T L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB

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1 Elecronic Companion EC.1. Proofs of Technical Lemmas and Theorems LEMMA 1. Le C(RB) be he oal cos incurred by he RB policy. Then we have, T L E[C(RB)] 3 E[Z RB ]. (EC.1) Proof of Lemma 1. Using he marginal cos accouning in Equaion(4) and sandard argumens of condiional expecaions, we express T L T L E[C(RB)] = E[H RB (Q RB )+Π RB (Q RB )+K 1(Q RB >0)] (EC.2) = T L = E [ E[H RB (Q RB )+Π RB (Q RB )+K 1(Q RB >0) F ] ] E[2Z RB T L +P K] 3 E[Z RB ]. The hird equaliy follows direcly from (11). To esablish he firs inequaliy in (EC.2) above, we shall show ha Z P K almos surely. Tha is, for each f F, z p K. Given any informaion se f, all he quaniies x, θ, ψ, φ and p defined above are known deerminisically. We spli he analysis ino wo cases: 1. If θ K, hen q RB The claim follows. 2. If θ <K, hen q RB = ˆq (he balancing quaniy) wih probabiliy p =1 implying z =θ K. = q (he holding-cos-k quaniy) wih probabiliy p and q RB 1 p. Thus, by Equaions (8) and (9), we have z =p K, and he claim follows. =0 wih This concludes he proof of he lemma. LEMMA 2. The overall holding cos and backlogging cos incurred by OPT are denoed by H OPT and Π OPT, respecively. Then we have, wih probabiliy 1, H OPT H RB 1( T 1H T2H ), Π OPT Π RB 1( T 1Π T2Π ). (EC.3) Proof of Lemma 2. The proof is idenical o Lemmas 4.2 and 4.3 in Levi e al. (2007). ec1

2 ec2 e-companion o Levi and Shi: Approximaion Algorihms for he Sochasic Lo-sizing Problem wih Order Lead Times LEMMA 3. The expeced holding cos and backlogging cos incurred by OP T plus he expeced amoun borrowed from he bank accoun A are a leas T L E[ZRB ]. Tha is, The following inequaliy holds E [( H OPT +Π OPT) +A ] T L E[Z RB ]. (EC.4) Proof of Lemma 3. Using lineariy of expecaion, i suffices o show E [ H OPT +Π OPT] T L E [ ] 1( T N ) Z RB. (EC.5) Using Lemma 2 and sandard argumens of condiion expecaions, we have [ ] E[H OPT ] E H RB 1( T 1H T2H ) = E = E [ E [ [ H RB 1( T 1H T2H ) F ]] Z RB 1( T 1H T2H ) ]. (EC.6) Similarly, we also have E[Π OPT ] E [ Z RB 1( T 1Π T2Π ) ]. (EC.7) Equaion (EC.5) follows from summing up Equaions (EC.6) and (EC.7). LEMMA 4. The following inequaliy holds E[A] E [ T L K 1(Q OPT >0) ]. (EC.8) In oher words, he expeced borrowing E[A] is less han he oal expeced fixed ordering cos incurred by OPT. Proof of Lemma 4. Firs we define he reduced informaion se f o be he informaion up o period excluding he randomized decisions of he RB policy over [1, 1]. In paricular, given he enire evoluion of demand f T, he sequence of orders placed by OPT is known deerminisically. Le 1 1 < 2 <...< n T L be he periods in which OPT placed n=n f T orders sequenially. Le 0 =0 and n+1 =T L+1. We shall show ha here are no problemaic periods wihin ( 0, 1 ) and ha, for each i=1,...n, he expeced borrowing wihin he inerval [ i, i+1 ) does no exceed K. Tha is,

