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Managing Inventory and Supply Performance in AssemblySystems with Random Supply Capacity and DemandRamesh Bollapragada, Uday S. Rao, Jun Zhang,

To cite this article:Ramesh Bollapragada, Uday S. Rao, Jun Zhang, (2004) Managing Inventory and Supply Performance in Assembly Systems withRandom Supply Capacity and Demand. Management Science 50(12):1729-1743. http://dx.doi.org/10.1287/mnsc.1040.0314

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MANAGEMENT SCIENCEVol. 50, No. 12, December 2004, pp. 1729–1743issn 0025-1909 �eissn 1526-5501 �04 �5012 �1729

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doi 10.1287/mnsc.1040.0314©2004 INFORMS

Managing Inventory and Supply Performance inAssembly Systems with Random Supply

Capacity and Demand

Ramesh BollapragadaCollege of Business, San Francisco State University, San Francisco, California 94132, and

Bell Laboratories, Lucent Technologies, Holmdel, New Jersey 07733, rbollapragada@yahoo.com

Uday S. RaoCollege of Business, University of Cincinnati, Cincinnati, Ohio 45221, uday.rao@uc.edu

Jun ZhangA. B. Freeman School of Business, Tulane University, New Orleans, Louisiana 70118, jzhang@tulane.edu

We consider stock positioning in a pure assembly system controlled using installation base-stock policies.When component suppliers have random capacity and end-product demand is uncertain, we characterize

the system’s inventory dynamics. We show that components and the end product play convex complemen-tary roles in providing customer service. We propose a decomposition approach that uses an internal servicelevel to independently determine near-optimal stock levels for each component. Compared with the optimal,the average error of the decomposition approach is 0.66% across the tested instances. Compared with currentpractice, this approach has the potential to reduce the safety-stock cost by as much as 30%. Our computationalanalysis on two-echelon systems also illustrates several managerial insights: We observe that the cost reductionfrom improving supply performance is high when demand variability or the number of components or tar-get customer service is high, or when the end product is more expensive relative to components. On average,(i) reducing the lead time of the more expensive component yielded higher benefit than reducing the lead timefor the less expensive component, and (ii) the benefit of improving one of the supply parameters (service levelor lead time) was higher when the value of the other parameter was already more favorable (lower lead timeor higher service level, respectively). Finally, we analytically show how a multi-echelon pure assembly systemmay be converted into an equivalent two-echelon assembly system to which all our results apply.

Key words : assembly system; installation base-stock policy; external and internal service levels; uncertainty;decomposition approach

History : Accepted by Wallace J. Hopp, operations and supply chain management; received October 16, 2000.This paper was with the authors 10 12 months for 6 revisions.

1. IntroductionOperations managers frequently face the problem ofdistributing the right amount of inventories at differ-ent levels in a supply chain. Typically, the goal of thisstock positioning is to meet management-set customerservice targets without violating budget constraintson the total inventory investment. In addition todetermining an effective stock-positioning approach,managers seek to identify which supply levers willbest improve system performance. The work reportedhere is motivated by coordination issues faced bya large telecommunications company in one of itsadvanced manufacturing facilities. The basic setting isa two-echelon assembly system in which components,ordered from outside suppliers, are assembled intoend products which are then sold to customers. Man-ufacturing lead times are comparable to component

supply lead times because the assembly process iscomplex. There is uncertainty in both component sup-ply and end-product demand; consequently, there issafety-stock inventory at both echelons. Currently,the firm sets component and end-product inventoriesindependently using a discrete time, decision supportsystem. The information system uses “node”-specificinput parameters, such as demand and supply vari-ability and lead times, to determine safety-stocklevels. The deployed system is quite robust and con-siders many important parameters. However, man-agers believe that the component and end-productlevels should be linked through an overarching modelthat takes a more integrated view of inventorymanagement. Hence, we study the interrelationshipbetween component and end-product inventories inthe supply chain.

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Bollapragada et al.: Managing Inventory and Supply Performance in Assembly Systems1730 Management Science 50(12), pp. 1729–1743, © 2004 INFORMS

In this paper, we present a supply model with ran-dom supply capacity, and characterize the steady-state inventory behavior of a component facing suchsupply uncertainty. We first focus on a two-echelonsystem consisting of components assembled into asingle end product. We prove that the problem ofdetermining base-stock levels to minimize total inven-tory investment, subject to meeting a specified tar-get customer service level, is a convex programmingproblem. We present bounds on the optimal stocklevels and present a simulation-based inventory opti-mization approach using perturbation analysis. Wepropose a quick decomposition approach for deter-mining near-optimal base-stock levels using an inter-nal service level (defined in §3.4) for components.This internal service level effectively coordinates thereplenishment ordering activities at different stagesof the supply chain. Across the tested instances, theaverage error of the decomposition approach is 0.66%.Compared with current practice, this decompositionapproach has the potential to reduce the safety-stockcost by as much as 30%. Through computationalexperiments, we identify when improvement in sup-ply performance will yield a significant reductionin total inventory investment. In particular, we seethat the cost reduction from improving supply perfor-mance is high when demand variability is high, thenumber of components is high, the target service levelis high, or the end product is more expensive relativeto components. However, the percentage reduction insafety-stock costs with supply lead time improvementis more or less constant across different values of eachof these parameters. On average, (i) reducing the leadtime of the more expensive component yielded higherbenefit than reducing lead time for the less expensivecomponent, and (ii) the benefit of improving one ofthe supply parameters (service level or lead time) washigher when the value of the other parameter wasalready more favorable (lower lead time or higher ser-vice level, respectively). We show how our analysisapplies to a multi-echelon pure assembly system byconverting it to an equivalent two-echelon system.In the remainder of this section, we briefly review

the literature. In §2, we develop a mathematical for-mulation using a new model of supply uncertainty.In §3, we present an analysis of two-echelon systems(including optimal and heuristic solutions). Resultsfrom computational experiments are detailed in §4.Multi-echelon pure assembly systems are consideredin §5. We conclude in §6 with a summary and exploremodel extensions in an appendix.

1.1. Literature ReviewMultiechelon inventory problems with demand un-certainty have been widely studied since the semi-nal work by Clark and Scarf (1960). See, for instance,

Axsäter (1993), Federgruen (1993), and Diks et al.(1996) for a review of related work, and Ettl et al.(2000) and Graves and Willems (2000) for solutionapproaches. A common heuristic procedure uses aninternal service level at each location to facilitatedecomposition (Lee and Billington 1993, Inderfurthand Minner 1998). For instance, Lee and Billingtonfirst estimate the mean and variance of lead-timedemand (for normal demand). Then, given the inter-nal service levels, they use a heuristic search to com-pute base-stock levels for Hewlett Packard’s printersupply chain. Similar to Lee and Billington, oursolution approach will decompose the multi-echelonproblem into single-echelon problems. However, weuse a novel search for optimal internal service levelsand we do not restrict demand to be normal. Instead,at each location, we use Monte Carlo simulation tofind the component base-stock level corresponding tothe internal service level. Another difference betweenour model and previous approaches is that we explic-itly incorporate both supply and demand uncertainty,in a multi-echelon assembly framework.Supply uncertainty in single-echelon systems has

been studied by Henig and Gerchak (1991) (periodicreview with random yield), Ciarallo et al. (1994) (ran-dom supply capacity), Anupindi and Akella (1993)(placement of replenishment orders with dual unre-liable suppliers), and Moinzadeh and Lee (1989)(�Q� r� policy with order delivery potentially in twoshipments). The above papers all consider supplyquantity uncertainty; random supply lead time hasbeen studied by Kaplan (1970), Song and Zipkin(1996), and Robinson et al. (2001). In the environ-ment that motivates this research, partial deliveryof an order is allowed and happens frequently. Wemodel the supply process using a random capacityas in Ciarallo et al. (1994) and in Bollapragada et al.(2004). However, we have several components fac-ing supply uncertainty, all of which are required tomake the end product. Furthermore, we assumethat unsatisfied demand for components is back-logged, whereas Ciarallo et al. (1994) assume thatthis demand is lost. The lost-demand model fits com-modity products well, whereas the backlog modelis more appropriate for high technology, i.e., spe-cialty products for which qualified component sup-pliers go through a lengthy certification process. Ourindustrial data suggest that, although there was oftena discrepancy between orders placed with the sup-plier and the corresponding shipments, the cumula-tive orders matched the cumulative shipments quiteclosely.When no inventories are held at the end-product

level, our system reduces to an assemble-to-order(ATO) system with component supply uncertainty.Extensive work has been done in studying such

