (or lack of a ransparen insigh) of is mechanics. This is why here has been significan ineres among invenory researchers in proposing and evaluaing simple policies for hese sysems. The TBS policy we sudy is a prominen choice for such a simple policy. Anoher prominen choice is he dual index policy sudied by Veeraraghavan and Scheller-Wolf (2008), among ohers. We have explained earlier he moivaion for TBS policies. A more deailed argumen in suppor of hese policies is provided in AVM. The dual sourcing model sudied by AVM is a coninuous ime model in which demand is a couning process and he supplies from boh sources follow renewal processes. The capaciy raes from boh sources are decision variables. Furhermore, given a pair of capaciy raes for he wo sources, heir TBS policy works as follows: R supplies coninuously a is capaciy rae (his base supply process is analogous o he consan quaniy ordered from R in every period in our discreeime TBS policy) and E supplies whenever he oal invenory posiion in he sysem falls below a base-sock level (our ordering policy for E is idenical). The auhors sudy he opimizaion of he wo capaciy raes and he base-sock level for E. The opimal base-sock level is obained from he newsvendor formula using he disribuion of he seady-sae overshoo (he amoun by which he invenory posiion exceeds he base-sock level). The dynamics of his overshoo process is idenical o ha of a GI/GI/1 queue. The opimizaion of he capaciy raes does no allow for a closed-form soluion, hough. Consequenly, hey perform an asympoic analysis of his opimizaion as he expeced demand rae grows infiniely large (he co-efficien of variaion of iner-arrival imes is held consan). A key resul of his analysis is ha he opimal capaciy rae from he regular supplier becomes close o he expeced demand rae in his regime. A consequence of his resul is 5

6 ha under he opimal choice of capaciy raes, he overshoo process resembles a queue in heavy raffic. This enables he auhors o perform a heavy raffic analysis o derive closed form expressions for asympoically opimal capaciy raes for he wo suppliers and an asympoically accurae expression for he opimal expeced cos of he sysem. These expressions are simple square roo formulas ha are boh inuiive and provide clear insighs abou he rade-offs beween coss and lead imes. While he model sudied in our paper is differen from ha sudied in AVM (e.g., heir model is in coninuous ime and heir supply processes are sochasic, whereas ours is in discree ime and supply is deerminisic), he mos imporan disincion beween our work and heirs is he following: AVM s focus is on opimizing wihin he class of TBS policies. Our focus is on analyzing he effeciveness of he bes TBS policy relaive o he opimal policy over all feasible policies. 4. Preliminary Resuls This secion is devoed o deriving preliminary resuls which we use in Secion 5 for proving our main analyical resuls. In addiion, Theorem 1 is useful in Secion 6 for numerically compuing he bes TBS policy. 4.1 Tailored Base Surge Policies: Opimizaion A ailored base surge policy is specified by wo parameers Q < µ and S. 6 In every period, an order for Q unis is placed on R. The ordering decision for E follows an order-up-o rule wih arge level S here he invenory posiion which is raised o he arge is he expedied invenory posiion which includes he ne-invenory and all he ousanding orders from boh suppliers which will be delivered wihin he nex l E periods. I should be noed ha in some periods, he expedied invenory posiion before ordering will exceed he arge level S, in which case no order is placed from E. The quaniy by which he expedied invenory posiion exceeds S in such a period is called he overshoo. Le C D,Q,S (h, b, c, l E, l R, F ) denoe he long run average cos of he TBS policy wih parameers Q (he quaniy ordered from R in every period) and S (he order-up-o level for E). Le S (Q) = arg min S C D,Q,S (h, b, c, l E, l R, F ). 6 If Q µ, he sysem is no sable in he sense ha he expeced invenory on hand approaches. 6

7 For a given Q, le O (Q) denoe he seady sae overshoo random variable which is defined as he seady sae version of he sochasic process {O } ha follows he recursion O +1 = max(0, O D + Q). (4.1) The exisence of he saionary disribuion denoed by O (Q) is guaraneed by Loynes Lemma (Loynes, 1962). Le D[1, ] denoe he demand over periods. Le G(y) = h E[(y D[1, l E + 1]) + ] + b E[(D[1, l E + 1] y) + ] denoe he holding and shorage cos incurred in a period given he expedied invenory posiion a lead ime earlier is y. Then, we can wrie C D,Q,S (h, b, c, l E, l R, F ) = c (µ Q) + E[G(S + O (Q))]. Therefore, S (Q), he opimal order-up-o level or base-sock level for a given Q is he soluion o he newsvendor equaion E[G (S + O (Q))] = 0, ha is, P (S (Q) + O (Q) D[1, l E + 1]) = b b + h. While his equaion gives a closed form expression for he bes order-up-o level for a given consan order quaniy, Q, he opimal value of Q iself can be found by a simple echnique like bisecion search using he following new resul. (The proofs of all he analyical resuls ha follow can be found in he appendix.) Theorem 1. The funcion C D,Q,S (Q) (h, b, c, l E, l R, F ) is convex in Q. Since he quaniy received from R is he same in every period under any TBS policy, boh S (Q) and C D,Q,S (Q) are independen of he regular lead ime, l R. However, he opimal cos (over all admissible policies), C D, is non-decreasing in l R. 7 Thus, he relaive performance of he bes TBS policy (in fac, any TBS policy) improves as he regular lead ime increases. In Secion 6, we use a numerical invesigaion o sudy his improvemen when all oher parameers are held consan (demand disribuion, cos parameers) while he regular lead ime alone increases. In Secion 5, we complemen ha sudy by analyically characerizing demand disribuions and cos parameer values under which he bes TBS policy is near-opimal. 7 The proof of he claim ha C D, is non-decreasing in l R is he following. Consider he opimal policy in a sysem wih a regular lead ime of k + 1 periods. Tha same policy is also admissible in a sysem wih a regular lead ime of k periods by simply delaying every order from R deliberaely by one period. Thus, he opimal cos in he former sysem is also achievable in he laer sysem and he opimal cos in he laer sysem is, by definiion, smaller han his cos. 7

8 4.2 Upper Bounding he Cos of he Bes TBS Policy In his secion, we will derive an analyical upper bound on he cos of he bes TBS policy. This bound will be useful laer when we derive a bound on he cos performance of he bes TBS policy relaive o he opimal policy. The sandard, muli-period, news-vendor sysem wih backordering is useful in he developmen of his bound. More specifically, B(h, b, l E, F ) is a sysem where here is a single supplier providing deliveries (wih zero procuremen coss) wih a replenishmen lead ime of l E periods, he uni holding and shorage cos parameers are h and b, respecively, and he demand disribuion is F for every period. The opimal policy in B is o se he invenory posiion afer ordering in every period o be S E, he news-vendor level wih a demand disribuion of D[1, l E + 1], ha is, S E solves P (S E D[1, l E + 1]) = b/(b + h). We use C B, (h, b, l E, F ) o denoe he opimal long run average cos of his sysem and C B,S (h, b, l E, F ) o denoe he long run average cos in B under he order-up-o S policy. Nex, consider he TBS policy of ordering Q unis from R in every period and using an orderup-o S E policy wih E. Now, he difference beween he cos of using his policy in D and he opimal cos in B can be bounded above by bounding he expeced seady sae overshoo random variable. The dynamics of he overshoo (see (4.1)) are he same as hose of he waiing ime in a single server queue. One of Kingman s bounds (Kingman, 1970) is hen used o bound he expeced seady sae overshoo. We noe ha AVM also use a similar bound in heir analysis. Lemma 2. The infinie horizon average cos of he bes TBS policy wih a given Q is bounded above as follows: ( C D,Q,S (Q) (h, b, c, l E, l R, F ) C B, (h, b, l E, F ) + h σ 2 2 (µ Q) ) + c (µ Q). We are now ready o esablish he main resul of his secion. We presen an upper bound on he cos of he bes TBS policy by minimizing boh sides of he inequaliy in he saemen of Lemma 2. This bound is used in our wors-case analysis of TBS policies. Theorem 3. The expeced cos per period of he bes TBS policy is bounded above as follows: min Q CD,Q,S (Q) (h, b, c, l E, l R, F ) C B, (h, b, l E, F ) + σ 2 h c. 8

