Optimal Policies for a Capacitated Two-Echelon Inventory System

OPERATIONS RESEARCH Vol. 52, No. 5, September October 2004, pp. 739 755 iss 0030-364X eiss 1526-5463 04 5205 0739 iforms doi 10.1287/opre.1040.0131 2004 INFORMS Optimal Policies for a Capacitated Two-Echelo Ivetory System Rodey P. Parker Yale School of Maagemet, New Have, Coecticut 06520-8200, rodey.parker@yale.edu Roma Kapusciski Uiversity of Michiga Busiess School, 701 Tappa Street, A Arbor, Michiga 48109-1234, roma.kapusciski@umich.edu This paper demostrates optimal policies for capacitated serial multiechelo productio/ivetory systems. Extedig the Clark ad Scarf (1960 model to iclude istallatios with productio capacity limits, we demostrate that a modified echelo base-stock policy is optimal i a two-stage system whe there is a smaller capacity at the dowstream facility. This is show by decomposig the dyamic programmig value fuctio ito value fuctios depedet upo idividual echelo stock variables. We show that the optimal structure holds for both statioary ad ostatioary stochastic customer demad. Fiite-horizo ad ifiite-horizo results are icluded uder discouted-cost ad average-cost criteria. Subject classificatios: ivetory/productio: multiechelo, stochastic ucertaity, policies, capacity expasio. Area of review: Maufacturig, Service, ad Supply Chai Operatios. History: Received May 2001; revisios received Jue 2003, September 2003; accepted September 2003. 1. Itroductio The productio, storage, ad delivery of goods betwee factories, warehouses, ad retailers is a rich area of study with may iterestig questios still usolved. These are issues of substatial importace, because the coordiatio of goods betwee members of supply chais costitutes a sigificat ivestmet i terms of maagerial attetio, ivetory costs, ad capital ivestmets. Careful aalysis of these problems ca icrease resposiveess to edcustomers eeds without ecessarily icreasig costs. I this paper, we aalyze the basic model that icludes two critical elemets: multiple stages (two echelos ad productio capacity costraits at both stages. The issue of optimal orderig ad ivetory policies i multiechelo productio ad ivetory systems without ay capacity costraits was the focus of Clark ad Scarf (1960. They cosider a purely serial supply chai (kow as a multiechelo system, with the lowest istallatio facig stochastic demad from the ed customer. I a fiite-horizo settig, Clark ad Scarf (1960 determie that the optimal orderig policy for the etire multiechelo system ca be decomposed ito decisios based solely o echelo ivetories. Federgrue ad Zipki (1984 exted the multiechelo result to a ifiite-time horizo. Roslig (1989 demostrates that a pure assembly system ca be reduced to a serial multiechelo system. The Clark ad Scarf (1960 results have bee reprove by Che ad Zheg (1994 usig lower bouds o the log-ru costs, ad by Muharremoglu ad Tsitsiklis (2003 usig a alterative approach based o item-customer decompositio. The research for systems with limited productio capacity (defied as a fiite upper limit o the amout that may be processed i a sigle period uder periodic review is mostly costraied to oe-echelo systems. Federgrue ad Zipki (1986a, b cosider a capacitated sigle istallatio for ifiite-horizo average-cost ad discouted-cost criteria, respectively. Followig Zipki s (1989 aalysis of the cyclical sigle-stage system without capacity limits, Kapusciski ad Tayur (1998 ad Aviv ad Federgrue (1997 study a sigle istallatio with limited capacity facig stochastic cyclical demad ad fid the optimal policy for such systems; Metters (1997 applies heuristics to the same problem with lost sales. There has bee very little research o a multiechelo system with limited capacity at each istallatio. Speck ad Va der Wal (1991a cosider a two-echelo system, preset a couterexample justifyig why a modified base-stock policy is ot optimal, ad suggest that umerically earby modified base-stock policies provide a good approximatio to the optimal policy. Speck ad Va der Wal (1991b provide a algorithm to determie parameter values for such a base-stock policy. Glasserma ad Tayur (1994, 1995 cosider this problem, but show aalysis for the capacitated supply chai assumig that the system operates uder a base-stock policy. Roudy ad Muckstadt (2000 also assume base-stock policy (for oe stage ad propose a efficiet approximatio. This paper demostrates that whe the smallest capacity is at the dowstream facility i a capacitated serial supply chai, the optimal policy is a simple modificatio of 739

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System 740 Operatios Research 52(5, pp. 739 755, 2004 INFORMS a echelo base-stock policy. The policy for lower echelo is uchaged we order up to a specific target (subject to the availability from the higher echelo. The policy for the higher echelo is modified. The higher echelo orders up to a specific echelo target, takig care ot to exceed specific istallatio (o-had ivetory. Due to the additioal costrait o the istallatio ivetory, we label the policy as the modified echelo base-stock (MEBS policy. This policy ca be iterpreted as havig a structure of geeralized kaba productio-ivetory policies. These are policies with release mechaisms that could ecompass costraits o the amouts of ivetory i a subset of cosecutive istallatios. Axsäter ad Roslig (1999 rak various policies, icludig geeralized kaba policies, i multistage systems with various release mechaisms, accordig to how geeral they are. Dallery ad Liberopoulos (2000 ad Liberopoulos ad Dallery (2000 also cosider geeralized kaba cotrol policies ad cotai refereces to other papers i this area. Eve though most of this literature deals with Poisso arrivals of demad ad expoetially distributed service times i cotiuous time, some of the observatios ca be related to our fidigs. Veatch ad Wei (1994 umerically show that kaba policies with uit-sized cotaiers are geerally superior to order-up-to policies whe the lower istallatio s capacity is smaller, ad this policy relatioship is reversed whe the capacity coditio is reversed. For the case whe lower istallatio has lower or equal capacity i a periodic settig, the policy we show to be optimal ca be iterpreted as geeralized kaba. Although ot exactly the same, it is similar to the oe cosidered i Veatch ad Wei (1994, which justifies the good performace of the Veatch ad Wei heuristics. Our mai result providig the structure of the optimal policy for a two-echelo system with the costraiig capacity closer to the customer is based o a few observatios. We show that odomiated orderig is limited to a bad of states (formally defied i Defiitio 1 ad that for the states i the bad, the cost fuctio is separable. Specifically, we demostrate that it will ever be optimal for the higher istallatio (farther from the customer to hold more ivetory tha ca be processed i a sigle period by the lowest stage, which is a bottleeck. This immediately implies that the covetioal (Clark ad Scarf 1960 echelo base-stock policy caot be optimal. Accordig to the covetioal policy, a huge spike of demad would geerate the same-size order at the higher istallatio, which may exceed the capacity costrait, thus geeratig uecessary holdig costs. Usig this bad domiace, i a two-stage system we substitute the costraits upo productio by a costrait o the ivetory, ad show separability of the cost fuctio. The resultig MEBS policy resembles the covetioal multiechelo policy, except for the additioal bad costrait. We permit geeral (multiple of period legth lead times leadig to the lower istallatio, but restrict the lead time at the higher istallatio to oe period. Without this limitatio, we would ot be able to guaratee that the ivetory remais i the udomiated bad. The limitatio to two stages is directly liked to the lead-time issue; i fact, Glasserma ad Tayur (1994 describe a techique of isertig dummy istallatios to act as a surrogate for positive lead times. The fial limitatio of our model is the requiremet that the costraiig capacity is at the lower istallatio. We address this issue with a example i 4.3.4. We show that our results exted to other system cofiguratios i 4.3.3. This paper is orgaized as follows. I 2, we describe the model i detail, statig the sequece of evets ad formulatig the model. Sectio 3 illustrates the deceptive ature of the umerical results where some behavior that is apparetly ot base-stock i ature is optimal. Sectio 4 cotais the paper s key results for the fiite horizo. I this sectio we prove the optimal policy ad exted the model to specific lead-time models ad to a model with ostatioary stochastic demads. Sectio 5 exteds the results to discouted-cost ad average-cost criteria for the ifiite horizo. Cocludig remarks are i 6. 2. Model Cosider a serial multiechelo supply chai with N istallatios (see Figure 1. Each of the istallatios, j>1, supplies its immediately lower istallatio, j 1, i the supply chai ad receives goods from istallatio j + 1. Istallatio N receives goods from a outside ucapacitated supplier. Each istallatio j, other tha N, is limited i its order by the available ivetory held at istallatio j + 1. Additioally, at istallatio j there is a capacity limit K j that serves as a upper limit o the amout that ca be processed i each period. Istallatio 1 supplies goods to the ed customer, whose demad, D, is stochastic ad idepedet from period to period. D 0 ad = ƐD <. The sequece of evets is as follows: (1 at the begiig of every period, istallatio 1 places a order with istallatio 2, istallatio 2 places a order with istallatio 3, ad so forth up to istallatio N ; (2 istallatio 2 the delivers the ordered amout to istallatio 1, istallatio 3 delivers to istallatio 2, ad so forth, util the outside supplier delivers to istallatio N ; ad (3 ed-customer demad is the realized, ad istallatio 1 attempts to satisfy this demad as closely as possible usig its available stock. The sequece of orders i Step (1 is listed i this way to demostrate the dyamics of the system, although these orderig decisios are take cocurretly. 1 Usatisfied demad is backlogged with a liear pealty cost per period, p. Echelo j icurs a icremetal holdig cost, h j 0, for each uit of ivetory i each period, ad Figure 1. Multiechelo system with N istallatios. K N 1 K 1 K N K 2