3 e-companion o Levi and Shi: Approximaion Algorihms for he Sochasic Lo-sizing Problem wih Order Lead Times ec3 E ( 0, 1 ) T 2M =, (EC.9) Z RB [ i, i+1 ) f T T 2M K. (EC.10) I is imporan o noe ha f T does no include he randomized decisions of he RB policy. Thus, he se T 2M is sill random and so is he amoun borrowed from he bank. In paricular, he expecaion in Equaion (EC.10) is aken wih respec o he randomized decisions of he RB policy. Equaions (EC.10) and (EC.9) imply ha, for each f T, E K n f T =K n, and herefore Z RB f T T 2M E[A] K E[N]=E [ T L K 1(Q OPT >0) ] (EC.11). (EC.12) Thus, i suffices o prove Equaions (EC.10) and (EC.9). Figure EC.1 gives a graphical inerpreaion of Equaion (EC.10), i.e., we wan o show ha he fixed ordering cos K incurred by OPT in period i will cover he expeced amoun borrowed from he bank in periods ha belong o se T 2M wihin he inerval [ i, i+1 ). Figure EC.1 Decomposiion of he problemaic periods in he se T 2M ino inervals beween ordering poins of OPT Proof of Equaion (EC.9). WefirsshowhaEquaion(EC.9)holds.Recallhedefiniion T 2M = { } :Θ <K and X RB <Y OPT X RB RB + Q. Since a he beginning of he planning horizon, i is assumed ha every feasible policy will have he same iniial invenory posiion, i follows ha if period is in T 2M, OPT mus have placed an order and overaken he invenory posiion of he RB policy. (The wo policies face he same sequence of demands.) However, ( 0, 1 ) denoes he se of periods in which OPT has no placed any order ye. Thus, he inersecion of hese wo ses is empy.

4 ec4 e-companion o Levi and Shi: Approximaion Algorihms for he Sochasic Lo-sizing Problem wih Order Lead Times Proof of Equaion (EC.10). Nex we show ha Equaion (EC.10) holds. Recall ha f T denoes an enire evoluion of he sysem excluding he randomized decisions of he RB policy. Given he enire evoluion of demands f T, consruc a decision ree based on he randomized decisions of he RB policy. The roo node corresponding o period 1 conains he informaion se f 1 =f 1 f T. The ree is buil in layers, each corresponding o a period, where he number of nodes in layer is 2 1 numbered l=1,...,2 1. In paricular, a node l in period (layer) corresponds o some informaion se f F which includes he realized reduced informaion se f f T, and he realized randomized decisions up o period 1 of he RB policy. Therefore i is known wheher under his sae period belongs o he se T 2M or no. The edges in he ree represen he differen (randomized) decisions ha he RB policy may make wih heir respecive probabiliies. Each pah from he roo o a specific node corresponds o a sequence of realized randomized ordering decisions made by he RB policy. For example, consider again some node l in period (layer) in which he RB policy will order q RB l unis wih probabiliy p l and nohing wih probabiliy 1 p l ; hen he node l in period (denoed by l) will have wo edges o wo children nodes in he nex period +1 each conaining is disincive ordering informaion. Concepually one can hink abou he decision ree as a collecion of independen coins, each corresponding o a node in he ree. The coin corresponding o node l a layer (period) has probabiliy of success (ordering) p l. Nex we pariion he nodes in he ree ino problemaic nodes (pn nodes), i.e., nodes ha correspond o a pair (,f ) for which T 2M, and non-problemaic nodes (nn nodes). An example of a general decision ree is illusraed in Figure EC.2. Focus now on a specific ime inerval [ i, i+1 ). Suppose we have consruced he ree from period 1 o T; he number of nodes and pahs are clearly finie (possibly exponenial). Le he se G o be he se of all possible oucomes of he randomized decisions in all nodes in layers wihin he inerval [1, i 1] and in all he nn nodes wihin he inerval [1,T]. In paricular, each g G corresponds o a specific se of oucomes in all nodes in layers (periods) wihin he inerval [1, i 1] and in all he nn nodes in he ree. Using he erminology of coins proposed before, g corresponds o he

5 e-companion o Levi and Shi: Approximaion Algorihms for he Sochasic Lo-sizing Problem wih Order Lead Times ec5 Figure EC.2 An example of a general decision ree oucome of he respecive subse of coins corresponding o all nodes wihin [1, i 1] and all nn nodes wihin [1,T]. Condiioning on some g G induces a pah from he roo of he ree (in period 1) up o he earlies pn node, say j, where j corresponds o he period (layer) of ha node. Here we abuse he noaion ignoring he index of he node wihin layer j. (Namely, he exac value will be je for some e.) I is sraighforward o see ha j i. If j falls ouside he inerval [ i, i+1 ), i.e., j i+1, i follows ha here are no pn nodes wihin he inerval [ i, i+1 ), and here is no borrowing over he inerval. Assume now ha j falls wihin he inerval [ i, i+1 ) (j can possibly be in period (layer) i ). We will show ha he expeced borrowing does no exceed K. Tha is, E s f T,g K. s [j, i+1 ) T 2M Z RB (EC.13) The proof of Equaion (EC.10) will hen follow. Recallhanodej correspondsosomeinformaionsef j F j.ifollowshahesaringinvenory posiion x RB j and he corresponding holding-cos-k quaniy q RB j are known deerminisically. Condiioning on g, he only uncerainy in he evoluion of he sysem depends on he randomized decisions made in pn nodes wihin [j, i+1 ). Consider he sub-ree induced by condiioning on g. The