9 We close his secion wih a second upper bound on he cos of he bes TBS policy. This bound is simply he cos of procuring exclusively from E opimally, which is achieved using an order-up-o policy. Thus, his policy is a TBS policy wih Q = 0, i.e. zero sourcing from R. Therefore, is cos is an upper bound on he cos of he bes TBS policy. Theorem 4. The expeced cos per period of he bes TBS policy is bounded above as follows: min Q CD,Q, (h, b, c, l E, l R, F ) C B, (h, b, l E, F ) + c µ. A quick comparison of he wo upper bounds derived in Theorem 3 and Theorem 4 (which corresponds o he TBS policy wih Q = 0) provides us a simple insigh abou TBS policies: In environmens in which he demand uncerainy is no excessive (mahemaically speaking, when σ/µ is no high), a non-rivial TBS policy (i.e., Q > 0) dominaes he policy of using E exclusively; hus he simple version of dual sourcing (i.e. using TBS policies) provides value. 4.3 Effec of Variabiliy on Opimal Cos We show in his secion ha if demand becomes more variable (in he formal sense of convex ordering), he opimal cos of he sysem increases. While his resul is obvious in he case of he single-period news-vendor model, a proof is necessary for more sophisicaed, muli-period models such as he dual sourcing model here. A recen example of such a proof for anoher muli-period model is ha of Zipkin (2008b) for los-sales invenory sysems. The resul is ineresing in is own righ since i confirms our inuiion abou he effec of variabiliy on coss. Moreover, i is used in he proof of an imporan resul in Secion 5.2. We say F cx F ( F is smaller han F in he convex order ) if E[g( D)] E[g(D)] for any convex funcion g and wo random variables D and D wih disribuions F and F, respecively. Theorem 5. Assume ha F cx F. Then, he opimal long run average cos when he demand disribuion is F is smaller han he opimal long run average cos when he demand disribuion is F. Tha is, C D, (h, b, c, l E, l R, F ) C D, (h, b, c, l E, l R, F ). We now presen wo relaed resuls which are boh inuiive and useful in our analysis in he following secion. The firs resul is ha he opimal cos is unaffeced if he demand disribuion is shifed by a consan while he second resul is ha if he demand disribuion is scaled by a consan, hen, he opimal cos is also scaled by he same consan. 9

10 Lemma 6. Consider hree disribuions F, F and ˆF (say, represening random variables D, D and ˆD) such ha F (x) = F (x A) (i.e., D D + A) and ˆF (x) = F (x/b) (i.e., ˆD = D B), where A 0 and B > 0 are consans. Then, C D, (h, b, c, l E, l R, F ) = C D, (h, b, c, l E, l R, F ) and C D, (h, b, c, l E, l R, ˆF ) = B C D, (h, b, c, l E, l R, F ). An immediae corollary of his lemma is he following. Corollary 7. Consider wo disribuions F and F (say, represening random variables D and D) such ha F (x) = F ( ) x A B (i.e., D = B D + A), where A 0 and B > 0 are consans. Then, C D, (h, b, c, l E, l R, F ) = B C D, (h, b, c, l E, l R, F ). 5. Main Analyical Resuls In his secion, we firs sudy he special case of a wo-poin demand disribuion (Secion 5.1). We characerize condiions on he cos parameers and he probabiliy describing he wopoin disribuion under which a TBS policy is an opimal policy (over all admissible policies). Subsequenly, we build on ha analysis o sudy a more general demand disribuion which we call a base-surge disribuion (Secion 5.2). In ha case, we derive a bound on he raio of he cos of he bes TBS policy o he opimal cos under some assumpions. 5.1 Two Poin Disribuions In his secion, we sudy he case in which demand comes from a wo-poin disribuion. Le D, he random variable represening demand in a period, have wo possible values d Low and d High and le i ake hese values wih probabiliies p Low and p High = 1 p Low, respecively. We now presen an assumpion ha ensures ha he probabiliy of a low demand is sufficienly high and hen explain is consequence. Assumpion 1. The probabiliy of low demands, p Low, exceeds γ γ+1, where γ := c+b (le +1)+h (l R +1) h. Consider firs he special case in which d Low = 0 and d High = 1. Consider a saring sae in which here are no backorders and here is no invenory anywhere in he sysem. The expeced ime unil he arrival of a cusomer is p Low /(1 p Low ) since he demand process is a Bernoulli process (which is memoryless). Now, if we order a uni from R in anicipaion of he firs cusomer s arrival, his uni will be held as invenory on hand from he ime i is delivered o he ime ha he cusomer arrives (assuming he laer happens afer he former). Thus, he expeced holding 10

11 cos incurred due o his uni exceeds h (p Low /(1 p Low ) (l R + 1) ). (Noe ha, since l R > l E, he expeced holding cos incurred by ordering a uni from E in anicipaion of a cusomer s arrival is even higher.) On he oher hand, waiing unil a cusomer arrives and immediaely ordering a uni from E o saisfy ha cusomer s demand upon delivery resuls in a procuremen cos plus backordering cos of c + b (l E + 1). Therefore, if he condiion h (p Low /(1 p Low ) (l R + 1) ) c + b (l E + 1), (5.1) is saisfied, i is never opimal o procure a uni when here is no cusomer who is already backordered; moreover, since c < b (l R l E ), i is opimal o procure a uni from E (as opposed o R) as soon as a cusomer arrives. This policy is essenially a TBS policy wih Q = 0 and S = 0 and he condiion in (5.1) is equivalen o Assumpion 1. An idenical argumen, along wih Lemma 6, shows ha a TBS policy wih Q = d Low and S = d Low (l E + 1) is opimal under any general wo-poin disribuion saisfying ha assumpion. The following heorem saes his formally and we supplemen he inuiion above wih a rigorous proof in he appendix. Theorem 8. For a wo-poin disribuion, F, saisfying Assumpion 1, a TBS policy wih Q = d Low and S = d Low (l E + 1) is opimal. Wha Theorem 8 demonsraes is ha when demand is usually a a base level and occasionally rises o a surge level, he TBS policy of using R o consanly mee he base-level of demand is opimal over all admissible policies as long as he probabiliy of a surge is sufficienly low. To show his resul, we have modeled he base and surge levels as demand poins (i.e., degenerae disribuions) raher han he more general model of associaing a probabiliy disribuion wih each of hese wo levels. We explore such a generalizaion in he nex sub-secion. 5.2 Base-Surge Disribuions In his secion, we model demands using base-surge disribuions which we describe as follows. Le X and Y be wo non-negaive, independen random variables. Here, X is he base demand disribuion and Y is he surge demand disribuion. Surges are rare and we model his using a 11