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Bollapragada et al.: Managing Inventory and Supply Performance in Assembly SystemsManagement Science 50(12), pp. 1729–1743, © 2004 INFORMS 1731

systems; see, for example, Gallien and Wein (2001),Gurnani et al. (1996, 2000), Song et al. (1999), Songet al. (2000), Song and Yao (2002), and Wang (2000). Inmost of these studies, supply uncertainty is modeledusing random component lead times. The exceptionsare the papers by Gurnani et al., where supply uncer-tainty is due to random yield at component suppliers.Typically, research on ATO systems assumes that thefinal assembly takes negligible time compared withcomponent procurement, which holds true in certainenvironments such as the personal computer industry.Our work is different, in that we explicitly considernonzero lead time for final assembly; these lead timesapply to environments such as assembly of complextelecommunications switching systems.

2. Problem ModelingConsider a pure assembly system in which com-ponents are purchased from external suppliers andassembled into a single end product through differentstages of subassembly. The assembly system can berepresented by a tree network with N nodes, whereeach node represents an item (component, subassem-bly, or end product). For i = 0� � � � �N −1 with 0 denot-ing the end-product, let ��i� denote the set of directpredecessors of node i. That is, if j ∈��i�, then item jis needed to assemble item i. We use li to denotethe assembly lead time for item i; when all items j ,j ∈ ��i�, are available, it takes li periods to completethe assembly of item i. Let � �i� denote the set of allsuccessors of node i. That is, if k ∈ � �i�, then item iis needed, either directly or indirectly through othernodes, to produce item k. Without loss of generality,one unit of node j is needed to produce one unit ofits successors. Let �n be the demand in period n. Weassume �n is stationary and use � to denote the perioddemand.

2.1. Supply UncertaintyEach component i has a unique supplier who quotesa delivery lead time li. However, suppliers may notalways deliver orders within their quoted lead times.Hence, li is the nominal lead time of component i.We model suppliers by assuming they have randomcapacity. In period n, a random delivery capacity�i

n is realized at the component supplier, who thendelivers either the amount of i on order or �i

n units(whichever is smaller). Unsatisfied orders are back-logged. To ensure system stability, we assume that �i

n

is a stationary process with E��n� < E��in�.

Another common way of modeling supply uncer-tainty is to use random lead times. When a base-stock policy is followed, there is a connection betweenthe random capacity and random lead-time models.This connection can be established by studying theinventory dynamics in both systems. The key random

variable corresponds to shortfall, where shortfalldenotes the quantity of inventory remaining to bedelivered by the supplier. In §A.3, we show that theshortfall for our random capacity model is relatedto the shortfall for the random lead-time modelwith stationary lead time. Consequently, our solutionapproach may be adapted to the random lead-timecase, from which similar results can be obtained.

2.2. Minimizing Cost Under a Service ConstraintThe system operates under a periodic-review instal-lation base-stock policy. Under this policy, when theinventory position of an item (i.e., on-hand plus on-order minus backlog) falls below some specified base-stock level, a replenishment order is placed to raisethe item’s inventory position back to the base-stocklevel. All events occur at the beginning of each periodand in the following sequence: (i) Obtain demandfor end-product assemblies and communicate it to allechelons. Use available inventory to satisfy as muchof this demand as possible; backorder any excess cus-tomer demand. (ii) Initiate new assembly activitiesand place component purchase orders. (iii) Receivecomponent deliveries from suppliers (based on ordersplaced at least the nominal lead-time periods earlier);add completed subassemblies and end products toinventory.Let si denote the base-stock level and let ci be

the corresponding cost per unit of node i. LetI 0�s0� � � � � sN−1� denote the steady-state inventory levelof the end product at the beginning of a period, beforedemand materializes. Let � be the desired Type-1 ser-vice level for the end product. That is, the chance ofsatisfying end-product demand from inventory mustbe at least �. Then the problem may be formulated asfollows:

minsi≥0

N−1∑i=0

cisi s.t. P[I 0�s0� � � � � sN−1�≥ �

]≥ �� (1)

Although the objective in (1) is not the traditionalinventory holding cost, it reflects the basic trade-off between keeping inventories at different locationsbecause a higher si typically implies more invento-ries of item i. An objective similar to (1) has beencommonly used when service-level constraints arespecified, for instance, in the component commonalityliterature (Baker et al. 1986) and in ATO systems(Wang 2000). Our numerical experiments (§4) showthat the inventory costs resulting from the objec-tive in (1) are quite close (within 0.5%) to the mini-mum inventory costs corresponding to a holding costobjective. We do not consider shortage penalty costsbecause the customer-service-level target is used tolimit shortages.

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Bollapragada et al.: Managing Inventory and Supply Performance in Assembly Systems1732 Management Science 50(12), pp. 1729–1743, © 2004 INFORMS

3. A Two-Echelon Assembly SystemIn this section, we study the steady-state inventorywhen a base-stock policy is followed. We exploit prob-lem convexity to characterize the relation betweendifferent base-stock levels, and develop optimal andheuristic solution approaches. To simplify notation,we consider an assembly system with only two com-ponents and assume that all lead times are zero.In §3.5, we demonstrate how our analysis may beapplied to the case of nonzero lead times; the exten-sion to more than two components is straightforward.

3.1. Steady-State InventoryLet Y i

n denote the outstanding orders of component iin period n that have not been delivered becauseof the limited supply capacity. Because an installa-tion base-stock policy is followed for each compo-nent, when demand �n materializes an order of �n

is placed with the supplier. The supplier then deliv-ers min�Y i

n +�n� �in�, where �i

n is the realized capacityof the supplier in period n. Hence, the outstandingorders of component i at the beginning of period n+1satisfy

Y in+1 = Y i

n + �n −min�Y in + �n��i

n�

= max�0�Y in + �n −�i

n�� for i = 1�2� (2)

The above recursion for the outstanding orders ofa component is similar to the inventory shortfallbehavior in a single-echelon capacitated production-inventory system under the base-stock policy. Hence,following the terminology in Glasserman and Tayur(1994), we call Y i

n the shortfall of component i. Simi-larly, we define the shortfall of the end product Y 0

n asthe difference between s0 and the inventory positionof the end product; Y 0

n is equal to the backlog of eachcomponent in period n.Clearly, the assembly quantity in period n is

restricted by the on-hand inventory of both compo-nents. Let I i

n denote the (on-hand) inventory level ofcomponent i at the beginning of period n. The inven-tory position of component i is, then, its on-handinventory, I i

n, plus its outstanding orders, Yin, minus its

backlog, Y 0n . Because a base-stock policy is followed

for each component, at the beginning of period n wehave

I in +Y i

n −Y 0n = si� for all i� (3)

and, hence, the on-hand inventory of component i isI in = si +Y 0

n −Y in. Because at most mini=1�2 I i

n productscan be assembled in period n, analogous to (2), theshortfall of the end product is

Y 0n+1 = Y 0

n +�n−min�Y 0n +�n�I

1n�I 2n�

�3�= Y 0n +�n−min�Y 0

n +�n�s1+Y 0

n −Y 1n �s2+Y 0

n −Y 2n �

= maxi=1�2

�0�Y in+�n−si�� (4)

Consequently, by Equations (2) and (4), we have arecursion for the shortfall vector Yn ≡ �Y 0

n �Y 1n �Y 2

n � asYn+1 =��Yn��n��n�, where ��·� is increasing and con-tinuous in Yn. Using Theorem 5.1 from Glassermanand Yao (1995), we can establish the existence of asteady-state distribution for the shortfall vector. Thisis stated below in Proposition 1, where d→ and d=denote “converges in distribution” and “equals in dis-tribution,” respectively. Missing proofs are availablein the electronic companion pages (http://mansci.pubs.informs.org/ecompanion.html).