6 ec6 e-companion o Levi and Shi: Approximaion Algorihms for he Sochasic Lo-sizing Problem wih Order Lead Times Figure EC.3 An example of a decision subree: focus on he inerval [ i, i+1) and some g G, j is he earlies period in which a problemaic node (pn) occurs. According o g, here are wo possible oucomes whenever a problemaic node (pn) is reached, and here is only one possible oucome whenever a non-problemaic node (nn) is reached. If a problemaic node (pn) orders, here will no be furher borrowing unil he nex order of OPT in period i+1. non-problemaic nodes (nn nodes) in he sub-ree have only one ougoing edge ha corresponds o he decision (order/no-order) specified by g o ha node. The problemaic nodes (pn nodes) have wo ougoing edges corresponding o he order/no-order decisions, respecively. (Recall ha g does no specify he decisions in hese nodes.) Moreover, each pn node s [j, i+1 ) is associaed wih he probabiliy p s of ordering. (We again abuse he noaion inroduced before and omi he index e of he node wihin he layer/period.) An example of a decision subree specified by some g G is illusraed in Figure EC.3. Any sequence of randomized oucomes corresponding o he decisions in he pn nodes induces a pah of evoluion of he sysem. The resuling cumulaive borrowing from he bank accoun A, corresponding o his pah, is equal o K imes he sum of probabiliies associaed wih he pn nodes in his pah. (For each pn node s in he pah, he borrowing is equal o p s K =z s.) Nexweclaimhahesub-reedefinedaboveincludesamosonepnnodeineachlayer(period).

7 e-companion o Levi and Shi: Approximaion Algorihms for he Sochasic Lo-sizing Problem wih Order Lead Times ec7 This follows from he fac ha any pah beween wo pn nodes r,s such ha j r <s< i+1 in he ree includes only no-ordering edges of pn nodes. To see why he laer is rue, observe ha if an order is placed by he RB policy in a pn node, he resuling invenory posiion of he RB policy is higher han OPT. Since boh policies face he same sequence of demands, he RB policy will no have higher invenory posiion han OPT a leas unil he nex order placed by OPT. This excludes he exisence of pn nodes in subsequen periods unil OP T places anoher order, i.e., beyond period i+1 1. In ligh of he laer observaion, we re-number all he pn nodes in he sub-ree as 1,2,...,M (where 1 corresponds o j, specified before). Moreover, i follows ha he probabiliy o arrive a node m=1,...,m and borrow p m K is equal o m 1 (1 p s). (This probabiliy corresponds o no-ordering decisions in all he pn nodes prior o m.) The oal expeced borrowing is hen K { p 2 1+ M m=2 m )}} (1 p s ) p m p k. (EC.14) {( m 1 Observe ha he probabiliy o borrow exacly K m p k is equal o ( m 1 ) (1 p s) p m. Moreover, we have already shown ha he expression in (EC.14) is bounded above by K (see Lemma 5). This concludes he proof of he lemma. LEMMA 5. Le {p l } l=1 saisfy he condiion 0 p l 1 for all l. Then he following inequaliy holds, p 2 1+ l=2 {( l 1 l (1 p s ) p l p k )} 1. (EC.15) Proof of Lemma 5. We consruc an increasing sequence {a m } where a m =p 2 1+ m l=2 For each m, if we replace p m by 1, we ge m 1 ā m =p 2 1+ l=2 {( l 1 {( l 1 l (1 p s ) p l p k )}+ l )} (1 p s ) p l p k. (EC.16) )( ) m 1 (1 p s ) 1+ p k, (EC.17) ( m 1 such ha a m ā m. Nex we will show by inducion ha ā m 1 for all m from which he proof of