12 probabiliy p ha here will be no surge (equivalenly, a probabiliy 1 p of a surge). Thus, he demand in a period is given by D = X + Z (Z is independen of X), where Z = 0 wih probabiliy p and Z = Y wih probabiliy (1 p). We derive a bound on he raio of he cos of he bes TBS policy o he opimal cos over all admissible policies when p is sufficienly large, under a reasonable assumpion on he cos parameers. We will use σ X o denoe he sandard deviaion of X and µ Y o denoe he expecaion of Y. Theorem 9. Under Assumpion 1, he raio of he cos of he bes TBS policy and he opimal cos (over all admissible policies) is smaller han ( 2 ) σ X h c + h b (l E + 1) 1 + (c + b (l E. + 1)) (1 p) µ Y Moreover, when he cos parameers h, b and c are such ha h c and c < b (l E + 1), his raio ) ( is bounded above by 1 + ( σ X (1 p) µ Y ). This bound suggess ha when he probabiliy of a surge is small bu he expeced surge (given ha here is a surge) is large relaive o he uncerainy in base demand, he bes TBS policy is near-opimal. We conclude his sub-secion wih a remark on wha his analysis implies for managing invenory and sourcing decisions for service pars. For a ypical service par, he demand disribuion is characerized by a large probabiliy mass a zero and some probabiliy disribuion condiional on demand being non-zero. We can now use Theorem 9 o make he following observaion. Remark 1. Le he demand disribuion F be such ha F (0) = p. Le F represen he demand disribuion condiional on demand being sricly posiive, i.e., F (0) = 0 and F (x) = P (D x D > 0). For any F, he policy of never ordering from R and ordering up o zero from E (i.e. reacing o all posiive demand by ordering he demanded quaniy from E) is opimal if he probabiliy of zero demand, i.e., p, is large enough for Assumpion 1 o be saisfied. This is a rivial TBS policy wih Q = 0 and S = 0. Thus, in his case, single sourcing is opimal. 6. Numerical Resuls In his secion, we evaluae he cos effeciveness of he bes TBS policy over a es bed of problem insances by numerically compuing, on each insance, boh he long run average cos of ha policy and he opimal cos (by solving he infinie horizon, average-cos dynamic program). 12

13 The raio of he wo coss menioned above is he performance measure of ineres here. The goal is o undersand how his raio depends on he b/(b + h) raio, he lead ime difference beween R and E and he emergency procuremen cos, c. We also wan o undersand wheher hese paerns are consisen across a variey of demand disribuions. Our iniial es bed consiss of he problems defined by he following parameer choices. The lead ime from E is zero, he lead ime from R is varied beween 2 and 7 periods, and h = 20, hroughou. (As an illusraive example, consider a review period of 1 monh and a holding cos of $20 per uni per monh. A a 20% annual cos of capial, his corresponds o a nominal or regular uni cos of $1200.) The value of b is eiher 80 or 180 corresponding o b/(b + h) raios of 80% or 90%, respecively. The value of c is eiher 20, 50 or 100. (In he example, our choices of b, i.e., $80 per uni per monh and $180 per uni per monh, ranslae o 6.7% and 15% of he regular uni cos per uni per monh. Our choices for he uni cos premium charged by he faser supplier, i.e., $20, $50 and $100, range from 1.7% - 8.3% of he uni cos.) The following six disribuions are chosen. We use p i for P (D = i). (1) Two-Poin: p 0 = 0, p 1 = 2/3, p 2 = 0, p 3 = 0, and p 4 = 1/3. (2) Unimodal-Symmeric: p 0 = 0.125, p 1 = 0.2, p 2 = 0.35, p 3 = 0.2, and p 4 = (3) Righ-Skewed: p 0 = 0.125, p 1 = 0.5, p 2 = 0.125, p 3 = 0.125, and p 4 = (4) Lef-Skewed: p 0 = 0.125, p 1 = 0.125, p 2 = 0.125, p 3 = 0.5, and p 4 = (5) Bi-modal: p 0 = 0.1, p 1 = 0.35, p 2 = 0.1, p 3 = 0.1, and p 4 = (6) Uniform: p 0 = 0.2, p 1 = 0.2, p 2 = 0.2, p 3 = 0.2, and p 4 = 0.2. (The reason for limiing our invesigaion over disribuions wih suppor on {0, 1, 2, 3, 4} is he compuaional effor required o compue he opimal cos by dynamic programming.) For each of hese insances, we repor he opimal cos, he cos of he bes TBS policy and he percenage difference beween he wo in a se of six ables (one for each disribuion) presened in he appendix. Some observaions based on hese resuls follow. In each of he ables, we see ha he performance of he bes TBS policy improves as l R increases (i.e., as he lead ime difference beween R and E increases). The improvemen from a one period increase o l R is quie dramaic when l R is small. In all our insances wih c = 20 or c = 50, he cos of he bes TBS policy is a mos 4.6% more han he opimal cos when l R = 7. Tha is, TBS policies emerge as an effecive choice for such lead ime differences. Nex, we would expec 13

14 ha TBS policies become a less effecive choice when c is large. (This is because, when c is large, using only R and, ha oo, using an order-up-o policy is a preferred policy. Tha policy passes he demands o R as is orders. The TBS policy on he oher hand sources a consan amoun from R every period.) This is observed in our numerical resuls. Our nex observaion is on he service level (more precisely, he b/(b + h) raio). As his level increases, he policy of sourcing exclusively from E (more generally, he policy of using R o source he deerminisic par of demand, if any, and using E o manage uncerainy) becomes relaively more aracive. Since his policy is a special case of TBS policies, we expec he effeciveness of he bes TBS policy o improve as service levels increase. Broadly speaking, such a rend can be observed in our resuls by comparing he percenages corresponding o b = 180 versus ha for b = 80 in each of our ables. Anoher effec we noice in our resuls is ha he bes TBS policy becomes more effecive for higher levels of demand uncerainy. More specifically, he average percenage cos gap beween he bes TBS policy relaive o he opimal decreases wih he sandard deviaion of demand on our es bed. This is likely because when demand uncerainy is high, he opimal policy devoes a larger fracion of he supply sourced o E, hus approaching he policy of sourcing only from E - his policy is he TBS policy wih Q = 0. Finally, we also presen some resuls of ess o invesigae wheher he observaions made above on he performance of TBS policies are affeced if he lead ime from E were no zero. To his end, we include a able in he appendix wih resuls for problems in which he difference beween he wo lead imes is held consan a hree (i.e., l R l E = 3), while l E is varied from one o hree. This able seems o indicae ha here is no noiceable effec of l E on he performance of TBS policies relaive o he opimal, for a fixed value of l R l E. I appears ha, on an average (over he insances esed), an increase in l E is accompanied by approximaely he same percenage increase in boh he cos of he bes TBS policy and he opimal cos, hus leaving he percenage difference beween he wo mosly unchanged. References Allon, G., and J. A. Van Mieghem Global Dual Sourcing: Tailored Base-Surge Allocaion o Near- and Offshore Producion. Managemen Science 56 (1): Gallego, G., and I. Moon The Disribuion Free Newsboy Problem: Review and Exensions. The Journal of he Operaional Research Sociey 44 (8):