Proposition 1. If the demands �n and the supplycapacities �i

n are stationary and ergodic, with E��� < E��i�for all i, then Yn

d→ Y and the steady-state distributionY = �Y 0�Y 1�Y 2� satisfies

Y i d= max�0�Y i + � −�i�� for i = 1�2�Y 0 d= max

i=1�2�0�Y i + � − si��

Note that the inventory level of the end product iss0−Y 0 (because the lead times are zero the inventorylevel is equal to the inventory position). Hence, theType-1 service level is P�s0 − Y 0 ≥ ��. Therefore, thetwo-echelon assembly system problem can be refor-mulated as

min2∑

i=0cisi s.t. P�s0−Y 0 ≥ ��≥ ��

where Y 0 is given by Proposition 1. Let A�s1� s2� s3�denote the feasible region for this problem.Proposition 2 below shows that the reformulationcorresponds to a convex programming problem.

Proposition 2. The feasible region for the two-echelonassembly system, A�s0� s1� s2�, is a closed convex set, if thejoint distribution of ���Y 1�Y 2� is quasi-concave.1

3.2. Relation Between Base-Stock LevelsWe now characterize the convex-decreasing relation-ship among the order-up-to levels s0, s1, and s2. Notethat the practice of setting stock levels independentlyignores these relationships.

Proposition 3. Let s0�s1�s2�=min�s0 �P�s0−Y 0≥��≥��. Similarly, let sk�s0�sj �=min�sk �P�s0−Y 0≥��≥��,for �k�j�= �1�2� or �2�1�. Then (i) s0�s1�s2� is decreasing2

and convex in �s1�s2�, and (ii) sk�s0�sj � is decreasing andconvex in s0, for each sj .

1 A probability measure is called quasi-concave if its probabilitydensity function is quasi-concave. Many common probability dis-tributions fall into this category—for example, the Gamma distri-bution and the normal distribution.2 In this paper, all comparisons between vectors are componentwise comparisons. Thus when we say �s1�s2� increases, we meanthat no component of �s1�s2� decreases.

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Bollapragada et al.: Managing Inventory and Supply Performance in Assembly SystemsManagement Science 50(12), pp. 1729–1743, © 2004 INFORMS 1733

By Proposition 3, components and end-productinventories play complementary roles in servicingcustomer demands. A reduction in the end prod-uct’s base-stock level must be compensated by anincrease in the components’ base-stock levels to main-tain the same customer service level. Moreover, themagnitude of this increase in component stock growsin a convex manner as the end-product inventorydecreases. Therefore, a good balance between com-ponent stocks and end-product stocks is needed tolower the total inventory investment. Propositions 4and 5 characterize the relationship between feasiblebase-stock levels of different components.

Proposition 4. sk�s0� sj � is decreasing in sj for all s0,with �k� j�= �1�2� or �2�1�.

The intuition behind Proposition 4 is as follows:More inventory of a component allows the inventoryof the other component to be reduced due to reducedcoordination pressures.

Proposition 5. If the supply capacity of Component 1is stochastically (Stoyan 1983) greater than that ofComponent 2 and l1 = l2, then the optimal base-stock levelssatisfy s1∗ ≤ s2∗.

Corollary 6. If �1 d= �2 and l1 = l2, then in the opti-mal solution, s1∗ = s2∗.

Propositions 7 and 8 below specify bounds on s0∗,s1∗, and s2∗. These bounds yield a starting point in oursearch for optimal stock levels.

Proposition 7. Let s0∗ denote the optimal base-stocklevel of the end product, then

s0�s1max� s2max�≤ s0∗ ≤ s0�s1min� s2min��

where simax and si

min denote, respectively, upper and lowerbounds on si∗, the optimal base-stock level for componenti; for instance, si

min = 0 and simax =�.

Proposition 8. Given a feasible base-stock level for theend product, s0, the corresponding feasible base-stock levelof each component satisfies:(i) si ≥ Si�s0� for i = 1�2, where Si�s0� is the base-

stock level for component i obtained by solving a two-stageserial system problem with infinite inventory of the othercomponent.(ii) s1 ≤ s1�s0� S2�s0�� and s2 ≤ s2�s0� S1�s0��, where

si�·� ·� is defined in Proposition 3.

Proposition 9. The optimal base-stock level, si∗, de-creases with ci, for i = 0�1�2.Proposition 9 is intuitively clear. As the cost of any

item increases, lower inventories should be kept forthis item.

3.3. Simulation-Based Inventory OptimizationBy Proposition 2, the two-echelon assembly systemproblem is a convex programming problem. There-fore, we can obtain optimal stocking levels using agradient-based search. We estimate the gradient usinginfinitesimal perturbation analysis (IPA). To overcomethe difficulties caused by the “noise” (Bashyam andFu 1998) in the nonlinear service-level constraint, wesolve the following related problem:

��C�=max P�s0−Y 0≥��

s.t.2∑

i=0cisi=C�si≥0� for i=0�1�2� (5)

The solution to the above problem provides the high-est customer service level, ��C�, for a given inventoryinvestment C. Because ��C� is increasing in C, ouroriginal problem may be solved using a binary searchfor the minimum C satisfying ��C� ≥ �. For each C,we solve the corresponding instance of (5) and deter-mine ��C� using a search procedure that is an adap-tation of Zoutendijk’s feasible direction method witha linear equality constraint (Bazaraa et al. 1993). Thissearch algorithm takes the following form: st+1 = st +atdt . st is the vector of base-stock levels �s0� � � � � sN−1�at the tth iteration. at is the step size and dt is animproving feasible direction vector found by solvingthe following linear program:

maxdt

"V � st�′dt

s.t.2∑

i=0cidi

t = 0$ −1≤ dit ≤ 1$ di

t ≥ 0� if sit = 0� (6)

where "V � st� is the gradient of the objective functionat base stock st . As in Glasserman and Tayur (1995),this gradient may be estimated using simulation andIPA. Given "V � st�, (6) is a variant of a continuouslinear knapsack problem that we solve to optimalityusing a greedy approach. Even after this procedureis accelerated using Proposition 3 and the bounds onbase-stock levels from Propositions 7 and 8, it canbe computationally intensive, especially as the num-ber of components increases. Hence, we develop aquicker heuristic approach.

3.4. A Decomposition Heuristic (Using an InternalService Level)

Let the internal service level, %i, of component i bedefined as the probability that its on-hand inventoryis greater than the demand for this component (aris-ing from assembly of the end product). That is, %i =P�si−Y i ≥ ��. This internal service level depends onlyon �, �i, and si. Hence, once %i is specified, the deter-mination of base-stock level si is independent of othercomponents.

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Bollapragada et al.: Managing Inventory and Supply Performance in Assembly Systems1734 Management Science 50(12), pp. 1729–1743, © 2004 INFORMS

From our computational experiments, we foundthat the total inventory cost is insensitive to the choiceof the internal service level, over a wide range ofnear-optimal values of %i. This observation moti-vated us to develop a heuristic decompositionapproach that sets %i to be a function of a singlecomponent-independent parameter (denoted as %)and component-specific problem parameters. That is,for component i we set %i = f i�%� ci� li��i�. Then wesolve the following problem:

min0≤%≤1

U�%�≡minsi≥0

c0s0+ c1s1+ c2s2

s.t. P�s0−Y 0 ≥ ��≥ ��

P�s1−Y 1 ≥ ��=%1�

P�s2−Y 2 ≥ ��=%2�

Given %, we first calculate %i, then we determinethe base-stock levels for both components by solv-ing two single-echelon problems, minsi≥0�cisi � P�si −Y i ≥ �� ≥ %i�, using Monte Carlo simulation. Sub-sequently, we compute the optimal end product’sbase-stock level corresponding to the calculated com-ponent base-stock levels, by solving mins0≥0�s0 � P�s0−Y 0 ≥ �� ≥ ��. Because the service-level measure ismonotone increasing in s0, an efficient binary searchcan be used to find s0. Proposition 10 below demon-strates that, when %i is monotone in %, a Fibonaccisearch over % finds the optimal %∗ that minimizesU�%�. Details of the algorithm are presented inFigure 1. In §4, we test this decomposition approachby setting %i =%, for all i, and comparing the result-ing heuristic solution with the optimal solution. Set-ting internal service levels of all components to beidentical is a commonly used practice in the industry(Agrawal and Cohen 2001).