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System Operatios Research 52(5, pp. 739 755, 2004 INFORMS 741 the istallatio holdig cost is H j = kj h k. The objective is to miimize the discouted sum of holdig ad pealty costs. I the fiite-horizo case, time is couted backwards ad represets the umber of periods remaiig util the ed of the horizo. Iitially, it is assumed that there are o delivery lead times, other tha delays caused by the dyamics of the system. I additio, we assume without loss of geerality that the shippig costs are zero. The variables cosidered here are: The ivetory at istallatio j at the start of period is x j ; the amout ordered by istallatio j i period is a j 0; ad yj is ivetory i istallatio j after shipmets are made i period. The ivetory dyamics are described by x 1 1 = y1 D = x 1 + a1 D x j 1 = y j = xj aj 1 + a j for j>1 A echelo ivetory is defied as the amout i trasit to ad i stock at a istallatio plus the amout i trasit to ad i stock at all lower istallatios. The correspodig echelo variables X = X jn j=1 RN ad Ỹ = Y jn j=1 R N are defied as X j = j i=1 x i ad Y j = j i=1 y i. Clearly, Y j = Xj + aj x j 1 = X j 1 X j 1 1 = Y j Y j 1 The equivalece of istallatio ad echelo variables is apparet, but the beefits of usig echelo variables may ot be obvious. Clark ad Scarf (1960 show that the optimal orderig policy for this model could be described by echelo base-stock levels, z 1 z 2 z N : If the echelo ivetory X j is below z j, order z j X j ; otherwise, order othig. (Whe there is isufficiet stock at a upstream istallatio, a partially filled order is preferable to o deliveries at all. Clearly, the echelo ivetory variables are odecreasig i echelo levels ad (see Figure 2 each potetial value of Y j is bouded below by Xj ad above by Xj+1, except for j = N. Capacity limits impose a additioal costrait o each Y j. Whe demad is realized, all Xs ad Y s are shifted leftwards by the size of the demad. We defie the etire model with the followig recursio statemets. Defiitio (Core Model V X = mi Ỹ X J Ỹ (1 J Ỹ = LỸ + Ɛ D V 1 Ỹ D (2 V 0 = 0 (3 Figure 2. X 1 Echelo variables provide atural orderig. Y 1 Y 2 X 2 X N 1 Y N 1 X N K 1 K 2 K N 1 K N Y N where LỸ =Ɛ D {( N j=1 N + h i Y i Y 1 i=2 h j Y 1 D + +pd Y 1 + } ( N N = h i Y i Ɛ D D + p+ h i Ɛ D D Y 1 + i=1 i=1 (4 X i X = Ỹ R N Y i Xi+1 Y i Xi K i i =1N 1 X N Y N Y N XN K N ad X = X 1 X2 XN Ỹ = Y 1 Y2 YN Ỹ D = Y 1 D Y 2 D Y N D V represets the expected discouted costs of operatig uder the optimal ivetory policy i this capacitylimited system for a time horizo of periods. L represets the periodic costs, or the costs (holdig ad backorder icurred i a sigle period. The otatio x + deotes maxx 0 ad a b + deotes maxa b. The discout factor assumes values 0. The costs of purchasig goods, c 0 (assumed liear i amout, from the exteral supplier, ad the reveues, r 0 (assumed liear i amout, collected from the ed customer are omitted from this model. It ca be easily demostrated that these amouts ca be absorbed ito the holdig ad pealty costs by redefiig p = p + r c1 ad h N = h N + c1. Whe p + r1 >c1, the derivatios ad the optimal policy structure (similar to Veiott 1966 are ot affected. 3. The Disguised Base-Stock Policy A base-stock policy attempts to brig a echelo ivetory up to its base-stock level if the ivetory is below this level ad orders othig otherwise. I this sectio, we demostrate some behavior of the capacitated model that could be costrued as o-base-stock. Table 1 shows optimal behavior of the capacitated model for the followig parameters. I this example, ƐD = 96 < 10 = K 1 = K 2. N =2 c=0 h 2 =005 h 1 +h 2 =1 p=10 r =0 =09 K 1 =10 K 2 =10 PrD =7=01 PrD =8=02 PrD =9=025 PrD =10=01 PrD =11=02 PrD =12=01 PrD =13=005 As ca be see i the last two colums of Table 1, for x 1 15 ad x 2 = 8, the model is orderig up to echelo

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System 742 Operatios Research 52(5, pp. 739 755, 2004 INFORMS Table 1. Example of the push-ahead effect. Iitial Iitial Edig istallatio echelo Istallatio echelo ivetory ivetory orders ivetory x 1 x 2 X 1 X 2 a 1 a 2 Y 1 Y 2 5 8 5 13 8 10 13 23 6 8 6 14 8 10 14 24 7 8 7 15 8 10 15 25 8 8 8 16 7 9 15 25 15 8 15 23 0 2 15 25 16 8 16 24 0 2 16 26 17 8 17 25 0 2 17 27 18 8 18 26 0 1 18 27 19 8 19 27 0 0 19 27 20 8 20 28 0 0 20 28 levels of 15 ad 25 for echelos 1 ad 2, respectively, but the policy the varies from this for higher iitial stocks at istallatio 1. The last four rows of Table 1 show the target level for echelo 2 icreasig above a level of 25, apparetly approachig aother base-stock level of 27. This type of behavior, labeled the push-ahead effect by Speck ad Va der Wal (1991a, appears to be a exceptio to the base-stock policy. The model optimally orders more tha a typical base-stock would suggest. However, this ca be show to be a disguised versio of a base-stock policy show to be optimal i 4. 4. Fiite-Horizo Results 4.1. Mai Case Cosider a case whe the capacity of istallatio 1 is the smallest i the supply chai. We demostrate that the a modified echelo base-stock policy is optimal for the capacitated serial supply chai with N = 2. Lemma 1. Let K 1 K j. For ay X, all optimal Ỹ satisfy y j maxk 1x j aj 1 for j>1. Proof. See the appedix. Corollary 1. Assume that K 1 K j ad X j Xj 1 K 1 for all j 2N. The, (a The optimal Y j satisfy Y j Y j 1 K 1 for all j>1. (b If the optimal policy is followed, the the ivetory positios satisfy Xm j Xj 1 m K 1 for all j>1 ad m<. (c All capacities may be replaced with capacities equal to K 1 without affectig costs. Corollary 1(c is true because oe of the capacity i excess of a level equal to K 1 is used. This is apparet for all istallatios j<n because all a j xj+1 K 1 (from Corollary 1(b. It is true for istallatio N also, because the resultig ivetory x 1 N K 1, ad thus a N K 1 also. Remark 1. If the bottleeck capacity is at echelo i 0, i.e., K i0 K i for all i, the Lemma 1 ad Corollary 1 hold for all echelos j>i 0. Lemma 1 ca be easily modified to iclude lead times: Remark 2. If there are positive lead times j for delivery of goods from istallatio j + 1toj, the the istallatio ivetories x j ad y j additioally iclude goods shipped from istallatio j + 1 that have ot yet arrived at istallatio j. Uder this redefiitio, For ay X, all optimal Ỹ satisfy y j max i+1k i0 x j aj 1 for j>i 0, where K i0 = mi j K j (i 0 is the bottleeck. Cosequetly, upstream of the bottleeck, Corollaries 1(a ad 1(b hold, but the limit of K 1 (or bad, see below eeds to be replaced by a limit of j + 1K 1. Corollary 1(c holds uchaged. For further discussio of the effect of lead times, see 4.3.1. It may be opportue to defie the followig regio we commoly refer to as the bad. Defiitio 1. Feasibility bad, or simply bad, is defied as = X R N X j X j+1 X j + K 1 j 1. This bad establishes the regio where the ivetory at all istallatios, except istallatio 1, does ot exceed K 1. Lemma 1 determies that whe followig a optimal policy from a ivetory positio outside the bad, the system will traverse to the bad i the most direct maer. Corollary 1(a implies that the istallatios upstream (i.e., away from the cosumer of the most costraied istallatio will ot hold more tha K 1 uits of ivetory oce the ivetory levels are withi the bad; i.e., oce i the bad, the system will remai withi the bad. Before showig the structure of the optimal policy, we prove some properties of the model. Property 1. (a J is cotiuous ad covex. (b V is cotiuous ad covex. (c V is odecreasig i. Proof. (a ad (b are proved by iductio. Because V 0 = 0, J 1 = L is clearly cotiuous ad covex because each term i L (see (4 is cotiuous ad covex. Usig Propositio B-4 i Heyma ad Sobel (1984, covexity of J ad set covexity of = X X implies that V is covex. Cotiuity of V follows from covexity of V for all iteral poits of the feasibility set ad is guarateed from the cotiuity of J ad compactess ad covexity of X for the border poits. Assume that V is cotiuous ad covex. Because expectatio ad liear trasformatio preserve both cotiuity ad covexity, ƐV Ỹ D is cotiuous ad covex. Therefore, J +1, as a sum of two cotiuous ad covex fuctios, is cotiuous ad covex. (c V caot decrease i from the oegativity of the oe-period fuctio, L. While the objective fuctio is ot geerally separable, it does separate withi the bad. Theorem 1. Assume N = 2, X 2 X1 K 1, ad K 1 K 2. The two-istallatio model ca be decomposed ito