15 Hernandez-Lerma, O., and J. B. Lassere Discree-Time Markov Conrol Processes: Basic Opimaliy Crieria. Springer-Verlag, New York. Huh, W., G. Janakiraman, J. Mucksad, and P. Rusmevichienong Asympoic Opimaliy of Order-up-o Policies in Los Sales Invenory Sysems. Managemen Science 55 (3): Huh, W., G. Janakiraman, and M. Nagarajan Average Cos Single-Sage Invenory Models: An Analysis Using a Vanishing Discoun Approach. Operaions Research 59 (1): Janssen, F., and T. de Kok A wo-supplier invenory model. Inernaional Journal of Producion Economics 59: Karlin, S., and H. Scarf Invenory Models of he Arrow-Harris-Marschak Type wih Time Lag. In Sudies in he Mahemaical Theory of Invenory and Producion, ed. K. J. Arrow, S. Karlin, and H. Scarf, Chaper 9, Sanford Universiy Press. Kingman, J Inequaliies in he Theory of Queues. Journal of he Royal Saisical Sociey. Series B (Mehodological) 32 (1): Loynes, R The sabiliy of a queue wih nonindependen iner-arrival and service. Proc. Camb. Philos. Soc. 58: Minner, S Muliple-supplier invenory models in supply chain managemen: A review. Inernaional Journal of Producion Economics 81-82: Rosenshine, M., and D. Obee Analysis of a sanding order invenory sysem wih emergency orders. Operaions Research 24 (6): Schäl, M Average Opimaliy in Dynamic Programming wih General Sae Space. Mahemaics of Operaions Research 18 (1): Sheopuri, A., G. Janakiraman, and S. Seshadri New Policies on he Sochasic Invenory Conrol Problem wih Two Supply Sources. Operaions Research 58 (3): Simchi-Levi, D., P. Kaminsky, and E. Simchi-Levi.. Designing and Managing he Supply Chain: Conceps, Sraegies, and Cases. Third ed. New York, NY, U.S.A.: McGraw-Hill Irwin. Song, J., and P. Zipkin Invenories wih Muliple Supply Sources and Neworks of Queues wih Overflow Bypasses. Managemen Science 55 (3):

16 Veeraraghavan, S., and A. Scheller-Wolf Now or Laer: A Simple Policy for Effecive Dual Sourcing in Capaciaed Sysems. Operaions Research 56 (4): Zipkin, P. 2008a. Old and New Mehods for Los-Sales Invenory Sysems. Operaions Research 56 (5): Zipkin, P. 2008b. On he Srucure of Los-Sales Invenory Models. Operaions Research 56 (4): A. Appendix wih Proofs Proof of Theorem 1 Proof. Le C D,Q, (h, b, c, l E, l R, F ) denoe he opimal cos over all admissible policies which order a consan amoun Q from R in every period (hese policies include policies oher han order-up-o policies for E). We firs claim ha among policies which order a consan amoun Q < µ from R in every period, here exiss an opimal policy of he order-up-o ype for E. Tha is, C D,Q, (h, b, c, l E, l R, F ) = C D,Q,S (Q) (h, b, c, l E, l R, F ). The proof of his claim is he following: The invenory process in a dual sourcing sysem ha orders Q from supplier R in every period is idenical o ha in a single sourcing sysem wih lead ime l E and in which he random variable represening demand in any period is given by D Q. I is well known ha he opimal policy in such a single sourcing sysem (even hough he random demand, D Q, in a period need no be non-negaive) is an order-up-o policy. This fac implies he claim. We will suppress he dependence of C D,Q, on he parameers (h, b, c, l E, l R, F ) in his proof. We will show he resul by demonsraing he following inequaliy: C D,Q 1, + C D,Q 2, 2 C D, Q 1 +Q 2 2, for all Q 1 and Q 2. (A.1) Le θ 1 (θ 2 ) denoe he opimal policy wihin he class of policies which order he consan quaniy Q 1 (Q 2 ) from R in every period. Tha is, C D,θ 1 = C D,Q 1, and C D,θ 2 = C D,Q 2,. Le us now consider a hird policy θ 3 defined as follows: I orders he consan quaniy (Q 1 +Q 2 )/2 from R every period and, in every period, i orders from E he average of he quaniies ordered by θ 1 and θ 2 from E. Tha is, q θ 3,R = Q 1+Q 2 2 and q θ 3,E qθ 1,E = 2 for all. This implies ha he procuremen cos +q θ 2,E under θ 3 in every period is he average of he procuremen coss under θ 1 and θ 2. Le us now examine he holding and shorage coss under θ 3. Assuming he same saring sae under all hree policies, 16

17 i is easy o see from he definiion of θ 3 ha x θ 3 = xθ 1 +x θ 2 2 for all. The holding and shorage cos in period is a convex funcion of he ne-invenory a he beginning of period. This convexiy and he equaliy above, along wih our earlier observaion abou procuremen coss, imply ha c θ 3 cθ 1 +c θ 2 2. Therefore, we obain C D, Q1+Q 2 2, C D,θ 3 CD,θ 1 +C D,θ 2 2 = CD,Q 1, +C D,Q 2, 2. This complees he proof of (A.1). Proof of Lemma 2 Proof. In his proof, we will use B o denoe B(h, b, l E, F ) and D o denoe D(h, b, c, l E, l R, F ). Le S B, denoe he opimal order-up-o level in B. Noe ha he order-up-o policy wih his arge level achieves a long run average cos of C B, (h, b, l E, F ) in B. Le us now sudy D under he TBS policy wih a consan order quaniy of Q from R and he order-up-o policy wih arge level S B, for E. Le us use θ o denoe his policy. Observe ha he expeced holding and shorage cos in D per period under θ is [ (S h E B, + O (Q) D[1, l E + 1] ) ] [ + (S + b E B, + O (Q) D[1, l E + 1] ) ]. Using he facs ha, for all (x, y) such ha y 0, (x + y) + x + + y and (x + y) x, we see ha his cos is bounded above by [ (S h E B, D[1, l E + 1] ) ] [ + (S + b E B, D[1, l E + 1] ) ] + h E[O (Q)]. Noice ha he sum of he firs wo erms in his expression is nohing bu C B, (h, b, l E, F ). Thus, we know ha he long run average cos in D under θ is bounded above by C B, (h, b, l E, F ) + h E[O (Q)] + c (µ Q), (A.2) where he las erm capures he procuremen cos (since Q unis are procured from R every period, µ Q is he average amoun procured from E per period). Nex, recall ha O (Q) represens he seady sae of he sochasic process O +1 = (O + Q D ) +, which is idenical o he waiing ime recursion in a GI/GI/1 queue wih D represening he iner-arrival ime and Q, he service ime. We know from Kingman (1970) (inequaliy (9) on page 104) ha he expeced waiing ime in a GI/GI/1 queue is bounded above by (σ 2 s + σ 2 )/[2( s)] where σ s (σ ) is he sandard deviaion of he service (iner-arrival) ime and ( s) is he mean service (iner-arrival) ime. In our case, σ s = 0, σ = σ 2, s = Q and = µ. Subsiuing hese quaniies ino Kingman s bound above, we obain E[O (Q)] σ 2 2 (µ Q). (A.3) 17