Proposition 10. U�%� is quasi-convex in %.

The internal service level provides a simple yeteffective instrument for coordinating inventory ofcomponents and the end product. In particular, it can

Figure 1 Decomposition Approach (Using Fibonacci Series F�)

Procedure Decomposition Assembly �(� c������� l�1. a= 0; d = 1�0;2. Find minimum ) such that F) > �d− a�/(;3. ratio= F)−2/F)

4. b = a+ ratio�d− a�; c = d− �b− a�= a+ d− b;5. For i from 1 to )− 2 {

sj =min�sj � P�sj −Y j ≥ ��≥ b�, j = 1�2s0 = s0�s1� s2�, U�b�= c0s0+ c1s1+ c2s2

sj =min�sj � P�sj −Y j ≥ ��≥ c�, j = 1�2s0 = s0�s1� s2�, U�c�= c0s0+ c1s1+ c2s2

If U�b� < U�c�, then d = c$ c = b$ b = a+ d− c;Else a= b$ b = c$ c = a+ d− b;}

6. If U�b� < U�c� then the desired % is in �a� c�Else % lies in the interval �b�d�.

be used when different components are processedin geographically dispersed facilities. Under such anenvironment, different material managers control thereplenishment-ordering activities of different compo-nents. In this case, it may be difficult for a centralplanner to specify how many components should beordered from its supplier in a particular period. Theplanner may prefer to specify an internal service level.In order to provide this service, the managers replen-ishing components will carry the desired componentinventories. Our analysis demonstrates that such ascheme can be used to achieve near-optimal inventoryinvestment across the entire system.Because our computational experiments in §4 will

use nonzero lead times for component procurementand product assembly, we briefly consider this casenext.

3.5. Analysis for Positive Lead TimesIn this section, we demonstrate how our analysismay be carried over to the case when the componentpurchasing and product assembly incur positive leadtimes. In particular, we derive the shortfall recursionsfor the end product. Because the component shortfallrecursions were based on its inventory position, theyremain the same as the zero lead-time case. Let IPi

n

be component i’s on-hand inventory plus inventoriesin transit from the supplier in period n. Due to thebase-stock policy, IPi

n−li +Y in−li

−Y 0n−li

= si holds for i =1�2. Because all in-transit inventory of component iin period n − li will be delivered by period n, theinventory level of component i at period n, I i

n, equalsIPi

n−li minus the total consumption of component ifrom period n− li to period n−1. This total consump-tion can be shown to be equal to �n−1 + · · · + �n−li −Y 0

n +Y 0n−li. Hence, the inventory level of component i

at period n for i = 1�2 satisfiesI in = IPi

n−li − �n−1− · · ·− �n−li +Y 0n −Y 0

n−li

= si −Y in−li − �n−1− · · ·− �n−li +Y 0

n �

Consequently, similar to (4), the end-product shortfallsatisfies

Y 0n+1 =max

i=1�2�0�Y i

n−li + �n−li + · · ·+ �n − si��

Based on these shortfall recursions, we can extend allour results to the case of positive lead times.

4. Computational ResultsIn this section, we first investigate performance ofthe decomposition heuristic. Next, we explore howthe base-stock objective compares with a holding-cost objective. Finally, we provide some managerialinsights on managing multi-echelon assembly sys-tems with supply uncertainty.

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Table 1 Lead-Time Settings for Different Number of Components,N − 1

N − 1 l1 l2 l3 l4 l5 l6 l7 l8 l9 l10 l0

2 1 2 �9�1�4 1 2 3 4 �7�3�6 1 2 3 4 5 6 �5�5�8 1 2 3 4 5 6 7 8 �3�7�10 1 2 3 4 5 6 7 8 9 10 �1�9�

4.1. Performance of Heuristic DecompositionApproach

We first compare the heuristic decomposition to theoptimal in terms of the inventory objective, base-stocklevels, and run time. Next, we compare the heuris-tic solution to current practice. In the comparisons,we exclude the costs of satisfying average demandand focus on safety-stock costs. That is, if si is thebase-stock level from the decomposition approach,the heuristic cost = ∑N−1

i=0 ci�si − E����. Similarly, wecalculate the optimal cost using the same formula,with the optimal si∗ replacing si. The accuracy of theheuristic is measured by its

Percentage Relative Error

= Heuristic Cost−Optimal CostOptimal Cost

× 100%�

The running time is measured in CPU seconds on aSUN Sparcstation 10.The test problem instances were generated by

adapting the parameter settings used by Houtum andZijm (1991). The number of components, N − 1, waschosen from �2�4�6�8�10�; larger instances could notbe effectively handled by the simulation-based opti-mization procedure. The lead-time settings are shownin Table 1; the lead time for the end product, l0, waseither . or 10− . for . = 1�3�5�7�9. We set the unitcost of component i as ci = li for i = 1� � � � �N − 1, andc0 = l0 ×∑N−1

i=1 ci. We tested the heuristic for Gammademand (�) with a mean of 10. The capacity of eachsupplier, �i, was chosen to be gamma distributed withthe same coefficient of variance as � and mean deter-mined using P�� ≤ �i� = /i. We call /i component i’ssupply service level. We fixed /i = / for all i. By set-ting demand cv = 0�7, customer service level � = 0�9,and /= 0�9, we obtained the base case, for each choicefor number of components �N −1� and lead time �l0�.We then varied the demand cv, �, and / individually

Table 2 Heuristic Performance for Different Service-Level Targets, �

� 70% 75% 80% 85% 90% 95% 96% 97% 98% 99%

Avg. error (%) 1.56 1.20 1.02 0.55 0.52 0.56 0.67 0.40 0.53 1.12Std. dev. error (%) 1.08 0.88 0.70 0.31 0.28 0.49 0.78 0.26 0.79 0.92Max. error (%) 3.40 2.92 1.91 1.07 1.21 1.50 2.59 1.13 2.57 2.69

while keeping the other two parameters unchanged.In particular, demand cv was varied from 0.1 to 1 insteps of 0.1; the target service level, �, was chosen asin Table 2. The set of / values tested included 0.70,0.75, 0.80, 0.85, 0.90, 0.95, 0.96, 0.97, 0.98, and 0.99. Thedifferent choices for N , l0, cv, �, and / yielded a totalof 5× 2× �10+ 10+ 10�= 300 test instances.4.1.1. Heuristic Decomposition vs. Optimal.