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System Operatios Research 52(5, pp. 739 755, 2004 INFORMS 743 programs depedet oly upo the echelo ivetories, as follows: V X 1 X2 = V 1 X1 + V 2 X2 (5 I additio, V 1 ad V 2 are covex. First, we state without a proof, a lemma used to show decouplig of the fuctio value. Lemma 2 (Karush 1959. (a If a fuctio fyis covex o ad attais its miimum at y, the mi fy= f L a + f U b ayb where f L a = mi ay fy = fmaxa y is covex odecreasig i a ad f U b = fb mi by fy = fb fmaxb y is covexoicreasig i b. (b If a fuctio fy is quasi-covexo ad attais its miimum at y, the mi fy= f L a + f U b ayb where f L a = mi ay fy = fmaxa y is odecreasig i a ad f U b = fb mi by fy = fb fmaxb y is oicreasig i b. Remark 3. Note that Lemma 2(a is take directly from Karush (1959, adjusted for otatio. Lemma 2(b also may be straightforwardly show. Porteus (2002 explais that whe gx = Ɛfx X, where X is a Pólya frequecy fuctio radom variable ad f is quasi-covex, the g is also quasi-covex. 2 However, the decompositio of the optimality results caot be exteded to the cotext with quasi-covex periodic costs for echelo 1. This is simply because the optimizatio occurs across both echelo ivetory decisios, ad the sum of quasi-covex fuctios is ot ecessarily quasi-covex. There appear to be o obvious coditios that could prove sufficiet to yield structural results i such a cotext. Proof of Theorem 1. The critical elemet of the proof is rephrasig the costraits ito simpler, but equivalet, coditios. Such rephrasig holds i all periods if the iitial poit is withi the bad. From the defiitio of X,we have X 1 Y 1 X2 (6 Y 1 X1 + K 1 (7 X 2 Y 2 X2 + K 2 (8 From Corollary 1(b, because the begiig ivetory is withi the bad (i.e., X ad K 1 K 2, Y 2 Y 1 + K 1 (9 Because K 1 K 2 ad Y 1 X2 (from Equatio (6, we achieve Y 1 + K 1 X 2 + K 2 ad (8 ca be replaced by (9. Because X 2 X1 K 1, the upper boud i (6 is less tha or equal to the upper boud i (7. Combiig these facts, uder the coditios i the theorem statemet, we ca restate the costrait coditios i period as X 1 Y 1 X2 Y 2 Y 1 + K 1 (10 By startig withi the bad i period, from Corollary 1, for all periods m<, Xm 2 X1 m K 1 ad thus the costrait set ca be expressed by the simplified coditios (10 for all m<. Most importatly, whe imposig (10, we do ot eed to impose the capacity costraits aymore. The proof of the theorem is by iductio, ad the claim holds trivially for = 0. Assume it holds for 1 (iductio assumptio. First, we eed to demostrate that V X 1X2 = V 1X1 + V 2X2. [( 2 ] Ɛ D h j Y 1 D + +pd Y 1 + j=1 V X = mi Ỹ X +h 2 Y 2 Y 1+Ɛ D V 1 2 Y 2 D +Ɛ D V 1 1 Y 1 D (11 Defie f 2 Y 2 = h 2Y 2 + Ɛ D V 2 1 Y 2 D Because f 2 is covex o R (based o the iductio assumptio, from Lemma 2 we get mi f 2 X 2Y 2Y 1+K Y 2 = f 2L X2 + f 2U Y 1 + K 1 1 where f 2L X2 = mi f 2 X 2x x ad f 2L ad f 2U are covex fuctios o R. Holdig Y 1 costat ad miimizig over Y 2, [( 2 ] Ɛ D h j Y 1 D + +pd Y 1 + j=1 V X = mi X 1 Y 1 X2 = mi X 1Y 1X2 h 2 Y 1 2U +f Y 1 +K 1 + Ɛ D V 1 1 Y 1 D +f 2LX2 ] Y 1 D + +pd Y 1 + [( 2 Ɛ D h j j=1 h 2 Y 1 2U +f Y 1 +K 1 + Ɛ D V 1 1 Y 1 D +f 2LX2 = mi f 1 X 1Y 1 Y 1 2L +f X2 (12 X2 Agai, applyig Lemma 2, V X = f 1L X1 + f 1U = V 1 X1 + V 2 X2 X2 + f 2L X2

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System 744 Operatios Research 52(5, pp. 739 755, 2004 INFORMS where V 1X1 = mi X 1Y 1 [( 2 ] Ɛ D h j Y 1 D + +pd Y 1 + j=1 h 2 Y 1 2U +f Y 1 +K 1 + Ɛ D V 1 1 Y 1 D { V 2X2 = mi f 1U X 2Y 2 X2 +h 2Y 2 +Ɛ D V 1 2 Y 2 D } ad both V 1 ad V 2 are covex from Lemma 2, which completes the iductio. Let 1 ad 2 be the miimizers of f 1 ad f 2, respectively. We ow formally defie ad discuss the iduced pealty fuctios. Defiitio 2 (Iduced Pealty Fuctios. The iduced pealty fuctios icurred i the two-echelo model are 0 x 2 f 2U x= h 2 x 2 +Ɛ D V 1 2 x D V 1 2 2 D x< 2 ad 0 x 1 lx l 1 h 2x 1 f 1U x = + f 2U x + K 1 f 2U 1 + K 1 + Ɛ D V 1 1 x D V 1 1 1 D x< 1 where [( 2 lx = Ɛ D h j x D + + pd x ] + j=1 Just as Clark ad Scarf (1960 determied that a iduced pealty fuctio acted upo the upper echelo as a puishmet whe it was uable to supply eough stock for the lower echelo to achieve its optimal ivetory positio (base-stock level, the capacitated case has similar iduced pealties. First, echelo 1 icurs a pealty (f 2U wheever its combiatio of order ad capacity level (Y 1 + K 1 fails to reach a sufficiet level. This ca be iterpreted as assumig (as a cost the additioal beefit that echelo 2 would have accrued if the lowest istallatio were ot capacity limited. (Corollary 1(a proves that the higher echelos orders are limited as a result of the lowest istallatio s capacity beig the bottleeck of the productio system. Secod, the higher echelo icurs a pealty (f 1U wheever its stock is isufficiet to supply echelo 1 with a amout eeded to achieve its base-stock level. This is aalogous to the iduced ad pealty i Clark ad Scarf (1960. Each istallatio potetially icurs a pealty for the limitatio it imposes upo the other istallatio. Note that while f 1U ad f 2U are ormalized, i.e., equal to 0 at 1 ad 2, f 1L ad f 2L are ot. Istead, f 1L mi f 1 ad f 2L mi f 2 are the costs of deviatig from the miimums. Let us ow costruct a capacitated versio of a basestock policy. This policy is withi the family of geeralized kaba policies, where K 1 is the umber of kabas at stage 2. Defiitio 3 (MEBS Policy. The modified echelo base-stock policy (MEBS ca be writte as (Y j Xj for all j Y 1 = miz1 X1 + K 1X 2 Y j = mizj Yj 1 + K 1 X j+1 for j = 2N 1 Y N = miz N 1 YN + K 1 Cosider the fuctio J, defied i (2. We defie the followig: = arg mi Y 1 J 1Y 1 ad z 2 = arg mi Y 2 J 2Y 2, where J 1Y 1 = h 1 Y 1 Ɛ D D + p + N i=1 h iɛ D D Y 1 + + Ɛ D V 1 1 Y 1 D ad J jy j = h j Y j Ɛ D D + Ɛ D V 1Y j j D for j>1. If z 2 K 1, the z 1 = arg mi Y J Y 1 Y 1 + K 1 1, else z 1 =. From these defiitios we see that if the itersectio of the two echelos miimizig poits is withi the bad, the each of them is the base-stock level. However, if the itersectio does ot occur withi the bad, the echelo 1 s base-stock level will be foud alog the upper edge of the bad, where Y 2 = Y 1 + K 1. Theorem 2. The modified echelo base-stock policy (MEBS with parameters z 1 z2 defied above is optimal for a N = 2 system where K 1 K 2 ad X 2 X 1 K 1. Proof. See the appedix. Recall that, based o Lemma 1, the optimal policy outside the above-defied bad,, is to order othig util the ivetory positio is draw ito the bad. While i the bad, the MEBS policy differs from the ucapacitated echelo base-stock policy (demostrated i Clark ad Scarf 1960 oly i that it costrais the system to operate withi the ivetory bad. Let us revisit the couterexample preseted i Table 1 ( 3. The parameters are foud to be z 1 = 15 ad z 2 = 27. Figure 3 illustrates this case. Note that set X 1 X 2 = Y 1 Y 2 X 1 Y 1 X 2 Y 2 Y 1 X 1 + K 1 Y 2 K 2 is trucated to the bad Y 1 Y 2 Y 1 + K 1 ; i.e., desired ivetory at echelo 2 is limited by mix 1 + K 1 z 2. Withi this feasible set, MEBS policy first chooses Y 1 closest to the target z 1 ad the Y 2 closest to z 2. Thus, z 1 = 15 implies that for X 1 15, optimal Y 1 15 ad, cosequetly, Y 2 15 + 10 = 25 <z 2. Higher startig levels of ivetory (X 1 > 15 allow, however, Y 2 to climb towards ad evetually reach the desired z 2 = 27. Note that this result idicates that the form of the basestock policy, which Glasserma ad Tayur (1995 explore umerically, could be suboptimal because it does ot icorporate the cesorig effect described i MEBS. Speck ad Va der Wal (1991b cosider a heuristic approach to determiig approximate values of the optimal average cost