18 Combining (A.2) and (A.3), we obain he following upper bound on he cos of θ and herefore on ) he cos of he bes TBS policy for a given Q: C B, (h, b, l E, F ) + h + c (µ Q). Proof of Theorem 3 ( σ 2 2 (µ Q) Proof. Le Q minimize he upper bound expression given in Lemma 2. I is easy o verify ha Q = µ σ h/2c which is non-negaive if µ σ h/2c. In his case, he cos of he bes TBS policy is bounded above by he cos of he TBS policy which uses Q = Q; i can be verified by subsiuing Q wih Q in Lemma 2 ha he cos of his policy is bounded above by C B, (h, b, l E, F )+σ 2 h c. In he case when µ σ h/2c, he cos of he bes TBS policy is bounded above by he cos of he TBS policy which uses Q = 0. The cos of ha policy is C B, (h, b, l E, F ) + c µ C B, (h, b, l E, F ) + σ h c/2 C B, (h, b, l E, F ) + σ 2 h c. Proof of Theorem 5 Proof. In order o avoid excessive noaion, we find i convenien o presen he elemens of he proof raher han hose deails of hese elemens which can be found in references ha we provide. While he performance measure used in his paper is he long run average cos, our proof requires considering a finie horizon, discouned cos dynamic program firs, an infinie horizon discouned cos dynamic program nex and finally he average cos dynamic program. Le us firs consider he finie horizon dynamic program which is characerized by a cos funcion say f α,t (x) when he demand disribuion is F and x represens he sae of he sysem, i.e., informaion on how much invenory is presen on hand, amoun on backorder if any, amouns on order from boh supply sources scheduled o arrive in he nex several periods (because of he lead imes). Here, α (0, 1) denoes a discoun facor o capure he ime value of money and T denoes he lengh of a finie planning horizon. Similarly, le α,t f (x) denoe he corresponding cos funcion when he demand disribuion is F. We define f α,t α,t T +1 (x) = f T +1 (x) = 0 for all x. I is a sandard exercise in dynamic programming o show ha f α,t and f α,t are convex funcions. Le us assume by inducion ha f α,t α,t +1 (x) f +1 (x) for all x. We can now use sandard argumens, he above inequaliy and he following facs (a) - (d) o show ha f α,t (x) f α,t (x) for all x. (A.4) The facs are: (a) The single period cos is a convex funcion of demand, for any given x. (b) The sae ransformaion (i.e. how he sae in period + 1 depends on he sae in period and he 18

19 demand in period ) is linear in (x, d), where d is he realizaion of demand. (c) F cx F. (d) The funcions f α,t and f α,t are convex. Nex, we appeal o sandard dynamic programming convergence argumens 8 (as T ) along wih (A.4) o conclude ha he infinie horizon discouned cos funcions f α, 1 (x) and also saisfy he inequaliy f α, 1 (x) f α, 1 (x) f α, 1 (x) for all x. (A.5) Furhermore, using he vanishing discoun approach in he dynamic programming lieraure, we know ha he funcions f α, 1 (x) and f α, 1 (x) converge o he opimal long run average coss C D, (h, b, c, l E, l R, F ) and C D, (h, b, c, l E, l R, F ), respecively, as he discoun facor α approaches 1. This convergence along wih (A.5) implies he desired inequaliy C D, (h, b, c, l E, l R, F ) C D, (h, b, c, l E, l R, F ). Proof of Lemma 6 Proof. We begin wih he proof of he second resul since i is sraighforward. Consider any policy θ which is used in he sysem ha faces he demand disribuion F. We modify ha policy o consruc anoher policy ˆθ as follows: q ˆθ,R = B q θ,r and q ˆθ,E = B q θ,e for all. Then, since he random demand in period under he disribuion ˆF is B D for all, i is easy o verify ha he long run average cos under ˆθ and ˆF is exacly B imes he long run average cos under θ and F. Since θ was an arbirary, feasible policy, his implies ha C D, (h, b, c, l E, l R, ˆF ) B C D, (h, b, c, l E, l R, F ). An argumen symmeric o he above leads o he opposie inequaliy hus yielding he desired equaliy. This complees he proof of he second resul. We now proceed o prove he firs resul. The logic is similar o ha used above. Again, we consider any policy θ which is used in he sysem ha faces he demand disribuion F. We modify ha policy o consruc anoher policy θ as follows: q θ,r = A + q θ,r and q θ,e = q θ,e for all. Tha is, in every period, θ follows θ idenically for orders from E and orders A unis in excess of ha ordered by θ from R. I is easy o he long run average cos under θ and F is exacly equal o he long run average cos under θ and F. Since θ was an arbirary, feasible policy, his implies ha C D, (h, b, c, l E, l R, F ) C D, (h, b, c, l E, l R, F ). 8 Please see Schäl (1993) and Hernandez-Lerma and Lassere (1996) for general resuls of his ype and Huh e al. (2011) for how hese resuls apply o a general class of invenory problems. More specifically, Sheopuri e al. (2010) commen on he implicaion of hese resuls o he dual sourcing problem sudied here. 19

20 I only remains o esablish he opposie inequaliy, which we focus on nex. Since our ineres is limied o he long run average cos performance measure, we assume wihou loss of generaliy ha boh he sysem facing he demand disribuion F and ha facing F sar period 1 wih an empy pipeline (i.e. no ousanding orders) and a ne-invenory of zero. Le us now consider any policy θ which is used in he sysem ha faces he demand disribuion F. We modify ha policy o consruc anoher policy θ as follows: l R u=1 l R u=1 l R u = ( q θ,r l E qu θ,r + q θ,e u=1 u=1 l R u = ( u=1 q θ,r u A) + and q θ,r l E u + q θ,e u A) +. u=1 Such a careful consrucion is necessary for policy θ o ensure ha is orders are non-negaive. In words, policy θ aemps o bring in he same amoun of cumulaive supply ino he sysem as policy π up unil any period excep for a deliberae reducion of A unis which is he excess cumulaive demand ha he sysem facing F experiences relaive o he sysem facing F. I is now easy o verify ha he long run average cos under θ and F is less han or equal o ha under θ and F. Since θ is an arbirary, feasible policy, his implies ha C D, (h, b, c, l E, l R, F ) C D, (h, b, c, l E, l R, F ). The desired equaliy follows from his inequaliy and is opposie which was proved earlier. A Preliminary Resul used in he Proof of Theorem 8 Recall ha, in he discussion preceding he saemen of Theorem 8, we had argued inuiively ha he TBS policy wih Q = 0 and S = 0 is opimal when d Low = 0 and d High = 1. We firs prove his claim formally before proving he heorem for he more general case of arbirary, non-negaive values of d Low and d High such ha d Low < d High. Lemma 10. Under Assumpion 1, he TBS policy wih Q = 0 and S = 0 is opimal when d Low = 0 and d High = 1. Moreover, he opimal cos and he cos of his TBS policy equal β (1 p Low ) where β := ( c + b (l E + 1) ). Proof. Assume ha he sysem sars wih zero backorders and no invenory anywhere in he sysem. This is wihou loss of generaliy since our focus is on he long run average cos performance measure. Le ˆθ denoe he TBS policy wih Q = 0 and S = 0. This policy does no incur any 20