Across the 300 tested instances, the average errorof the decomposition approach was 0�66% withstandard deviation 0.59% and maximum error 3�40%.(Similar performance was observed across the over2 thousand other test instances used in §4.3 to investi-gate supply uncertainty.) We report the relative errorsin Table 2 for different � with / = 0�9 and demandcv = 0�7. Similar results were obtained when cv or /was varied. From these experiments we noted thatthe heuristic’s accuracy was, on average, between0.3% and 1.6% and was insensitive to factors such asdemand variance, target service level, and the supplycapacity. The highest errors occurred when �= 0�7 or0.99. Given the mean and the standard deviation ofthe errors, we could identify instances where the rel-ative error was more than three standard deviationsbeyond the mean. All these instances (6 out of 300)occurred when cv = 0�7, / = 0�9, and � = 0�7 (2 outof 6) or � = 0�99 (4 out of 6). This suggests that thedecomposition approach might perform better forintermediate levels of customer service (in the range�= 0�75 to 0.98).In Table 3, we show that the heuristic’s error was,

on average, within 1 percent of the optimal, indepen-dent of the number of components. An explanationfor the observed “stable” performance of the heuristicas the number of components changes is as follows:The error in the heuristic results from the fact that weforce all internal service levels to be identical. As thenumber of components increases, we are effectivelyadding more constraints, and consequently we mayexpect the performance of the heuristic to deteriorate.However, when the number of components becomesvery large the objective function becomes flat and theincremental impact of adding one more constraint issmall. Consequently, the magnitude of the absoluteerror could, on the one hand, increase or decrease asthe number of components increases. On the otherhand, the total safety-stock cost always increases withnumber of components. Because this cost is used in

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Table 3 Heuristic Performance for Different Number of Components

Components (#) 2 4 6 8 10

Avg. error (%) 040 070 075 093 055Std. dev. error (%) 049 053 053 075 049Max. error (%) 269 242 197 340 221Decom. time 11378 27001 46852 69651 1�03145

(CPU sec.)Opt. time 3�02153 2�95205 6�13810 7�18289 8�57735

(CPU sec.)

the denominator to calculate the relative error, thiserror could decrease even when the absolute error(numerator) increases. Thus, we see that due to thecombined effect of the above-mentioned factors, it ispossible for the percentage error of the heuristic todecrease as the number of components increases.Base-Stock Levels: We use four-component problem

instances (with l0 = 7, �= 0�95, and /= 0�90) to illus-trate how inventories are distributed under differ-ent values of demand variance. Table 4 shows theresults for a subset of tested demand cv values. Wesee that (i) as cv increases, the total inventory invest-ment increases; (ii) the individual item base-stock lev-els from the heuristic are close to the correspondingoptimal values; and (iii) for an external service levelof �= 0�95 with supply service level /= 0�9, the inter-nal service levels, %∗, obtained from the decomposi-tion approach were between 91% and 97%; %∗ wasnot monotone in demand cv. Given the high valueof %∗, it may appear that the decomposition does wellbecause there are few component backorders. How-ever, as illustrated below, there are instances in whichthe heuristic performs well even though the internalservice level is less than 50%.Example 1. Consider a two-component case with

�l1� l2� l0� = �1�2�1�, �c1� c2� c0� = �1�2�3�, � = 0�99,demand cv = 0�7, and / = 0�7. The base-stock lev-els computed by the decomposition approach are�s1� s2� s0� = �8�6�18�5�49�8� with a safety-stock costof 114.9 and an internal service level of %∗ = 47%.The corresponding optimal stock levels are (13.0, 24.0,44.4) with a safety-stock cost of 114.1.Internal Service Levels: Our decomposition approach

determines near-optimal base-stock levels by choosinga proper internal service level (%) for all components.

Table 4 Sample Base-Stock Levels of Four-Component Problems

Heuristic solution Optimal solution

cv s1 s2 s3 s4 s0 ∗ s1 s2 s3 s4 s0

0.2 1323 2434 3503 4581 7705 091 1479 2515 3551 4699 7690.4 2000 3257 4507 5747 8425 096 2222 3145 4267 5390 8440.6 2512 3918 5246 6498 9306 096 2889 3784 4942 6213 93210.8 2726 4357 5691 7059 10099 094 3327 4453 5429 6656 101061.0 3895 5555 7262 8654 10657 097 4166 5131 6097 8263 10738

Such an approach raises the following question: Whendoes the choice of the internal service level matter? Toaddress this issue, we evaluated the total safety-stockcost for the two-component instance in Example 1with different values of internal service levels, %i,i = 1�2. In particular, we varied %i from zero to 1steps of 0.1. For each of the 10 × 10 combinations ofinternal service levels, we computed the correspond-ing base-stock levels and the total safety-stock cost;this safety-stock cost varied from 115 to 150. We showthe safety-stock cost versus internal service levels inTable 5, where costs are reported as a percentageabove the minimum possible. From this table, we seethat, within a wide range of internal service-level val-ues, the safety-stock costs were within a few percentof the minimum possible. For example, if we fixed %1to be 0.5, and varied %2 from 0.2 to 0.7, the total safety-stock cost never increased by more than 5% above115. However, when %2 = 0�9 (closer to the customerservice level � = 0�99), the total safety-stock cost was135 (or 17% higher than 115). Using Table 5, we seethat the choice of internal service level does matter,particularly if the chosen value is very large or verysmall. However, there is a (%1, %2) region of internalservice levels in which the total safety stock is insen-sitive to the choice of (%1, %2). This zone of insensitiv-ity typically includes the case of %1 = %2 used by thedecomposition heuristic. Note that a similar behav-ior of inventory cost versus internal service level hasbeen observed for a serial system in Bollapragadaet al. (2004), for different values of problem param-eters such as �. In our experience, when there is sup-ply uncertainty the impact of problem parameters onthe choice of the optimal internal service levels iscomplicated by the complementary roles played bycomponent stocks and end-product stocks in achiev-ing customer service. (See Bollapragada et al. 2004 fora computational study and Shang and Song 2003 foran analysis using approximation.)

4.1.2. Decomposition vs. Practice. We also testedthe performance of the heuristic against practicesused at the environment that motivated this research.As discussed in §1, managers currently set base-stocklevels using node-specific parameters without tak-ing into consideration the interrelationship between

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Table 5 Safety-Stock Cost for Different Internal Service Levels as a Percentage Above the Minimum Value of 115

1

2 00 01 02 03 04 05 06 07 08 09

0.0 76 92 105 116 129 143 159 179 204 2470.1 2 4 2 4 3 4 46 59 73 89 107 122 1640.2 03 2 5 14 2 4 3 4 48 65 85 104 1420.3 3 0 06 18 18 17 2 7 44 64 89 1190.4 55 2 3 00 11 2 3 13 2 4 44 68 1090.5 77 44 3 0 09 06 19 14 2 4 49 870.6 108 70 59 41 3 0 01 14 2 1 2 8 710.7 153 106 85 73 56 44 2 7 10 24 470.8 191 166 139 115 100 84 71 45 53 440.9 299 246 220 197 191 172 147 121 104 104

Note. Column minimums are highlighted in bold font; cases with safety-stock cost suboptimality of 5% or lower are italicized.

stocks of different items. We quantify the potentialbenefits of the decomposition for the following com-mon practice: The manager determines componentstock levels by setting the internal service level equalto the external target customer service level of theend product. (Axsäter 2003 observes similar prac-tices.) For the 300 instances discussed above, wecomputed the base-stock levels corresponding to thispractice, and compared the resulting safety-stockcost to the safety-stock cost from the decomposition(see Table 6).From Table 6, we see that the reduction in safety-

stock cost ranges from 2% to 13%, on average. Theaverage cost reduction is higher when there are two orten components. Furthermore, higher cost reductionswere observed when the heuristic internal servicelevel deviated significantly from the external ser-vice level. For instance, consider setting the inter-nal service level equal to the external service levelof 99%, as per the practice, for the instance inExample 1. This yields base-stock levels of �s1� s2� s0�=�39�4�52�9�33�8� with a safety-stock cost of 166.6. Forthis example, the base-stock levels and safety-stockcost from the decomposition are (8.6, 18.5, 49.8)and 114.9, respectively, which corresponds to a costreduction of 31%. In this case, the heuristic solutioncarries higher end-product stock than the practice-based solution, with correspondingly lower compo-nent stocks. In our experience, the practice of settinginternal service levels equal to or greater than the cus-tomer service level makes more sense when no end-product inventory is carried. When it is cost effectiveto carry a high level of end-product inventory, wecan expect corresponding low component inventories