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System Operatios Research 52(5, pp. 739 755, 2004 INFORMS 745 Figure 3. Illustratio of MEBS policy with target z 1 z 2 = 15 27. Startig ivetory is at the lower-left corer of the feasible regio ad the optimal ivetory Y 1 Y 2 is deoted by a black circle. Echelo 2 ivetory (7,15 (8,16 Target (z 1, z 2 Optimal poit (15,25 Feasible regio Echelo 1 ivetory Echelo 2 ivetory Target (z 1, z 2 (16,24 (17,25 (18,26 Echelo 1 ivetory Note. Left: Startig from either 7 15 or 8 16 it is optimal to raise ivetory levels to 15 25. Right: The optimal decisios for 16 24, 17 25, ad 18 26 are 16 26, 17 27, ad 18 27, respectively. value fuctios. They do ot, however, cosider the possibility that the gap betwee the base-stock levels (i.e., z 2 z 1 exceeds K 1 while K 1 K 2. While there is similarity i the directio of adjustmets, we ca also see that the Speck ad Va der Wal (1991a heuristic is differet i that they suggest a chage i echelo 1 s base-stock level based o istallatio 2 s ivetory positio. That is, at some level as istallatio 2 s ivetory positio icreases, the basestock level at echelo 1 is also forced to icrease. This is a cotrast to the optimal policy where there is a icrease i echelo 2 s base-stock level whe there is a icrease i istallatio 1 s ivetory positio. Theorem 3. The optimal base-stock levels, z 1 ad z2,are (a odecreasig i period ; ad (b oicreasig i K 1 so log as K 1 K 2 is maitaied. Proof. See the olie appedix at http://or.pubs.iforms. org/pages/collect.html. The optimal base-stock levels icrease i the horizo legth, which is cosistet with other ivetory models. Later, we show that these levels stabilize to steady-state values. If a startig ivetory is above the base-stock levels, but still withi the bad,, o material will be ordered ad demad will progressively draw dow the echelo ivetory levels equally util the ivetory levels fall below the base-stock levels, whereupo it will order up to the base-stock levels, if possible. All ivetory paths will the cotiue to be below the base-stock levels, ad the ivetory territory above the base-stock levels withi the bad will ot be revisited. 3 4.2. Discussio of the Model Assumptios The restrictios that N = 2 ad istallatio 2 has o lead times other tha those derived aturally from the periodicity of the model are somewhat limitig, but ecessary for our proof to hold. The proof we provide is based o the equivalece of the two capacity costraits to a costrait o istallatio 2 ivetory. The same equivalece does ot hold for more tha two echelos or whe arbitrary lead times are itroduced (but we do ot claim that MEBS or a optimal policy similar to MEBS is ot optimal i geeralizatios of the model cosidered i this paper. The coditios uder which Theorem 1 is prove iclude X 2 X 1 K 1, which caot be guarateed if lead times are permitted at istallatio 2 uless the base stocks differ by K 1 or less. Especially with log lead times, the basestock levels are ulikely to differ by less tha K 1. Let us demostrate the difficulty of echelo 2 lead times with a simple sceario. Suppose a situatio has arise where the pipelie ivetory leadig to istallatio 2 sums to a level greater tha K 1. Now a successio of very low cosumer demad realizatios occurs so that istallatio 1 s pipelie ivetory is sufficiet to cater for expected demads, ad o additioal orders are placed with istallatio 2. Iexorably, all the pipelie ivetory arrives at istallatio 2 ad the istallatio ivetory stock exceeds K 1, thus cotraveig the coditio X 2 X 1 K 1 ad ot allowig the capacity costraits to be omitted. The costrait that N = 2 is closely related to the leadtime restrictios, as demostrated by Glasserma ad Tayur (1994. Aother restrictio imposed is that K 1 K 2. While this is represetative of may real systems, especially where we ca also imbed this two-stage system i loger systems of a specific type (see 4.3.3, it is worth cosiderig the opposite for the two-stage system, i.e., K 2 <K 1. A umerical example i 4.3.4 illustrates a behavior with multiple plateaus for each echelo, which does ot resemble MEBS policy. It is iterestig to ote that our MEBS policy may be iterpreted as a special case of geeralized kaba policies. Axsäter ad Roslig (1993 described kaba policies as a special case of base-stock policies. Such a characterizatio holds whe cosiderig the chace of backorders as beig extremely low. However, Veatch ad Wei (1994 show that there do exist cases whe kaba ad base-stock policies perform sigificatly differetly. Later, Axsäter ad Roslig (1999 exted their previous classificatio of ivetory policies ad allow kaba-like costraits to be imposed o top of base-stock policies. The MEBS policy proposed here is a special case of their structure. It is also the same as that described i Buzacott ad Shathikumar (1992. The relatioship with the results of Veatch ad Wei is the most iterestig. As oted i 1, Veatch ad Wei (1994 cosider a capacity-limited, two-istallatio serial system uder the assumptios of exogeous Poisso demad ad cotrollable productio rates. Usig simulatio, they fid that kaba policies ted to be superior (although ot ecessarily optimal to base-stock policies whe the dowstream istallatio is the bottleeck. Whe the bottleeck is at the upstream istallatio, base-stock policies are superior. While the settig modeled i their paper differs from

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System 746 Operatios Research 52(5, pp. 739 755, 2004 INFORMS our model, the spirit of the heuristic is the same limit the ivetory at the upstream istallatio. While kaba policy seems to be superior i cotiuous settigs with Poisso arrivals, we fid that a geeralized kaba policy is optimal i a discrete-time geeral-demad settig. It is also importat to ote that kaba systems sometimes justified as a suboptimal ivetory policy but a good icetive mechaism i situatios we cosider, actually are exact applicatios of the optimal ivetory policy. Despite the specific assumptios we make i this paper, some geeralizatios are possible. I 4.3.1 ad 4.3.2 we show, amog other thigs, how some of the lead times ca be icorporated ad that the MEBS policy is optimal i a Markov-modulated demad model. Clearly, the optimal base-stock levels will fluctuate accordig to the Markov-chai state ad ivetory levels above base-stock levels may occur give such fluctuatios. I additio, 4.3.3 demostrates how our core results may be applied i other supply chai cofiguratios. 4.3. Extesios 4.3.1. Lead Times. The core model has o lead times, other tha those that occur through the atural dyamics of the supply chai. Here we expad the model to iclude aalysis of delivery lead times. We assume that the holdig cost is costat throughout the pipelie at the level of the higher istallatio. We could choose a separate pipelie holdig cost, higher tha h 2 but lower tha h 1 + h 2, to reflect the reality that additioal costs have bee icurred by this delivery (e.g., trasportatio cost, isurace coverage, etc., but the value-added costs at the delivery destiatio have yet to be icurred. (Aalytically, ay liear holdig cost for pipelie ivetory, eve outside the iterval h 2 h 1 + h 2, ca be icorporated ito the model. This would ot, however, add particular isight ito the iclusio of lead times ito the core model. Cosider, as before, a two-stage multiechelo ivetory system with capacity limits K 1 K 2. There is a delivery lead time of (iteger periods from istallatio 2 to istallatio 1. Istallatio 2 has o lead times. Let ã = a 1 +1 a1 +2 a1 + The trasitio equatios are x 1 1 = x1 + a1 + D x 2 1 = x2 + a2 a1 Let the pipelie ivetory costs be h 2 per uit. Istallatio 1 assumes the cost of the stock, h 1 + h 2, oce it arrives at the site. The systemwide ivetory at the ed of period is x 2 + a2 + a1 +1 + a1 +2 + a1 +3 + +a 1 + 1 + x1 + a1 + D + = X 2 + a2 x1 a1 + + x1 + a1 + D + We accout for all costs whe they occur except for the costs at istallatio 1. (This is similar to the developmet of Federgrue 1993. At istallatio 1 the costs are icurred i the period i which the order is delivered, i.e., periods after the order is triggered ad, therefore, are discouted by to brig the costs back to the orderig period. The cost equatios become [ ] h 2 X 2 + a2 h 2 Ɛ x 1 + a1 + a 1 +i D i i=0 i=1 { [ + Ɛ h 1 + h 2 x 1 + a 1 +i D i i=0 ] + [ ] + p x 1 + } a 1 +i D i = h 2 X 2 + a2 h 2 ƐX 1 + a1 D + Ɛ { h 1 + h 2 X 1 + a1 D+1 + + pd +1 X 1 a1 +} =L 2 X 2 + a2 + L1 X 1 + a1 where X 1 = x1 + i=1 a1 +i ad X 2 = x1 + x2 + is the covolutio of the demad radom variable over D j i=1 a1 +i. j periods; the expectatio operators apply to these covolutios. Evidetly, the terms X 1 + a1 ad X2 + a2 are isolated ad ca be labeled Y 1 = X1 + a1 ad Y 2 = X2 + a2, respectively. The actio set for this model is x 1 x2 ã = { a 1 a2 0a1 K 1a 1 x2 0a2 K } 2 or equivaletly X = { Ỹ R 2 X 1 Y 1 X2 Y 2 X2 + K 2 Y 1 X1 + K 1} With respect to the dyamic programmig recursio, the last periods will be costats because o decisios made i those periods will have ay cost effects durig the time remaiig. Note that we cotiue to have X 1 2 X1 1 = x 1 2 = Y 2 Y 1, ad the basis upo which Corollary 1 sits remais valid: It is uprofitable to order up to a amout above a level of K 1 at istallatio 2 because istallatio 1 caot order ay more tha this amout i a sigle period. Formal traslatio of the lead-time model to the origial model is as follows: Defiitio 4 (Lead-Time Model. V X = mi Ỹ X J Ỹ (13 J Ỹ = { L 1 Y 1 + L2 Y 2 + Ɛ D V 1 Ỹ D } (14 where X = { Ỹ R 2 X 1 Y 1 X2 Y 2 X2 +K 2 Y 1 X1 +K 1} L 2 Y =h 2 Y L 1 Y = h 2 ƐY D + Ɛh 1 +h 2 Y D +1 + +pd +1 Y +