21 holding coss. Moreover, every cusomer is backordered for exacly l E + 1 periods and is saisfied by a uni ordered from E a a cos of c. Thus, he long run average cos of ˆθ is C ˆθ = β µ = β (1 p Low ). We now claim ha, for any feasible policy, θ, he long run average cos C θ is a leas as large as C ˆθ, ha is, C θ β µ. (A.6) This claim implies he desired resul ha ˆθ is an opimal policy. We now proceed o prove (A.6). Consider any feasible policy, θ. Le C θ [1, T ] denoe he coss incurred under θ during he firs T periods. Similarly, we use H θ [1, T ] o denoe he holding coss and B θ [1, T ] o denoe he sum of procuremen coss and backordering coss under θ during ha inerval. Tha is, C θ [1, T ] = H θ [1, T ] + B θ [1, T ]. The long run average cos C θ can be wrien as C θ = lim sup T E [ C θ [1, T ] ] T. We now claim ha o show (A.6), i is sufficien o show he following saemen: [ ] For any ɛ > 0, here exiss T (ɛ) < s.. E C θ [1, T + T (ɛ)] β µ (1 ɛ) T. (A.7) The proof of his claim is he following: C θ = lim sup T = lim sup T = lim sup T E [ C θ [1, T ] ] T E [ C θ [1, T + T (ɛ)] ] T + T (ɛ) E [ C θ [1, T + T (ɛ)] ] T lim sup β µ (1 ɛ) T = β µ (1 ɛ). T T + T (ɛ) T [from (A.7)] T + T (ɛ) Since ɛ is an arbirary consan, his implies ha C θ β µ, which is he desired inequaliy (A.6). Thus, i only remains o prove ha he saemen in (A.7) is rue. 21

22 We proceed o prove (A.7). Since he demand in every period is zero or one, we refer o a uni of demand as a cusomer in his proof. We use F o denoe he demand hisory up unil he beginning of period, ha is, he realizaion of (D 1, D 2,..., D 1 ). Le n θ denoe he number of unis ha are ordered (from eiher R or E) in period wihou a corresponding cusomer ready a he beginning of ha period. While a mahemaical expression for n θ is no used in he proof, we find i useful o presen one in order o help he reader undersand he meaning of n θ beginning of period, hen, n θ = q θ,r where IP,sar θ,r ordering decisions are made. exceeds commied supply. more precisely. For example, if here are zero cusomers backordered a he + q θ,e. More generally, we have ( n θ = max 0, q θ,r + q θ,e (IP,sar θ,r ) ), is he oal invenory posiion in he sysem a he sar of period before any Thus, (IP θ,r,sar ) is he number by which backordered demand Therefore, if he oal order quaniy in period, i.e., q θ,r + q θ,e exceeds (IP θ,r,sar ), heir difference is he number of unis ha are ordered (from eiher R or E) in period wihou a corresponding cusomer ready a he beginning of ha period. Le N θ = n θ. u=1 We now derive a lower bound on he expeced number of periods each of hese unis will say in invenory and use ha o derive a lower bound on E [ C θ [1, T + T (ɛ)] ] for a suiably chosen T (ɛ) and complee he proof of (A.7). We accomplish his by assembling several ideas. Le ζ denoe a geomeric random variable wih success probabiliy (1 p Low ). Tha is, for any, given F, he disribuion of he number of periods unil he nex cusomer arrives is ζ. Noe ha E[ζ] = p Low /(1 p Low ). Le [ (ζ τ = E (l R + 1) ) ] +. The meaning of his quaniy is he following. Consider a uni which is ordered from R wihou a corresponding cusomer ready a ha ime. Then, his uni incurs a holding cos from he ime i 22

23 is delivered and a leas 9 unil he nex cusomer arrives. Le [ { (ζ τ(u) = E min (l R + 1) ) }] +, u This quaniy can be undersood as follows: Consider a uni which is ordered from R in some period wihou a corresponding cusomer ready a ha ime. Then, τ(u) is a lower bound on he expeced holding cos incurred by his uni during he inerval [, + u].. I is easy o verify ha τ(u) is an increasing funcion of u and ha lim τ(u) = τ. u Therefore, for any ɛ > 0, we know ha τ(u) τ (1 ɛ) for sufficienly large u - we choose such a value of u as T (ɛ); more formally, T (ɛ) = min{u N : τ(u) τ (1 ɛ)}. Nex, observe from he definiion of τ ha τ E[ζ] (l R + 1) = p Low /(1 p Low ) (l R + 1). This implies ha τ(t (ɛ)) ( p Low /(1 p Low ) (l R + 1) ) (1 ɛ). Recall ha Assumpion 1 (please see he discussion on ha assumpion in Secion 5.1) is equivalen o (5.1), ha is, h (p Low /(1 p Low ) (l R + 1) ) c + b (l E + 1). Combining he las wo inequaliies, we obain h τ(t (ɛ)) (c + b (l E + 1)) (1 ɛ) = β (1 ɛ). (A.8) We are now ready o examine E [ C θ [1, T + T (ɛ)] ], he lef hand side of (A.7). We have [ ] E C θ [1, T + T (ɛ)] [ ] [ ] = E H θ [1, T + T (ɛ)] + E B θ [1, T + T (ɛ)]. (A.9) 9 We say a leas because when a uni is released from R in a period wihou having a corresponding cusomer ready, his uni migh have o wai in invenory for a period longer han τ if a uni is subsequenly released from E and is delivered before he former uni. We also noe ha we use discree erms such as unis in his par of he proof purely for he sake of exposiional clariy. We are no assuming ha order quaniies are inegers. 23

24 We will firs derive a lower bound on he holding cos and hen do he same for he procuremenplus-backorder cos. From he definiion of n θ and he discussion surrounding i, i can be observed ha he expeced holding cos E [ H θ [1, T + T (ɛ)] ] exceeds = T E[holding coss incurred in [, T + T (ɛ)] by he n θ unis ordered in ] =1 T [ E F E[holding coss incurred in [, T + T (ɛ)] by he n θ unis ordered in F ] ]. =1 From he definiion of τ(t (ɛ)), we can see ha, given F, he expeced holding coss incurred in [, T + T (ɛ)] by he n θ unis ordered in exceeds n θ h τ(t (ɛ)). Thus, we have [ ] E H θ [1, T + T (ɛ)] T E[n θ h τ(t (ɛ))] =1 = E[N θ T ] h τ(t (ɛ)) E[N θ T ] β (1 ɛ) [from (A.8)]. We now bound he procuremen and backorder coss, i.e., E [ B θ [1, T + T (ɛ)] ]. Le D[1, T ] := T =1 D. Firs, we use he definiion of NT θ o make he following observaion: Of he D[1, T ] cusomers who arrive in he firs T periods, a leas (D[1, T ] N θ T )+ cusomers do no find a corresponding uni waiing for hem immediaely upon arrival. Given our assumpion ha c < b (l R l E ), his implies ha [ ] E B θ [1, T + T (ɛ)] E [(D[1, T ] NT θ ) +] (c + b (l E + 1) ) [ = E (D[1, T ] NT θ ) +] β [ ] E (D[1, T ] NT θ ) β [ ] E (D[1, T ] NT θ ) β (1 ɛ) (since ɛ > 0). Summing he lower bounds on he holding coss and he procuremen-cum-backorder coss, we obain [ ] E C θ [1, T + T (ɛ)] [ ] E[NT θ ] β (1 ɛ) + E (D[1, T ] NT θ ) β (1 ɛ) = β µ (1 ɛ) T. Thus, we have shown (A.7). This complees he proof of he lemma. 24