Table 6 Decomposition Heuristic Compared to Practice

Components (#) 2 4 6 8 10

Avg. cost reduction (%) 13.25 202 293 353 674Std. dev. cost reduction (%) 11.55 294 322 474 384Max. cost reduction (%) 31.04 1759 1515 2723 2421

with internal service level smaller than the customerservice level.It has been observed in the literature that the

internal service level of components will be likelylower than the external servicel level; see, for exam-ple, Graves (1996), Shang and Song (2003). In suchcases, if the external service level is used as theinternal service level to determine the base-stock lev-els, the resulting inventories at the component levelwill be higher than optimal. From Proposition 9, si

decreases with ci for i = 1�2. Therefore, a higher ci

implies a lower internal service level for i = 1�2. Sim-ilarly, a lower c0 implies a higher s0, which in turnimplies lower base-stock levels at the component level(Proposition 3). Therefore, a lower c0 also implieslower internal service levels. Hence, we expect that,as c0/�c1 + c2� becomes smaller, the internal servicelevel becomes smaller. Furthermore, when c0/�c1+ c2�is smaller, the common practice of using the exter-nal service level as the internal service level todetermine component base-stock levels would holdhigher than optimal inventories for items (compo-nents) that are relatively more expensive. Therefore,one would expect that the decomposition approach,which chooses an appropriate internal service level,will yield a larger improvement when c0/�c1 + c2�is smaller. Our computational study supported thisintuition. For instance, consider the two-componentinstances we tested. Of the 60 such instances, half hada c0/�c1+c2� cost ratio of 1.0, and the remainder had acost ratio of 9.0. Among these instances, the decompo-sition approach consistently improved the safety stockby at least 18.5% in the 30 instances with smaller costratio, for an average improvement of 24.4%. By con-trast, the improvement for the 30 instances with largercost ratio averaged only 2.1%.Note that, in practice, managers who set inter-

nal service levels equal to the external service levelmay not adjust the end-product base-stock level bytaking its relationship with component stock intoconsideration. If so, the potential cost reduction maybe higher than that illustrated above.

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4.2. Justification of the Objective FunctionWe now present results comparing the effects of opti-mizing inventory investment with optimizing inven-tory holding cost. Let hi be the unit holding cost perperiod for item i. Assuming that inventory costs areincurred after demand materializes, the problem witha holding-cost objective can be formulated as

minN−1∑i=0

hiE�I i�·��+ s.t. P�I 0�·�≥ ��≥ ��

Observe that this objective is increasing in �s0� � � � �sN−1�, hence it is quasiconvex. Thus, our convex pro-gramming optimization and heuristic decompositionapproaches still apply.For the inventory holding cost objective, we found

the corresponding optimal base-stock levels through agradient search with the solution from the base-stocklevel objective (§3.3) as the starting point. The stepsize was initially set equal to 0.1 and reduced by aconstant multiplier (0.75) every time the cost functiondid not decrease for a fixed number of iterations (weused 15 iterations) or the service level oscillated backand forth four times. We terminated the search whenthe step size was less than 0.001.We solved all 300 problem instances from §4.1 with

the inventory holding-cost objective. For 258 out of300 instances, the resulting base-stock levels did notincur lower inventory holding costs than the base-stock levels from the inventory investment objective.Among the 42 instances in which improved base-stock levels were found, the maximum reduction ininventory holding cost was 0.39%. Consequently, weconclude that replacing inventory holding costs withinventory investment as the objective function doesnot change the optimal base-stock levels significantly,in most cases.

4.3. Effects of Supply UncertaintyIn this section, we identify when improvement in sup-ply performance will yield a significant reduction inthe safety-stock cost. We measure supply performanceusing the component’s supply lead time and supplyservice level, /i = Prob�� ≤ �i�. We set /1 = /2 andvaried /i from 0.75 to 0.95 in steps of 0.1. (Resultsfor /1 �= /2 are presented at the end of §4.3.) We usedli = 1�2�3 with l2 ≥ l1. We used Gamma demand (�)with a mean of 10. The coefficient of variation ofdemand (cv) was either 0.2, 0.6, or 1.0. The customerservice level varied from � = 0�75 to 0.95 in steps of0.05. The end-product unit cost (c0) was chosen suchthat c0/�c1+ c2� varied from 1.5 to 3.0 in steps of 0.5.This full-factorial experimental design yields a total of3× 6× 3× 5× 4= 1�080 instances, which were solvedto optimality.Our results are summarized in Tables 7–12. Each of

these tables presents the optimal average safety-stock

Table 7 Impact of Demand cv on Safety-Stock Costs for Differ-ent Supply Service Level

� cv = 02 cv = 06 cv = 10

0.75 509 1605 2947100% 100% 100%

0.85 460 1395 2536902% 869% 86%

0.95 428 1281 2327841% 801% 789%

costs for values of two chosen parameters, where thesafety stock is averaged over instances with differentvalues of the other parameters. For example, Table 7shows average safety-stock costs for different valuesof demand cv and supply service level /; the upperleftmost number 50.9 in Table 7 corresponds to thesafety-stock costs for cv = 0�2 and / = 0�75, averagedover instances with different values of lead times, cus-tomer service levels, and unit costs. In Tables 7–12, wepresent safety-stock costs as an absolute value and asa percentage relative to a base case. We do so because,in our experience, some firms choose supply improve-ments based on the absolute reduction in safety-stockcosts, while other firms prefer to use the percentageimprovement criterion, instead.Impact of Demand cv. Tables 7 and 8 illustrate the

impact of demand cv on the safety-stock costs fordifferent levels of supply performance. As expected,the total safety-stock costs increase with demand cvand decrease when the supply performance improves.Over all the tested instances, the absolute decreasein costs as / or l1 improves has a higher magnitudewhen demand cv is higher. This is because, at higherdemand cv, a much larger safety stock becomesnecessary to compensate for poor supply perfor-mance. Interestingly, we note from Table 8 that therelative percentage benefit of reducing lead timeis nearly the same for different cv. In addition tothe above impact of demand cv, we see that thedecrease in safety-stock costs exhibits diminishingreturns as the supply service level / improves. Bycomparison, the decrease in safety-stock costs whensupply lead time l1 improves seems more stable. Forinstance, at cv = 0�6, the safety-stock costs decrease by

Table 8 Impact of Demand cv on Safety-Stock Costs for Differ-ent Supply Lead Time

l1 cv = 02 cv = 06 cv = 10

3 716 2206 4020100% 100% 100%

2 526 1617 2943735% 733% 732%

1 342 1043 1904649% 645% 647%

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Table 9 Impact of � on Safety-Stock Costs for Different SupplyService Levels

� �= 075 �= 085 �= 095

0.75 1121 1590 2466100% 100% 100%

0.85 944 1381 2165843% 868% 878%

0.95 843 1262 2031752% 794% 844%

�160�5− 139�5�= 21 when / improves from 75% to85% and by only 11.4 when / goes from 85% to 95%.The corresponding decrease in safety-stock costs atcv = 0�6 when lead time l1 improves from 3 to 2 andfrom 2 to 1 is 58.9 and 57.4, respectively.Impact of Target Service Level, �. Tables 9 and 10

demonstrate the impact of target service, �, onthe safety-stock costs when supply performance isimproved, for �= 0�75, 0.85, or 0.95. We note that, onaverage, the absolute decrease in costs when supplyperformance improves has higher magnitude when� is higher. However, the relative percentage benefitof improving / is slightly higher when � is lower,because the safety-stock cost used in the denominatorto calculate the relative benefit is smaller when � islower.Impact of Cost of End Product. From our computa-

tional experiments with different values of c0/�c1+c2�,we noted that the absolute benefit of improving sup-ply performance is higher when the end-product costis higher. This is because high end-product costs shiftinventories from end-product stocks to componentstocks. When component stocks are larger, improvingsupply performance yields greater benefits in termsof absolute stock reduction. We also noted that therelative percentage benefit is higher when the end-product cost (and therefore the total safety-stock cost)is lower.Impact of Number of Components. We generated 27

instances corresponding to the three values for eachof demand cv, �, and /, specified earlier. Lead timesfor all items were the same and either 1, 2, or 3. Thenumber of components N was either 2, 4, or 6, withci = li for i = 1� � � � �N and c0 = 2×∑N

i=1 ci. This yields

a total of 243 instances for which we computed the

Table 10 Impact of � on Safety-Stock Costs for Different SupplyLead Times

l1 �= 075 �= 085 �= 095

3 1500 2183 3413100% 100% 100%

2 1099 1597 2506733% 732% 734%

1 706 1030 1632471% 472% 478%

Table 11 Impact of Number of Components and Supply ServiceLevel on Safety-Stock Costs