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System Operatios Research 52(5, pp. 739 755, 2004 INFORMS 747 The proof of the followig theorem is idetical to that of Theorems 1 ad 2. Theorem 4. The lead-time model follows the MEBS policy for the coditios stated i Theorem 1. 4.3.2. Markov-Modulated Demad. Cosider a Markov-modulated demad. That is, there are M states with Markovia trasitios i matrix P ad a demad distributio associated with each of the states. Che ad Sog (2001 show that demad-distributio-depedet echelo base-stock policies are optimal for a ucapacitated serial multiechelo system facig ostatioary demads. Their proof was ecessarily complicated by the fact that they were illustratig a algorithm that had desirable ecoomic iterpretatios. We ow demostrate a similar result, but use the decompositio proof used i previous sectios. I the Markov-modulated demad model, the demad expectatio is augmeted by p mm, the probability that the ext state will be m give the curret-period state is m. Defiitio 5 (Markov-Modulated Model. The value fuctio of the ostatioary demad model is N N h j Y i i 1 Y V X m=mi Ỹ where i=2 j=i + [ N p mm Ɛ Dm h j Y 1 D m + m j=1 ] +pd m Y 1+ + p mm Ɛ V Dm 1Ỹ D m m m X = { Ỹ X i Y i Xi+1 Y i Xi K i i = 1N 1X N Y N XN + K } N (15 m 1 2M is the state of the Markov chai. It is easy to verify that Corollary 1(a remais valid for this model (because it does ot require that the realizatios be draw from the same distributio. It is possible to demostrate that the MEBS policy is optimal for this model. Because the proof is very similar to the proofs of Theorems 1 ad 2, it is omitted here. Theorem 5. Assume that the coditios stated i Theorem 1 hold. For the ostatioary demad model for N = 2, the MEBS policy is optimal. The parameters of the optimal MEBS policy deped, however, o both the period umber ad state of the system. 4.3.3. Other Capacity Coditios. While the assumptio that the lowest istallatio must have the lowest capacity level may appear restrictive, there exist other situatios for which a base-stock orderig policy is optimal. It ca be easily demostrated that a base-stock policy remais optimal for (a a serial multiechelo system where oly the two highest istallatios have capacity costraits K N 1 K N with geeral lead times permitted at all stages except the highest istallatio, ad (b a serial multiechelo system, with geeral lead times at all istallatios, where the oly capacity restrictio is at the uppermost istallatio, K N <. These results are summarized i the followig theorem. Theorem 6. (a Cosider a N -stage system without capacity limits at stages j<n 1 ad fiite capacities K N 1 K N <. Uder the ivetory coditio X N 1 XN K N 1, V X = N j=1 V jxj. The optimal policy is as follows: MEBS holds for the capacitated istallatios ad the remaiig istallatios follow a echelo base-stock policy. (b Cosider a N -stage system without capacity limits at stages j<n ad fiite capacity K N. For this system, V X = N j=1 V jxj. A echelo base-stock policy is optimal for this system. Proof. See the appedix. Now cosider a N -stage system with capacity limits at each stage ad idetical holdig costs. Note that whe holdig costs for all istallatios are equal, the problem becomes very easy for all stages 1 <i N, oe caot be worse off by forwardig the ivetory to echelo 1. Thus, the system is equivalet to a oe-stage capacitated system with lead time N, for which optimal policy is a modified base-stock (MBS policy. Corollary 2. Cosider a N -echelo model with istallatio holdig costs H i = h 1 for all i ad K 1 K j for all j>1. The optimal policy is MBS. 4.3.4. Lower Capacity at the Higher Echelo: K 2 < K 1. Cosiderig that we have dealt with the K 2 K 1 cofiguratio i the N = 2 case, the obvious questio is whether base-stock policies are optimal i the K 2 <K 1 cofiguratio. Table 2 cotais a umerical couterexample detailig the optimal orders for a umber of startig ivetory positios. This 10-period example is achieved with the followig parameter values: K 1 =11 K 2 =10 c=0 r=0 p=10 h 1 +h 2 =1 h 2 =005 =09 PrD =2=01 PrD =3=02 PrD =9=025 PrD =10=01 PrD =13=02 PrD =18=01 PrD =22=005 The mea demad for this distributio is 9.55, which is less tha the lowest capacity level, K 2 = 10. It appears that echelo 1 is strivig to order up to levels of 22, 23, ad 24, while echelo 2 appears to be tryig to reach levels of 43, 47, 48, ad 49 (ot show. If this does demostrate a base-stock policy, it is oe more orate ad itricate tha we ca evisio.

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System 748 Operatios Research 52(5, pp. 739 755, 2004 INFORMS Table 2. Couterexample for K 2 <K 1 i the N = 2 case. Iitial Iitial Edig istallatio echelo Istallatio echelo ivetory ivetory orders ivetory x 1 x 2 X 1 X 2 a 1 a 2 Y 1 Y 2 10 15 10 25 11 10 21 35 11 15 11 26 11 10 22 36 12 15 12 27 10 10 22 37 13 15 13 28 10 10 23 38 14 15 14 29 9 10 23 39 15 15 15 30 9 10 24 40 16 15 16 31 8 10 24 41 17 15 17 32 7 10 24 42 18 15 18 33 6 10 24 43 24 15 24 39 0 4 24 43 25 15 25 40 0 3 25 43 26 15 26 41 0 3 26 44 27 15 27 42 0 3 27 45 28 15 28 43 0 3 28 46 29 15 29 44 0 3 29 47 30 15 30 45 0 2 30 47 31 15 31 46 0 2 31 48 32 15 32 47 0 1 32 48 4.3.5. Effect of Variace upo Costs ad Recommeded Capacity Levels. I this subsectio, we illustrate the effect demad variace may have upo costs ad optimal capacity levels with a umerical example. We umerically solve the system with a discrete demad distributio for various coefficiets of variatio. As expected, the costs at the best operatig level 4 icrease with the coefficiet of variatio ad, as K 1 icreases (maitaiig the coditio, K 1 K 2, this best operatig level cost also decreases i a covexlike maer see Figure 4. The parameters used for this example are p = 5, h 1 + h 2 = 1, h 2 = 05r = 0c= 0 ad = 09, with K 1 10 11 12 15 20. The demad distributios were desiged so that the mea was kept at 9.8 ad the coefficiet of variatio was 0.1, 0.2, ad 0.4. Usig these data, the algorithm was ru util the differece of successive value fuctios was less tha 0.005. This resulted i horizo legths that differed for differet K 1 s ad demad distributios. A summary of these horizo legths appears i Table 3. Cosider the three curves deotig the best operatigcost curves for three levels of coefficiet of variatio (cv. We observe that doublig the cv from 0.1 to 0.2 has a smaller effect upo costs tha doublig the cv from 0.2 to 0.4, ad safety stocks are ot proportioal to stadard deviatios, as they are for oe-stage systems. Lastly, the percetage labels i Figure 4 idicate the percetage cost above the ucapacitated system. Cofirmig ituitio, the umber of uits of additioal capacity required to get withi 1% of ucapacitated costs is smaller for systems with less variace. Now, give a appropriate capacity acquisitio-cost fuctio, we ca draw coclusios about optimal oe-time capacity ivestmet. Two elemets of such a cost fuctio could be a fixed part (idepedet of the level of K 1 ad a variable part (depedet o K 1. If the variable part is covex icreasig (icludig a liear fuctio, the clearly the sum of the best operatig-cost fuctio (V ad the capacity acquisitio costs is also covex i K 1 with a fiite miimizig poit, K1, ad recommedatios about optimal capacity ivestmets to miimize total costs (operatig ad ivestmet costs may be made. The method of umerical computatio is based o the value-iteratio algorithm. The aalytical results offer Figure 4. The effect of icreasig capacity ad the coefficiet of variatio o the best operatig level. 180 160 60.40% cv=0.1 cv=0.2 cv=0.4 Best Operatig Cost 140 120 100 28.90% 21.20% 5.30% 80 3.40% 0.50% 14.70% 0.40% 0% 60 10 11 12 13 14 15 16 17 18 19 20 Capacity, K 1