25 Proof of Theorem 8 Proof. Le F denoe he wo poin disribuion wih a mass of p Low a 0 and a mass of (1 p Low ) a 1. Le F denoe he wo poin disribuion wih a mass of p Low a d Low and a mass of (1 p Low ) a d High. Then, an applicaion of Corollary 7 leads o he following relaionship: C D, (h, b, c, l E, l R, F ) = (d High d Low ) C D, (h, b, c, l E, l R, F ). We know from Lemma 10 ha C D, (h, b, c, l E, l R, F ) = β (1 p Low ). Thus, we have C D, (h, b, c, l E, l R, F ) = (d High d Low ) β (1 p Low ). I only remains o show ha C θ = (d High d Low ) β (1 p Low ), where θ is he TBS policy wih Q = d Low and S = (l E + 1) d Low. We proceed o esablish his claim. Assume wihou loss of generaliy (due o he consideraion of long run average coss) ha he sysem sars period 1 wih d Low unis of invenory on hand and exacly d Low unis o be delivered k periods from now for every k {1, 2,..., l R 1}. Inuiively, his is he ideal sae for he sysem o sar from if he demand in every period is d Low. In his sae, he expedied invenory posiion a he beginning of period 1 is exacly d Low (l E + 1) - hus, no order is placed on E in period 1 under θ. In fac, i is easy o see ha he firs sricly posiive order is placed on E by θ in he firs period ( 2) such ha D 1 = d High. Moreover, he expedied invenory posiion a he beginning of such a period is d Low (l E + 1) (d High d Low ) and, herefore, θ orders he quaniy d High d Low from E o raise he expedied invenory posiion o is arge level d Low (l E + 1). Furhermore, i can also be verified ha, subsequenly, a sricly posiive order is placed on E by θ in a period if and only if he demand in he previous period is d High ; ha order quaniy will also be d High d Low. Thus, he expeced procuremen cos in a period under θ is (1 p) c (d High d Low ). By design, θ never holds invenory on hand since he demand in any inerval of (l E +1) periods is a leas d Low (l E + 1) which is equal o he expedied invenory posiion afer ordering in any period (noe ha here is no overshoo under policy θ since Q = d low D for all ). So, θ does no incur holding coss. In oher words, θ is equipped o handle a demand of d Low in every period. On he oher hand, in every period (say ) ha he demand is d High, he excess demand d High d Low is backordered and his amoun says on backorder for exacly (l E + 1) periods (unil he order placed on E in period +1 arrives, i.e., unil period +1+l E ). Thus, he expeced backorder cos incurred by θ in a period is (1 p Low ) b (d High d Low ) (l E + 1). Summing he procuremen and backorder coss, we see ha he long run average cos under θ is C θ = (1 p Low ) (d High d Low ) (c+b (l E +1)) which equals (d High d Low ) β (1 p Low ). This complees he proof. 25

26 Proof of Theorem 9 Proof. The firs saemen we are required o prove is ha ( 2 ) min Q C D,Q, (h, b, c, l E, l R, F ) σ X h c + h b (l E + 1) C D, (h, b, c, l E, l R 1 +, F ) (c + b (l E. (A.10) + 1)) (1 p) µ Y Le F X, F Y and F Z denoe he disribuion funcions of he random variables, X, Y and Z, respecively. Le Q X minimize C D,Q, (h, b, c, l E, l R, F X ). Clearly, Nex, we claim ha min Q CD,Q, (h, b, c, l E, l R, F ) C D,Q X, (h, b, c, l E, l R, F ). (A.11) C D,QX, (h, b, c, l E, l R, F ) min Q CD,Q, (h, b, c, l E, l R, F X ) + c (1 p) µ Y + C B, (h, b, l E, F Z )(A.12). The proof is he following: Le S X = arg min S C D,Q X,S (h, b, c, l E, l R, F X ), i.e., he opimal orderup-o level for E given ha he demand disribuion is F X and a consan order of Q X is placed on R every period. Le S Z = arg min C B,S (h, b, l E, F Z ). Le I X, denoe he expedied invenory-posiion in D(h, b, c, l E, l R, F X ) a he beginning of period (afer ordering) under he (Q X, S X ) policy. Assume ha I X,1 = S X and ha hose ousanding orders which are no a par of his expedied invenory posiion are of size Q X each (his is wihou loss of generaliy because our ineres is in he long run average coss of he invenory sysems analyzed). Similarly, le I Z, denoe he expedied invenory-posiion in B(h, b, l E, F Z ) a he beginning of period (afer ordering) under he order-up-o S Z policy. Assuming ha I Z,1 = S Z, we have he rivial ideniy I Z, = S Z for all. Consider he following policy θ in D(h, b, c, l E, l R, F ) defined as follows. The consan quaniy Q X is ordered from R in every period. The order quaniy from E under θ (in period ) is he sum of he quaniy ordered in D(h, b, c, l E, l R, F X ) under he (Q X, S X ) policy (in period ) and he order quaniy in B(h, b, l E, F Z ) (in period ) under he order-up-o S Z policy. Le I X+Z, denoe he expedied invenory posiion in D(h, b, c, l E, l R, F ) a he beginning of period afer ordering under policy θ. Under he assumpion ha I X+Z,1 = S X + S Z, he definiion of θ implies he following relaionship: I X+Z, = I X, + I Z, for all. Le us define e X,, e Z, and e X+Z, o be he ne-invenories a he end of period in he hree sysems, D(h, b, c, l E, l R, F X ) (under he (Q X, S X ) policy), B(h, b, l E, F Z ) (under he order-up-o S Z policy), and D(h, b, c, l E, l R, F ) (under policy θ). These quaniies also have a similar relaionship; 26