Number of components

� 2 4 6

0.75 1642 3602 5479100% 100% 100%

0.85 1401 3025 4563855% 840% 833%

0.95 1269 2657 3960773% 738% 723%

optimal base-stock levels. Tables 11 and 12 show thereduction in safety-stock costs with improvements insupply performance for different number of compo-nents. We note that the absolute magnitude and therelative percentage benefit of improving supply per-formance is higher when the number of componentsis higher. Note that this benefit requires improvementof all component suppliers, which may be harder toimplement as the number of components increases.Miscellaneous Observations. The relative percentage

benefit of improving supply service level (/) varieswith demand cv, target service level �, and numberof components; however, the corresponding benefitof lead time improvement is more or less constantacross different values of each of these parameters.(For instance, the percentages in the three columnsof lead-time improvement in Table 8 are nearly thesame, but this is not the case in the corresponding /improvement in Table 7.)The benefit of improving the supply service level

/ exhibits diminishing returns, that is, the benefit islower at higher values of the service level. However,the corresponding benefit of reducing lead times doesnot always exhibit similar diminishing returns (seeTable 10 or 12).Different Combinations of Supply Performance Levels.

We now consider a two-component system in whichthe supply service levels and lead times of the twocomponents may be different. This allows us to esti-mate the benefits of improving only one supplier. Thequestions we explore here include: Which supplier

Table 12 Impact of Number of Components and Supply LeadTime on Safety-Stock Costs

Number of components

l1 2 4 6

3 2250 4821 7253100% 100% 100%

2 1411 3046 4595627% 632% 634%

1 650 1418 2155289% 294% 297%

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Bollapragada et al.: Managing Inventory and Supply Performance in Assembly Systems1740 Management Science 50(12), pp. 1729–1743, © 2004 INFORMS

should we improve? What should we improve—supply service level or lead time? How much shouldwe improve? To address these questions we con-ducted another full-factorial experiment. Componentlead time l1 and component lead time l2 each take onvalues 1, 2, or 3, and end-product lead time is l0 = 2.Unit costs values were fixed at c1 = 1, c2 = 2, andc0 = 6 for all instances. We chose three values eachfor demand cv, �, /1, and /2. This yielded a totalof 36 = 729 test instances. We also conducted experi-ments with other parameter values and obtained sim-ilar results. We make the following observations fromour experimental results:1. On average, reducing the lead time of the more

expensive component yielded higher benefit thanreducing the lead time for the less expensive com-ponent. Hence, if z�l1� l2� denotes the average safety-stock costs with lead times (l1� l2), then z�3�3� >z�1�3� > z�2�2� > z�3�1�. So, starting with (l1� l2� =�3�3�, it was better to improve Component 2’slead time by two units than to either improveComponent 1’s lead time by two units or improveboth components’ lead times by one unit. This obser-vation was not true for all test instances: It appearsto be valid for instances with lower values of sup-ply service level. It was not true for high / values—for example, when supply service level was 95% andsymmetric. So one may conclude that high service lev-els partially compensate for the cost differentials topush the lead times to be symmetric; alternately, thethreshold cost differentials needed to keep the solu-tion asymmetric increase with service levels.2. On average, the benefit of improving supply ser-

vice level / was higher when lead times were lower.The benefit of lead time reduction was higher whenthe corresponding / values were higher. For any fixed/, the benefit of decreasing lead time l2 (or alter-nately l1) was higher when l1 (correspondingly l2) waslower. Illustrative safety-stock costs supporting someof these observations are presented in Table 13.3. Based on a least-squares regression of costs

against the main parameters (�, cv, /1, /2, l1, l2) wenoted that � and demand cv have the largest positiveeffect on costs. The supply service level for the moreexpensive component, /2, had the strongest negativeeffect on cost; the impact of /1 was also significant

Table 13 Safety-Stock Costs for Different Supply Service Levels andLead Times

Lead time, l1 = l2Supply service level��1 = �2� 1 2 3

75% 1183 1253 132085% 994 1086 115795% 893 1000 1083

Note. Symmetric case with �1 = �2 and l1 = l2.

but about 25% smaller in magnitude than the impactof /2.The above observations provide managerial guide-

lines for improving supply performance. In additionto these observations we note the following: (i) Solv-ing the model for different parameter settings seemsto be the most accurate way of assessing the poten-tial benefits of any supply performance improvement.This is because of the complex nonlinear relation-ship between optimal stock levels of the differentitems and the customer-service-level constraint. (ii) Inpractice, the benefits of supply improvements mustbe matched against the costs of achieving theseimprovements.

5. Multi-Echelon Pure AssemblySystems

In this section, we demonstrate how a multi-echelonassembly system can be converted into an equivalenttwo-echelon assembly system. We say two systemsare equivalent when they incur the same inventoryinvestment and provide the same service level to thecustomers. For simplicity, we first illustrate how asimple three-echelon system can be converted into atwo-stage assembly system. We then show how thisconversion procedure may be applied recursively toany multi-echelon pure assembly system to yield anequivalent two-echelon assembly system.Consider the two pure assembly systems, System �

and System�, in Figure 2. System � is a three-echelon“Y”-shaped assembly system with four nodes. Node0 is the end product, and Node 1 the subassemblywhich is made by combining Components 2 and 3,each of which has an unreliable supplier. System � isa two-echelon assembly system with four nodes; 1, 2,and 3 are components, and 0 denotes the end prod-uct. The suppliers of Nodes 2 and 3 are the same as inSystem �; Node 1 has a 100% reliable supplier. Let Ci,Si, and Li denote, respectively, the unit cost, the base-stock level, and the lead time of node i in System �.We assume C1 > C2 + C3. Lemmas 11 and 12, statedbelow, demonstrate that it is possible to select base-stock levels and cost parameters in such a way thatboth Systems � and � provide the same service levelunder the same inventory investment.

Figure 2 Assembly Systems � and �

001

System B

1

3

2

3

2

System A

„

„

„

„ „

„

„

„

„

„

„

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Bollapragada et al.: Managing Inventory and Supply Performance in Assembly SystemsManagement Science 50(12), pp. 1729–1743, © 2004 INFORMS 1741

Lemma 11. When s0 = S0, s1 = S1, s2 = S2 + S1, ands3 = S3+S1 and l0 = L0, l1 = L1, l2 = L1+L2+1, and l3 =L1+L3+1 , the steady-state shortfall of the end product inSystems � and � are equal in distribution, if both systemsface the same demand. Therefore, both systems provide thesame customer service level.

Lemma 12. Under the conditions in Lemma 11, Sys-tems � and � incur the same inventory investment if c0 =C0, c1 =C1−C2−C3, c2 =C2, and c3 =C3.

Now consider an N node multi-echelon assemblysystem with supply uncertainty. By induction downthe branches of the multi-echelon assembly network,it is possible to prove that the following procedureconverts the multi-echelon system into an equivalenttwo-echelon assembly system to which our decompo-sition approach may be applied. By solving the corre-sponding N node, two-echelon system, we can obtainbase-stock levels for the multi-echelon system.