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System Operatios Research 52(5, pp. 739 755, 2004 INFORMS 749 Table 3. Example of horizo legth required for the covergece of up-to levels. Horizo legth for Capacity cv = 0.1 cv = 0.2 cv = 0.4 10 33 33 38 11 31 29 33 12 31 29 31 15 31 29 31 20 31 29 31 additioal opportuities for computatioal efficiecies. Namely, the covexity result permits us to search for the miimums efficietly, while Lemma 1 eables us to restrict the search to a subset of the state space, the bad. Usig a zero salvage value fuctio ad a discretized state space, the covexity of the value fuctio is exploited to determie the optimal decisios. 5. Ifiite-Horizo Results I this sectio, we demostrate that the key results foud for the fiite-horizo model i 4 also hold for the ifiitetime horizo. Federgrue ad Zipki (1984 exted the decompositio of Clark ad Scarf (1960 ito separate echelo-based dyamic programs ad show that the optimal policy holds for the ifiite horizo by demostratig it for each of the decomposed programs. We show that the origial combied fiite-horizo model described i 4 coverges i cost ad policy i the ifiite horizo. For otatioal simplicity we demostrate these for the basic model, but the same results ca be derived for the extesios described i 4.2. (The results achieved i Federgrue ad Zipki 1984 could be achieved more easily usig the techiques illustrated here. The closure is easily demostrated by boudig the optimal base-stock levels. Despite ot havig atural bouds as i case of the capacitated problem, Theorem 8 below establishes a boud o udomiated target levels, ad the problem ca be traslated ito a equivalet oe with the feasible actios limited to the states withi this bouded area. I this sectio, we additioally assume that ƐD<K 1. 5.1. Discouted Cost Let 0 <1. Cosider the followig ifiite-horizo cost for policy, which defies the order quatities, a i k, i 1N: V X 0 H 1 xk 1 +a1 k D k + +pd k xk 1 a1 k + = k 1 Ɛ Dk N + H i x i k +ai k ai 1 k k=1 i=2 where x j k+1 = xj k aj 1 k + a j k for j>1, xk+1 1 = x1 k + a 1 k D k, H i = ji h j (as defied before, ad all other dyamic relatioships are as before. (Note that we cout time forward i this sectio, usig k rather tha. Let us defie the miimal ifiite-horizo cost as V X = if V X Theorem 7. The fiite-horizo fuctio, V, coverges to a fiite-valued ifiite-horizo couterpart; that is, V X = lim V X < for all X S Proof. The cost of ay feasible policy i the ifiite horizo is bouded from above, V X 0 = [ N N k 1 Ɛ Dk H i x i k + h i a i k H 2a 1 k k=1 i=2 i=2 + H 1 x 1 k + a1 k D k + + pd k x 1 k a1 k + ] [ N N k 1 Ɛ Dk H i x i k + h i a i k k=1 k=1 i=2 i=2 i=2 + H 1 x 1 k + a1 k D k + pd k x 1 k a1 k ] [ N N k 1 H i x i 0 + kk i + h i K i < i=2 + H 1 x 1 0 +kk 1 + px 1 0 +kɛd ] The bouds are justified based o k=1 kk =/1 2 <, k=0 k = 1/1 < for <1, ad the assumptio that ƐD<, ad are idepedet of policy. Clearly, the value of the optimal policy, if it exists, is also bouded. Now cosider the set B X = { Ỹ X Ɛ D LỸ+ T V 0 Ỹ D } where T is the oe-period mappig deoted by (1 ad (2. We have V = TV 1 = T V 0. Because X is bouded, B must be bouded. The fuctio i the expectatio is cotiuous i Ỹ (from the cotiuity of fiite sums; see 4.1 ad X is closed. Therefore, B must be closed. Based o boudedess of values ad the fact that B is closed, ivokig Propositio 1.7 of 3.1 of Volume II of Bertsekas (1995 (equivaletly, Bertsekas ad Shreve 1996, Propositio 9.17, lim V X = V X for all X S We are iterested as to whether the optimal policy from the fiite-horizo problem coverges to the optimal policy for the ifiite horizo.

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System 750 Operatios Research 52(5, pp. 739 755, 2004 INFORMS Theorem 8. (a There exists a fiite upper boud o ivetory targets for V X, which is idepedet of 0 1; ad (b there exists a fiite upper boud o ivetory targets for ƐVỸ D, which is idepedet of 0 1. A ivetory target is ay YX>X (i.e., a up-to level where we actually icrease ivetory. Proof. See the appedix. Theorem 9. The optimal policy for the fiite-horizo fuctio, V, coverges to its ifiite-horizo couterpart. Cosequetly, Ỹ = lim Ỹ exists, ad Ỹ miimizes V. Proof. I Theorem 7, we demostrated that there exists V X = lim V X X S, which i tur implies JỸ= lim J Ỹ for all Ỹ X. The limit exists because, from Property 1(c i 4.1, J +1 Ỹ J Ỹ for each ad Ỹ X, ad, from Theorem 7, J +1 Ỹ are bouded. X is covex ad compact X S, ad J is covex (see Property 1(a i 4.1. The proof of Theorem 8 verifies that the optimal base-stock levels are bouded, ad hece so are the optimal decisios. Cosequetly, all the coditios of Theorem 8 15 i Heyma ad Sobel (1984 are satisfied ad we get Ỹ = lim Ỹ exists ad Ỹ = arg mi V Corollary 3. The modified echelo base-stock policy is optimal i the discouted-cost ifiite-horizo settig i the N = 2 system whe K 1 K 2 ad the begiig ivetory satisfies X 2 X 1 K 1. Proof. Theorem 9 demostrates that the optimal decisios coverge i the ifiite horizo. However, we must also establish that the base-stock levels also coverge. Because the base-stock levels are mootoically odecreasig i (Theorem 3 ad they are bouded (Theorem 8, z coverge (due to poitwise covergece. From Theorem 9, this results i the MEBS policy structure beig optimal i the discouted-cost ifiite horizo. Note that other tha ƐD<K 1, there have bee o additioal restrictios o the demad distributios to demostrate that the MEBS policy exteds to the ifiite horizo. To umerically evaluate these models, due to the aalytical results of Theorem 9, we ca use value-iteratio algorithms util the differeces of the value fuctios coverge to a predefied quatity. The case of Markov-modulated demad may be easily exteded to the ifiite horizo. As for the fiite-horizo case ( 4.3.2, the cost fuctio depeds additioally o the state m 1 2M of the uderlyig Markov chai. All theorems withi this sectio cotiue to hold. For Theorem 8 to apply, the uderlyig Markov chai eeds to be ergodic, each demad distributio must satisfy 0 < ƐD m < for all m, ad m p m ƐD m <K 1, where p m is the log-term probability for state m. Theorems 7 ad 9 are easily modifiable. 5.2. Average Cost I this sectio, we demostrate that the base-stock policy structure, optimal for the discouted expected cost model, is also optimal uder a expected average-cost criterio i the ifiite horizo. To satisfy this, the state space is ow the set of all itegers, ad therefore the actio space for each iitial ivetory is fiite. We also assume that the demad is i.i.d. oegative ad iteger ad ƐD 2 <. The previous results remai valid for this more restrictive model. Note that for problems that have fiite state ad actio spaces, demostratio of the covergece of the discouted-cost optimal policy to the average-cost case is quite stadard (see Seott 1989, 1999; Bertsekas 1995. Due to the iclusio of backloggig of usatisfied demad, we caot assume a fiite state space, or boud the state space. Whereas the discouted-cost model has ifiite odeumerable state sets, we assume ifiite deumerable state sets i the average-cost model. This is differet tha Federgrue ad Zipki (1984, who do ot require discrete demad, except for the computatio of the optimal policies. Discreteess of demad does simplify the aalysis, ad our proof does apply directly to a capacitated sigle-stage case aalyzed by Federgrue ad Zipki (1986a, who also assume discrete demad. (For the ucapacitated case, however, to boud the ivetory at a higher istallatio, istead of K 1, a boud based o Theorem 8 eeds to be used. There exist a few versios of sufficiet coditios that guaratee the covergece for the average-cost criterio see Seott (1999 for a excellet review. We focus o the coditios i Schäl (1993, which are based o the optimal discouted value fuctio for a ifiite-state Markov decisio process with ubouded costs. Schäl (1993 suggests two sets of sufficiet coditios for the covergece of the optimal discouted value fuctio ad policy to the average-cost equivalets. We use the first set, which cosists of two coditios. The first coditio is straightforward. To satisfy the secod coditio, some otatio is useful. For the sake of clarity, the ifiitehorizo value fuctio uder the discouted-cost criterio is ow labeled V. X = average cost of policy give state X. = set of radomized policies. g= if X S if X <. m = if X S V X. V X = ifỹ ƐV X Ỹ D. m = if X S V X. Schäl s secod coditio is the followig: (B sup w X < <1 for X S where w X = V X m. Let us restate a lemma from Schäl (1993, rephrased with our otatio ad simplified (geeral 0 is replaced with = 0. Use of this lemma allows us to boud w X i coditio (B.