8 ec8 e-companion o Levi and Shi: Approximaion Algorihms for he Sochasic Lo-sizing Problem wih Order Lead Times he lemma follows. I is sraighforward o verify ā 1,ā 2 1. Assume ha ā m 1 for some m Z +, we will show ha ā m+1 1. ā m+1 = p 2 1+ m = a m 1 + = a m 1 + = a m 1 + l ( m )( ) m (1 p s ) p l p k )}+ (1 p s ) 1+ p k m ( m )( ) m (1 p s ) p m p k )+ (1 p s ) 1+ p k {( l 1 l=2 ( m 1 ( m 1 ( m 1 )[( ) m m ] (1 p s ) 1+ p k (1 p m )+p m p k m 1 )( (1 p s ) 1+ p k )=ā m 1. (EC.18) Hence he claim follows by inducion. EC.2. Performance of he proposed algorihms The firs wo columns specify he es insances, namely, fixed ordering cos K, per-uni holding cos h, per-uni backlogging cos p and demand rae vecor λ. The hird column shows he cos incurred by he opimal policy. The fourh column shows he opimal parameers of paramerized RB policy. The fifh column shows he cos incurred by he parameerized RB policy. The sixh column shows he cos raio of he parameerized RB policy o he opimal policy. The sevenh column shows he cos of unparameerized RB policy (i.e., he original policy wihou parameer opimizaion). The eighh columns shows he cos raio of he unparameerized RB policy o he opimal policy.

9 e-companion o Levi and Shi: Approximaion Algorihms for he Sochasic Lo-sizing Problem wih Order Lead Times ec9 Demands Cos of Opimal Cos of Cos Cos of Cos (K,h,p) (λ 0,λ 1,λ 2 ) OPT (β,γ,η ) param. RB Raio unparam. RB Raio (0,1,9) (4,1,4) (*,2,*) (0,1,9) (4,1,2) (*,2,*) (0,1,9) (4,1,1) (*,2,*) (0,1,9) (3,1,2) (*,2,*) (0,1,9) (2,1,3) (*,2,*) (0,1,9) (1,1,4) (*,2,*) (5,1,9) (4,1,1) (0.2,2,9) (5,1,9) (1,1,4) (0.2,2,9) (5,1,1) (4,1,1) (0.4,1,1) (100,1,9) (5,1,0) (0.9,*,9) (100,1,9) (4,1,1) (0.9,*,9) (100,1,9) (3,1,2) (0.9,*,9) (100,1,9) (2,1,3) (0.9,*,9) (100,1,9) (1,1,4) (0.8,*,9) (100,1,9) (0,1,5) (0.8,*,9) Table EC.1 Numerical resuls wih lead ime L=0 and finie horizon T =12. Demands Cos of Opimal Cos of Cos Cos of Cos (K,h,p) (λ 0,λ 1,λ 2 ) OPT (β,γ,η ) param. RB Raio unparam. RB Raio (0,1,9) (4,1,4) (*,2,*) (0,1,9) (4,1,2) (*,2,*) (0,1,9) (4,1,1) (*,2,*) (0,1,9) (3,1,2) (*,2,*) (0,1,9) (2,1,3) (*,1.5,*) (0,1,9) (1,1,4) (*,1.5,*) (5,1,9) (4,1,1) (0.2,2,9) (5,1,9) (1,1,4) (0.2,2,9) (5,1,1) (4,1,1) (0.4,1,1) (100,1,9) (5,1,0) (0.9,*,9) (100,1,9) (4,1,1) (0.9,*,9) (100,1,9) (3,1,2) (0.9,*,9) (100,1,9) (2,1,3) (0.9,*,9) (100,1,9) (1,1,4) (0.9,*,9) (100,1,9) (0,1,5) (0.9,*,9) Table EC.2 Numerical resuls wih lead ime L=2 and finie horizon T =12.

10 ec10 e-companion o Levi and Shi: Approximaion Algorihms for he Sochasic Lo-sizing Problem wih Order Lead Times Demands Cos of Opimal Cos of Cos Cos of Cos (K,h,p) (λ 0,λ 1,λ 2 ) OPT (β,γ,η ) param. RB Raio unparam. RB Raio (0,1,9) (4,1,4) (*,2,*) (0,1,9) (4,1,2) (*,2,*) (0,1,9) (4,1,1) (*,2,*) (0,1,9) (3,1,2) (*,2,*) (0,1,9) (2,1,3) (*,2,*) (0,1,9) (1,1,4) (*,2,*) (5,1,9) (4,1,1) (0.2,2,9) (5,1,9) (1,1,4) (0.2,2,9) (5,1,1) (4,1,1) (0.4,1,1) (100,1,9) (5,1,0) (1.1,*,9) (100,1,9) (4,1,1) (1.1,*,9) (100,1,9) (3,1,2) (1.1,*,9) (100,1,9) (2,1,3) (1.1,*,9) (100,1,9) (1,1,4) (1.0,*,9) (100,1,9) (0,1,5) (1.0,*,9) Table EC.3 Numerical resuls wih lead ime L=0 and finie horizon T =15.