Multi-Echelon to Two-Echelon Conversion(METEC) AlgorithmStep 1. Create a two-echelon assembly system with

Node 0 denoting the end product and the remainingnodes corresponding to N − 1 components. Let ci, si,and li be the per unit cost, base-stock level, and (nom-inal) lead time of node i.Step 2. Set

li = Li+ ∑j∈� �i� and j �=0

�Lj +1�� si=Si+ ∑j∈� �i� and j �=0

Sj�

andci=Ci− ∑

j∈��i�

Cj� for all i�

Step 3. For each node i ∈ �1� � � � �N − 1�, if ��i� isan empty set, the supplier of node i is the same asin the original system. Otherwise, let node i have a100% reliable supplier in the new system.By Step 2, lead times in the two-echelon system are

equal to the total3 lead time from node i until theend-product echelon in the original system. Further-more, the equivalent base-stock level of node i in thetwo-echelon system is just its echelon base-stock level(which is the sum of base-stock levels of all nodesalong the unique path from the i to the end-product)in the original system. Finally, for any subassembly inthe original system, its equivalent unit cost is just theincremental cost of the subassembly above the cost ofthe immediate components that are used to assemblethis item. The following result follows by a recursiveapplication of Lemmas 11 and 12.

Proposition 13. The two-echelon assembly systemgenerated by the METEC algorithm is equivalent to the

3 A unit lead time is added to certain nodes because in our modelthe replenishment decisions are made after demand materializes.

original multi-echelon pure assembly system in terms ofinventory investment and customer service.

Note that when the supply of components is per-fect, Rosling (1989) shows that an assembly systemmay be converted into an equivalent serial system.Based on the conversion, he establishes the opti-mality of base-stock policies for assembly systems.When component supply is not perfect, it is not obvi-ous whether such a conversion exists. Under theassumption that a base-stock policy is followed,Proposition 13 shows that a multi-echelon assemblysystem with demand and supply uncertainty is equiv-alent to a two-echelon assembly system for whichour results apply. In particular, this conversion alsoapplies to the special case when the component sup-ply is deterministically capacitated. Rosling convertedan assembly system under a holding and shortagecost objective to a thinner (serial) system, but we con-verted an assembly system under an inventory invest-ment objective into a fatter (but shorter) system. In thenext section we conclude with a summary and someinteresting problem variants.

6. SummaryThis work presents a first attempt to find base-stock inventory levels under both supply anddemand uncertainty in a multi-echelon system witha customer-service-level constraint. Under generalassumptions on the demand distribution, we showedthat inventory investment can be minimized usingconvex programming, and we characterized theconvex-decreasing relationship between the base-stock levels of components and end products. To solveindustrial-sized assembly problems, we proposed adecomposition approach based on an internal servicelevel. Through this approach, the base-stock levelsfor all components can be determined independently,which permits distributed computing of optimal stocklevels. In our computational study, the cost of theheuristic solution from the decomposition was within0.66% of the optimal, on average. The internal servicelevel provides a central planner with a simple toolfor coordinating the replenishment ordering activitiesat different stages in a supply chain. Compared withcurrent practice, it has the potential to reduce thesafety-stock cost by as much as 30%.Through computational experiments with two-

echelon systems, we identified when improvementin supply performance would yield a significantreduction in total inventory investment. In particular,we noted that that the cost reduction from improv-ing supply performance was higher when demandvariability was high, the number of components washigh, the target service level was high, or the endproduct was more expensive relative to components.

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Bollapragada et al.: Managing Inventory and Supply Performance in Assembly Systems1742 Management Science 50(12), pp. 1729–1743, © 2004 INFORMS

The behavior of absolute cost reduction is notalways the same as relative percentage cost reduc-tion. For instance, the absolute decrease in costs fromimproving supply performance has higher magni-tude when target customer service is higher, on aver-age; however, the relative percentage benefit is higherwhen target customer service is lower. On average,(i) reducing the lead time of the more expensive com-ponent yielded higher benefit than reducing the leadtime for the less expensive component, and (ii) thebenefit of improving one of the supply parameters(service level or lead time) was higher when the valueof the other parameter was already more favorable(lower lead time or higher service level, respectively).We showed that our analysis applied to a multi-echelon pure assembly system by converting it to anequivalent two-echelon system.Future work could extend the decomposition ap-

proach to multiproduct assembly systems with com-ponent commonality and the case when demand isnonstationary. Our results will apply to any environ-ment in which the following conditions hold: R1: Thefeasible region is convex, and R2: The objective func-tion is quasi-convex in the base-stock levels. In theappendix, we show that R1 and R2 hold true, when(i) a Type-2 fill rate service measure is used, (ii) whenassembly is capacitated and there are maximum orderquantity limits for components, and (iii) when supplyuncertainty is modeled using stationary random leadtimes instead of random supply capacity.An online supplement to this paper is available at

http://mansci.pubs.informs.org/ecompanion.html.

AcknowledgmentsThe authors thank Professor Wallace J. Hopp, the associateeditor, and two anonymous referees whose comments sub-stantially improved the contents and presentation of thispaper.

Appendix

A.1. Type-2 Service Level RequirementThe Type-2 service level (or fill rate) is the long-run averageproportion of demands met from on-hand inventory. In thissection, we show that the feasible region for the problem isstill convex (R1 holds), so our analysis may be extended tothe fill rate case.For the two-echelon assembly system discussed in §3.2,

the fill rate is

1−E[�max�0�Y 1+ � − s1�Y 2+ � − s2�− s0�+

]/E����

where E��maxi�0�Y i +�− si�− s0�+� is the expected backlogin each period. Let / denote the required Type-2 servicelevel. We can formulate the problem as,

min2∑

i=0cisi s.t. E

[(max

i�0�Y i+�−si�−s0

)+]≤ �1−/�E����

Because maxi�0�Y i +�− si� is a convex function, the left-hand side of the constraint is convex in �s0� s1� s2�. Thus, the

feasible region for the above problem is a closed convex set.Note that the logarithmic-concavity assumption on demandis no longer required for this result to hold true. Due to thisconvexity, the Type-2 version of Proposition 3 (decreasingconvex relation between base-stock levels) is valid. Further-more, similar to Proposition 10 the objective function U�%�is unimodal in the Type-1 internal service level. Thus, ourdecomposition approach using an internal service level canbe applied.

A.2. Capacitated ModelConsider the case when there are known assembly capac-ities for the end product and maximum ordering quantityrestrictions on the components. Let Mi denote the max-imum purchase or assembly quantity at node i in eachperiod, for i = 0�1� � � � �N . We assume that a modified base-stock policy (Glasserman and Tayur 1994) is followed by allitems in the system. For the two-echelon assembly system,using an analysis similar to the one in §3, we can show thatthe shortfall of the end-product evolves as

Y 0n+1 = Y 0

n + �n −mini

{M0�Y 0

n + �n� si +Y 0n −Y i

n

}

= maxi

{Y 0

n + �n −M0�0�Y in + �n − si

}�

In steady state, the end-product shortfall satisfies Y 0 d=maxi�Y

0+�−M0�0�Y i+�−si� and the component shortfallY i d=max�0�Y i + � −min��i�Mi��. Consequently, the feasi-ble region for this problem is convex and a decompositionapproach is valid.

A.3. Random Lead TimesIn this section, we draw a connection between random lead-time models and our random capacity model, operatingunder a base-stock policy. Let Ri

n denote the quantity deliv-ered by the supplier of item i in period n for the randomlead-time environment. For instance, in Kaplan (1970) alloutstanding orders placed at or before a randomly gener-ated time period are delivered, which yields Ri

n. Note thatour definition of Ri

n does not preclude partial deliveries.With this notation, the shortfall evolves as follows:

Y in+1 =max�0�Y i

n + �n −Rin�� (A1)

When the lead-time distribution is stationary, the steady-state shortfall Y i = max�0�Y i + � − Ri�. For this case,Robinson et al. (2001) present a method to computemoments of the steady-state shortfall. They use the first twomoments of shortfall to compute base-stock levels. By (2)and (A1), the inventory dynamics in a random lead-timemodel behave as in our random capacity model with sup-ply uncertainty �i

n replaced with Rin. Because our analysis is

based on shortfall recursions, our results may be extendedto the random lead-time case.

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