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System Operatios Research 52(5, pp. 739 755, 2004 INFORMS 751 Lemma 3 (Schäl 1993. For <1, X S: [ 1 ] w X if Ɛ X L X + L X =0 where is the policy miimizig ƐV Ỹ D, X 0 = X, ad X is the ivetory positio periods later, whe policy is used ad radom variable is ay upper boud o = if 0V X m. The followig theorem is the mai result of this sectio. Theorem 10. There exists a statioary policy that is average optimal i the sese that x= lim sup 1/Ɛ X 1 m=0 L X m = if X S if X =g for all X. is limit discout optimal i that for ay X S for all 1 such that X = lim 1 X. Additioally, g = lim 1 1 m = lim 1 1 V. The uderlyig idea of the proof is to show that for ay startig poit ad ay discout factor, the extra cost for ot startig at the best poit is bouded ad the boud is idepedet of the discout factor. The proof of this theorem cosists of several steps. Utilizig Lemma 3, we establish a fiite upper boud o the relative cost differece o its right-had side. The boud is determied by costructig a alterative policy that deliberately visits every poit withi a fiite subset of the state space i which the optimal base-stock levels are guarateed to reside. The cost of the policy, while reachig each of the poits, is a upper boud o the cost util the optimal poit is reached. We show that the cost of this policy is fiite ad that the boud is idepedet of. The complete proof is i the olie appedix at http://or.pubs.iforms.org/pages/collect.html. As for the discouted cost ifiite-horizo sectio, umerical evaluatio of the average cost may be achieved usig the value-iteratio algorithms (see, for example, Puterma 1994, justified by the results i this sectio. Markov-modulated demad requires a simple modificatio, addig the state of the uderlyig Markov chai, ad a coectiveess requiremet similar to that i Kapusciski ad Tayur (1998. 6. Coclusios We have aalyzed a two-echelo supply chai with capacity costraits. Uder quite geeral coditios lower capacity at the lower istallatio we have show that the cost fuctio ca be decomposed ito echelo-depedet compoets, which leads to full characterizatio of the optimal ivetory policy. The optimal policy, the modified echelo base-stock (MEBS policy, is straightforward to describe; the lower istallatio attempts to reach a base-stock level, if possible, as it is costraied by capacity, K 1, ad availability of stock at its immediate supplier, istallatio 2. Echelo 2 also attempts to reach a base-stock level, but is limited by istallatio 1 s capacity. That is, the optimal echelo orderig decisios are restricted to the bad,. The result of this is that there are iduced pealty fuctios applied to the system oce the value fuctio is decomposed. These iduced pealty fuctios are aalogous to those of Clark ad Scarf (1960, except that each istallatio receives a separate fuctio. Echelo 1 accrues a iduced cost for potetially limitig the system by possessig the bottleeck operatio. Echelo 2 accrues a iduced cost for potetially ot providig sufficiet materials to keep its immediate customer sufficietly stocked. We exted this structural result to models icorporatig lead times, Markov-modulated demad (which may have a variety of ostatioary demad processes imbedded ito it, ad to other system cofiguratios. We also exted the mai result to the ifiite-time horizo for discouted-cost ad average-cost criteria. This is doe without sigificat restrictios upo the demad process. Appedix. Additioal Proofs Lemma 1. Let K 1 K j. For ay X all optimal Ỹ satisfy y j maxk 1x j aj 1 for j>1. Proof. Assume that there exists a optimal policy, such that for certai ad j, y j > maxk 1x j aj 1. Without loss of geerality, we choose miimal ad j, i.e., assume that is the shortest horizo for which there exists such a j, ad for that, j is the smallest amog the cadidate istallatios. Let be a alterative policy such that a j = 1 ad aj a j 1 = a j 1 + 1, but otherwise follows policy. Clearly, x j+1 1 = x j+1 1 + 1 ad x j 1 = x j 1 1. Because is optimal, it is feasible. Before comparig the costs of the two policies, we eed to check the feasibility of. It is easy to justify that the sufficiet coditios are: (i a j K j, (ii a j xj+1, (iii a j 1 x j+1 1, 1, ad (v a j 1 K j. (iv a j 1 1 x j Because a j <aj K j, we get (i. Because a j, we get (ii. x j+1 (iii a j 1 = a j 1 + 1 x j+1 1 + 1 = x j+1 1. (iv x j 1 = y j >K 1 implies x j a j 1 1. <aj 1 = x j 1 1 K 1 (v x j 2 = x j 1 a j 1 1 + a j 1 = x j 2 = x j 1 a j 1 a j 1 ad (from (iv, a j 1 1 x j Thus, a j 1 K 1 K j. 1, sox j The differece i cost betwee the policies is 1 + 1 a j 1 1 0. ( N Cost Cost = h i x j 1 x 1 j i=j ( N + h i x j+1 1 x j+1 1 i=j+1 ( N ( N = h i a j + h i a j i=j i=j+1 = h j 1>0 which is a cotradictio with optimality of.

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System 752 Operatios Research 52(5, pp. 739 755, 2004 INFORMS Theorem 2. The modified echelo base-stock policy (MEBS, with parameters z 1 z2 defied above, is optimal for a N = 2 system where K 1 K 2 ad X 2 X 1 K 1. Proof. Theorem 1 states that the value fuctio of the system is separable ito idividual value fuctios, each depedet upo the echelo startig ivetory. It also states that each of these separate value fuctios is covex with (urestricted miima at ad z 2. I additio, Corollary 1(a states that for these coditios, Y 2 Y 1 K 1 for optimal Y 1 ad Y 2. There are two possible cases: (a z 2 K 1 ad (b z 2 >K 1. (a I this case, the global miimizig poit is withi the bad ad z 1 z2 = z 2. Depedig upo the startig ivetory, this poit may ot always be achievable. If X 1 z 1 X2, the we would order to z 1 ;ifz1 lies beyod these limits, we would order the closer limit due to the aforemetioed covexity. Likewise, if X 2 z 2 Y 1 + K 1, we would order up to z 2. However, if z2 lay outside these limits, we would choose the closer limit due to the covexity of the fuctio. Therefore, MEBS is optimal for case (a. (b z 2 >K 1. Due to the result of Corollary 1(a, Y 2 Y 1 + K 1. Now, because z 2 >K 1, ad due to the joit covexity ad separability, there will exist z 1 such that z 1 z2 K 1, which miimizes the value fuctio alog Y 1 Y 1 + K 1. The miimizig z 1 occurs withi this iterval usig the followig logic. J is covex decreasig i Y 1 ad covex decreasig i Y 2 + K 1, implyig that J Y 1 Y 1 + K 1 is covex decreasig i Y 1. Similarly, J is covex icreasig i Y 1 z 2 K 1 ad covex icreasig i Y 2 z 2, implyig that J Y 1 Y 1 + K 1 is covex icreasig i Y 1 z 2 K 1. Whe X 1 z 1 X2, the miimizig poit z 1 z1 +K 1 is reachable. Whe X 2 <z 1, the covexity implies that the poit X 2 X 2 + K 1 gives the lowest cost uder the costrait set. z 1 z2 may ot always be achievable; this poit will ot be achievable if z 1 <z2 K 1.IfX 1 >z 1, the lower istallatio will order othig, Y 1 = X 1 ; echelo 2 will order up to either X 1 +K 1 or z 2, whichever is the smaller, due to the covexity. Thus, MEBS is optimal for case (b. Theorem 6. (a Cosider a N -stage system without capacity limits at stages j<n 1 ad fiite capacities K N 1 K N. Uder the ivetory coditio X N 1 XN K N 1, V X = N j=1 V jxj. The optimal policy is as follows: MEBS holds for the capacitated istallatios, ad the remaiig istallatios follow a echelo base-stock policy. (b Cosider a N -stage system without capacity limits at stages j<n ad fiite capacity K N. For this system, V X = N j=1 V jxj. A echelo base-stock policy is optimal for this system. Proof. (a The form of this proof follows similarly to that of Theorem 1, but the bottleeck is foud at echelo N 1. Ivokig Lemma 2, the sequece of decompositio of the value fuctio begis from echelo 1. The proof mirrors that of Clark ad Scarf (1960 up util echelo N 2. At this poit, the value fuctios for these istallatios are similar to those i Theorem 1, ad the proof is idetical from this poit oward. We utilize Corollary 1(a for these fial two istallatios, because the decompositio of istallatio N 2 from istallatios N 1 ad N merely adds a covex fuctio to the value fuctio for the fial two istallatios. (Essetially, it is show that the decomposed dyamic programs for all echelos cotai iduced pealty fuctios for potetially ot supplyig eough material, ad echelo N 1 has the additioal iduced pealty cost for limitig the ability of echelo N to reach a desirable ivetory level as a result of the capacity limitatio, K N 1. The optimality of the ivetory policy follows from the fact that the covexity of each of the idividual value fuctios of the lowest N 2 istallatios implies that a echelo base-stock level is desirable, idetical to that of Clark ad Scarf (1960. The optimal policy of the fial two istallatios follows idetically from Theorems 1 ad 2. (b The proof of this elemet of the theorem is eve simpler because the impositio of a capacity limitatio upo echelo N ca be icorporated ito the origial Clark ad Scarf (1960 proof without ay problem. This ca be viewed from the perspective of the costraits upo the echelo order-up-to decisio variables, Y j for j 1 2N. The upper boud of each decisio variable except the highest echelo is the stock availability at the ext higher echelo; that is, X j Y j X j+1 for j<n. The limitatio upo echelo N is the capacity limitatio, X N Y N X N + K N. The proof cotiues as i Clark ad Scarf (1960, decomposig the dyamic program from the lowest echelo to the highest. Oce the highest echelo is reached, the resultig dyamic program is a covex operad beig miimized with the decisio variable Y N bouded betwee X N ad X N + K N ; that is, it is a fuctio of X N oly. Cosequetly, the structure of the optimal policy is idetical to that of Clark ad Scarf (1960, amely a echelo base-stock policy, although the actual base-stock levels will differ from those of Clark ad Scarf. All the proofs for Techical Lemmas A1, A2, A3, ad A4 appear o the Operatios Research website at http:// or.pubs.iforms.org/pages/collect.html. Lemma A1. Cosider two fuctios, J A ad J B, each joitly covexad separable i their variables, Y 1 ad Y 2, which satisfy Y J B Ỹ 1 Y J A Ỹ ad 1 If K B <K A, the z 1A z 1B. Y J B Ỹ 2 Y J A Ỹ 2 Lemma A2. Let a N = 2 supply chai with K 2 K 1 operate uder a base-stock policy with levels z such that z 2 z 1 = K 1, ad let X z. If the lower echelo base-stock level is reached, the the upper echelo ca also achieve its base-stock level i the same period.