27 ha is, e X+Z, = e X, + e Z, for all. This relaionship along wih he sub-addiiviy of he newsvendor cos funcion h(e) + + b(e) implies ha he holding and shorage cos in D(h, b, c, l E, l R, F ) (under policy θ) is smaller han he sum of he holding and shorage coss in D(h, b, c, l E, l R, F X ) (under he (Q X, S X ) policy) and in B(h, b, l E, F Z ) (under he order-up-o S Z policy). Moreover, since he policy θ orders Q X from R in every period while he expeced demand is E[X +Z], he procuremen cos in D(h, b, c, l E, l R, F ) is equal o he procuremen cos in D(h, b, c, l E, l R, F X ) (under he (Q X, S X ) policy) plus c E[Z] which is nohing bu c (1 p) µ Y. These observaions abou he holding, shorage and procuremen coss yield he following inequaliy: C D,θ (h, b, c, l E, l R, F ) C D,Q X,S X (h, b, c, l E, l R, F X ) + c (1 p) µ Y + C B,S Z (h, b, l E, F Z ). The following hree relaionships follow direcly from definiions: (i) C D,Q X, (h, b, c, l E, l R, F ) C D,θ (h, b, c, l E, l R, F ) (since θ is jus one specific feasible policy ha orders Q X from R in every period), (ii) min Q C D,Q, (h, b, c, l E, l R, F X ) = C D,Q X,S X (h, b, c, l E, l R, F X ), and (iii) C B, (h, b, l E, F Z ) = C B,S Z (h, b, l E, F Z ). Combining hese relaionships wih he previous inequaliy, we obain he claimed inequaliy of (A.12). Nex, observe ha using an order-up-o zero policy in B(h, b, l E, F Z ) leads o an average cos of b (l E + 1) E[Z] = b (l E + 1) (1 p) µ Y. This implies he fac ha C B, (h, b, l E, F Z ) b (l E + 1) (1 p) µ Y. Using his fac in (A.12), we obain C D,Q X, (h, b, c, l E, l R, F ) min Q CD,Q, (h, b, c, l E, l R, F X ) + ( c + b (l E + 1) ) (1 p) µ Y.(A.13) We know from Theorem 3 ha min Q C D,Q, (h, b, c, l E, l R, F X ) C B, (h, b, l E, F X ) + σ X 2 h c. Since F X has a sandard deviaion of σ X, we can use a well known Disribuion-Free News-vendor resul (Gallego and Moon, 1993) ha C B, (h, b, l E, F X ) σ X l E + 1 h b. Thus, ( 2 ) min Q CD,Q, (h, b, c, l E, l R, F X ) σ X h c + h b (l E + 1) (A.14) We can now use (A.13) o obain ( 2 ) C D,QX, (h, b, c, l E, l R, F ) σ X h c + h b (l E + 1) + ( c + b (l E + 1) ) (1 p) µ Y. (A.15) 27

28 This inequaliy along wih (A.11) implies ha min Q CD,Q, (h, b, c, l E, l R, F ) σ X ( 2 ) h c + h b (l E + 1) + ( c + b (l E + 1) ) (1 p) µ Y. (A.16) The inequaliy above provides an upper bound on he cos of he bes TBS policy. Nex, we derive a lower bound on he opimal cos over all admissible policies. Our lower bound derivaion makes use of Theorem 5 on convex ordering. Le F denoe he disribuion of a random variable D defined as D = µ X + Z. I is easy o see ha F cx F. Thus, C D, (h, b, c, l E, l R, F ) C D, (h, b, c, l E, l R, F ). Le us now examine C D, (h, b, c, l E, l R, F ). Since D is he sum of a deerminisic quaniy µ X and he random variable Z, he opimal policy will procure µ X unis from R every period o handle he deerminisic par of demand. So, he only procuremen, holding and shorage coss incurred are hose incurred on he sochasic par of demand, Z. Noe ha Z has a probabiliy mass of, a leas, p a zero. The mean of Z is (1 p) µ Y. We can now use an analysis ha is idenical o ha presened in Secion 5.1 o see ha he opimal policy, when he demand is given by he random variable Z (µ X + Z), is o order zero (µ X ) from R in every period and order-up-o zero from E in every period. The cos of his policy, i.e. he opimal cos, is ( c + b (l E + 1) ) (1 p) µ Y. To summarize, we have now esablished ha C D, (h, b, c, l E, l R, F ) ( c + b (l E + 1) ) (1 p) µ Y. (A.17) Combining (A.16) and (A.17), we obain min Q C D,Q, (h, b, c, l E, l R, F ) C D, (h, b, c, l E, l R, F ) which is he desired resul in (A.10). 1 + ( 2 ) σ X h c + h b (l E + 1) (c + b (l E, + 1)) (1 p) µ Y We now proceed o esablish he second resul. When h c and c < b (l E + 1), he righ hand side in he above inequaliy (i.e., (A.10)) is bounded above by which, in urn, is smaller han 1 + σ X ( 2 + 1) h b (l E + 1) (h + b (l E + 1)) (1 p) µ Y, 1 + ( 1 + ) ( 2 σ X 2 (1 p) µ Y due o he fac ha he arihmeic mean exceeds he geomeric mean. This complees he proof of he second desired resul. ). 28

29 Two Poin Dis.: P(D = 1) = 2/3 and P(D = 4) = 1/3 L_R Op TBS % Diff L_R Op TBS % Diff L_R Op TBS % Diff h = h = h = b = b = b = c = c = c = h = h = h = b = b = b = c = c = c = Unimodal Symmeric Dis.: P(D=0) = 0.125, P(D = 1) = 0.2, P(D = 2) = 0.35, P(D = 3) = 0.2, P(D=4) = L_R Op TBS % Diff L_R Op TBS % Diff L_R Op TBS % Diff h = h = h = b = b = b = c = c = c = h = h = h = b = b = b = c = c = c = Righ Skewed Dis.: P(D=0) = 0.125, P(D=1) = 0.5, P(D=2) = 0.125, P(D=3) = 0.125, P(D=4) = L_R Op TBS % Diff L_R Op TBS % Diff L_R Op TBS % Diff h = h = h = b = b = b = c = c = c = h = h = h = b = b = b = c = c = c =

30 Lef Skewed Dis.: P(D=0) = 0.125, P(D=1) = 0.125, P(D=2) = 0.125, P(D=3) = 0.5, P(D=4) = L_R Op TBS % Diff L_R Op TBS % Diff L_R Op TBS % Diff h = h = h = b = b = b = c = c = c = h = h = h = b = b = b = c = c = c = Bi Modal Dis.: P(D=0) = 0.1, P(D=1) = 0.35, P(D=2) = 0.1, P(D=3) = 0.1, P(D=4) = 0.35 L_R Op TBS % Diff L_R Op TBS % Diff L_R Op TBS % Diff h = h = h = b = b = b = c = c = c = h = h = h = b = b = b = c = c = c = Uniform Dis.: P(D=0) = 0.2, P(D=1) = 0.2, P(D=2) = 0.2, P(D=3) = 0.2, P(D=4) = 0.2 L_R Op TBS % Diff L_R Op TBS % Diff L_R Op TBS % Diff h = h = h = b = b = b = c = c = c = h = h = h = b = b = b = c = c = c =

31 Uniform Dis.: P(D=0) = 0.2, P(D=1) = 0.2, P(D=2) = 0.2, P(D=3) = 0.2, P(D=4) = 0.2. Two Poin Dis.: P(D = 1) = 2/3 and P(D = 4) = 1/3. Unimodal Symmeric: P(D = 0) = 0.125, P(D = 1) = 0.200, P(D = 2) = 0.350, P(D = 3) = 0.200, P(D = 4) = Righ-Skewed Dis.: P(D=0) = 0.125, P(D=1) = 0.500, P(D=2) = 0.125, P(D=3) = 0.125, P(D=4) = Lef-Skewed Dis.: P(D=0) = 0.125, P(D=1) = 0.125, P(D=2) = 0.125, P(D=3) = 0.500, P(D=4) = Bi-Modal Dis.: P(D=0) = 0.10, P(D=1) = 0.35, P(D=2) = 0.10, P(D=3) = 0.10, P(D=4) = (l E, l R ) = (1, 4) (l E, l R ) = (2, 5) (l E, l R ) = (3, 6) h b c OPT TBS %Diff OPT TBS %Diff OPT TBS %Diff Table showing he effec of changing l E while keeping l R -l E consan a hree periods.