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System Operatios Research 52(5, pp. 739 755, 2004 INFORMS 753 Lemma A3. Cosider a N -stage serial supply chai operatig uder a MEBS policy, where z j z j 1 = K 1 ad K j K 1 for all j>1. Assume for a give period X j X j 1 = K 1 for all j>1. The, (a X j X j 1 = K 1 for all j > 1 i all future periods, ad (b oce z 1 K 1 X 1 z 1, the z is reachable i a sigle period for all echelos. Lemma A4. Cosider a N -stage serial multiechelo system with capacity limits of K i K 1 at stage i ad 0 < ƐD<K 1. Cosider the echelo base-stock policy, K, orderig up to i 1K 1 at echelo i. Defie T as the radom variable represetig the umber of periods betwee subsequet visits to these base-stock levels. The, (a ƐT <, ad (b there exists A R +, such that J 0K 1 2K 1 N 1K 1 A. Theorem 8. (a There exists a fiite upper boud o ivetory targets for V X, which is idepedet of 0 1, ad (b there exists a fiite upper boud o ivetory targets for ƐVỸ D, which is idepedet of 0 1. A ivetory target is ay YX>X (i.e., a up-to level where we actually icrease ivetory. Proof. We will show that for ay fiite-horizo problem ad for ay 0 1, the fiite-horizo up-to levels are uiformly bouded. The proof for part (b is similar to that of part (a below, but the modified policy defied below would operate K for a differet umber of periods related to differet target ivetory levels; however, the priciple of operatig for lk periods will be the same but the actual umber will differ. This proof follows the stadard type of proof see i Kapusciski ad Tayur (1998. Based o Lemma A4, there exists a policy, K, with base-stock levels 0K 1 2K 1 N 1K 1, such that there is a fiite expected umber of periods util this basestock vector is revisited (regeeratio. We demostrate that there is a fiite upper boud o the optimal base-stock values by cotradictio. Assume that lim sup ỹ =. Therefore, a icreasig sequece k N such that y k ad y k >y for all < k. The cost of followig this optimal policy will be compared to a alterative policy. The alterative, or modified policy, m, is described by operatig K for lk periods, followed by, the optimal policy. Let y N lk = k N 1K 1 ƐD where deotes roudig dow to the earest iteger. Let m deote the radom umber of periods defied as m = mii N ỹ k lk i =ỹ k lk i m or k lk if such a i does ot exist. Usig Wald s theorem, we ca boud Ɛm, N i=1 Ɛm yi k K 1 ƐD Now cosider the cost of policy for the lk periods. Give that the cost fuctio is covex, we ca employ Jese s iequality, so lk ƐL k ix k i i=1 lk[( N ( h j y 1 i=1 j=1 ( N N + j=2 lk[ N ( = h j y 1 i=1 j=1 N lk h j y j k j=2 k k j= k i+1 h p y j p=j k k j= k i+1 ( y N k N 1K 1 1 ƐD ƐD k y j 1 k ƐD N + ] h j y j k j=2 N j=2 ( y N = k ƐD N 1K 1 + ƐD N h ƐD j y j k j=2 = h ( N y N ƐD N 2 1 k + j=2 N 1K 1 + ƐD ƐD h j ƐD yj k y N k N j=2 h j y j k ] h j y j k (A1 (A2 Iequality (A1 is justified from (a the defiitio of lk, ad (b Corollary 1(a. Specifically, it is show i Corollary 1(a that the differece betwee edig ivetory levels at eighborig echelos higher tha echelo 1 will ot be greater tha K 1. This is the reflected i the defiitio of lk, where N 1 uits of K 1 are subtracted from echelo N s target level; this differece is esured to be lower tha y 1 k, thus justifyig (A1. Let us ow focus o (A2. Because y 1 k y 2 k N 1 y k y N (due to state space, the term growig most quickly as k icreases is the y N k 2 term. The coefficiet of this term is positive because h N > 0 ad 0 < ƐD<K 1. We ow have a expressio that acts as a lower boud o operatig for lk periods. The cost of operatig K for periods is bouded above by A (Lemma A4. Give that policy m acts as policy from period lk + 1 to period lk + m, the process operatig uder policy m is the same process as the process operatig uder policy startig from differet iitial states i period lk + 1. Glasserma ad Tayur (1994 demostrate that this process is Harris ergodic if the stability coditio ƐD <K 1 mi i K i (our otatio is satisfied, which our model does. Cosequetly, our model admits couplig, a direct result of the regeerative structure of a Harris ergodic Markov chai. This meas that the processes startig from differet iitial states will coicide after a fiite

Parker ad Kapusciski: Optimal Policies for a Capacitated Two-Echelo Ivetory System 754 Operatios Research 52(5, pp. 739 755, 2004 INFORMS (radom time; that is, m<. The cost of operatig uder either m or durig these m periods is, therefore, fiite. The cost of operatig policy m durig periods lk + 1 to lk + m cosists of pealty ad holdig costs. Because will be coducted durig this iterval, the pealty costs will ot be larger tha those of policy K. Similarly, the ivetory levels at echelo 1 achieved durig this iterval will be lower tha the echelo 1 ivetories realized from the origial policy, the cost of which forms a boud o the echelo 1 holdig cost. The boud for the holdig costs at higher echelos ca be regarded as a costat i every period: Corollary 1(a demostrates that the optimal policy will require o more tha K 1 uits at these higher istallatios ad, thus, N N i=2 j=1 h jk 1 = N 1 N j=1 h jk 1 serves as a boud i each period. Therefore, the cost of operatig m durig the first lk + m periods is bouded by N Alk + Ɛm + N 1 h j K 1 Ɛm j=1 ( N ( 1 A lk + Ɛm A y i k i=1 ƐD + 1 K 1 ƐD (A3 Cosequetly, the differece i costs betwee policies ad m over period lk + m is at least (A2 mius (A3, which is quadratic i y N k. So, as k icreases, this term domiates ad the cost differece goes to ifiity. This suggests that m is less costly tha, cotradictig our iitial assumptio. Edotes 1. Clark ad Scarf (1960 assig orderig decisios to the ed of the curret period istead of the begiig of the followig oe. This is equivalet to our sequece, which is more commoly used. 2. Porteus (2002, p. 137 shows a more geeral result. If a fuctio f chages sig j<times, whe the Pólya radom variable of order traslates f resultig i gx = Ɛfx X, the g chages sig at most j times. 3. 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