Select Papers on Supply Chain Management

Select Paperson SupplyChainManagement

Institute for Operations Research and the Management Sciences

INFORMS Select Papers on Supply Chain Management

Table of Contents

Article 1:Extended Enterprise Supply-Chain Management at IBM Personal Systems Group andOther DivisionsInterfaces, Vol. 30, No. 1, January-February 2000 (pp. 7-25).

Article 2:Xilinx Improves Its Semiconductor Supply Chain Using Product and ProcessPostponementInterfaces, Vol. 30, No. 4, July-August 2000 (pp. 65-80).

Article 3:Stock Positioning and Performance Estimation in Serial Production-TransportationSystemsManufacturing & Service Operations Management, Vol. 1, No. 1, 1999 (pp. 77-88).

Article 4:Quantity Flexibility Contracts and Supply Chain PerformanceManufacturing & Service Operations Management, Vol.1, No. 2, 1999 (pp. 89-111).

Article 5:Optimizing Strategic Safety Stock Placement in Supply ChainsManufacturing & Service Operations Management, Vol. 2, No. 1, Winter 2000 (pp. 68-83).

Article 6:A Dynamic Model for Requirements Planning with Application to Supply ChainOptimizationOperations Research, Vol. 46, Supp. No.3, May-June 1998 (pp. S35-S49).

Article 7:Development of a Rapid-Response Supply Chain at CaterpillarOperations Research, Vol. 48, No. 2, March-April 2000 (pp. 189-204).

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Copyright � 2000 INFORMS0092-2102/00/3001/0007/$05.001526–551X electronic ISSN

INVENTORY/PRODUCTION—APPLICATIONSINDUSTRIES—COMPUTERSMANUFACTURING—SUPPLY CHAIN MANAGEMENT

This paper was refereed.

INTERFACES 30: 1 January–February 2000 (pp. 7–25)

Extended-Enterprise Supply-ChainManagement at IBM Personal Systems Groupand Other DivisionsGrace Lin, Markus EttlSteve Buckley, Sugato Bagchi

IBM T. J. Watson Research CenterYorktown Heights, New York 10598

David D. Yao Columbia UniversityNew York, New York 10027

Bret L. Naccarato IBM Printing Systems CompanyEndicott, New York 13760

Rob Allan, Kerry KimLisa Koenig

IBM Personal Systems GroupResearch Triangle Park, North Carolina 27709

In 1994, IBM began to reengineer its global supply chain. Itwanted to achieve quick responsiveness to customers withminimal inventory. To support this effort, we developed anextended-enterprise supply-chain analysis tool, the Asset Man-agement Tool (AMT). AMT integrates graphical process mod-eling, analytical performance optimization, simulation,activity-based costing, and enterprise database connectivityinto a system that allows quantitative analysis of extendedsupply chains. IBM has used AMT to study such issues as in-ventory budgets, turnover objectives, customer-service targets,and new-product introductions. We have implemented it at anumber of IBM business units and their channel partners.AMT benefits include over $750 million in material costs andprice-protection expenses saved in 1998.

As the world’s largest company pro-viding computer hardware, soft-

ware, and services, IBM makes a widevariety of products, including semicon-ductors, processors, hard disks, personalcomputers, printers, workstations, andmainframes. Its manufacturing sites are

linked with tens of thousands of suppliersand distribution channels all over theworld. A single product line may involvethousands of part numbers with multilevelbills of materials, highly varied lead timesand costs, and dozens to hundreds ofmanufacturing and distribution sites

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linked by different transportation modes.Facing the challenges of increasing compe-tition, rapid technology advance, and con-tinued price deflation, the companylaunched an internal reengineering effortin 1993 to streamline business processes inorder to improve the flow of material andinformation. The reengineering effort fo-cused on improving customer satisfactionand market competitiveness by increasingthe speed, reliability, and efficiency withwhich IBM delivers products to themarketplace.

In 1994, IBM launched an asset-management reengineering initiative aspart of the overall reengineering effort.The objectives were to define the supply-chain structure, to set strategic inventoryand customer-service targets, to optimizeinventory allocation and placement, and toreduce inventory while meeting customer-service targets across the enterprise. Thecompany formed a cross-functional teamwith representatives from manufacturing,research, finance, marketing, services, andtechnology. The team identified five areasthat needed modeling support for decisionmaking: (1) design of methods for reduc-ing inventory within each business unit;(2) development of alternatives for achiev-ing inventory objectives for senior-management consideration; (3) develop-ment and implementation of a consistentprocess for managing inventory andcustomer-service targets, including tooldeployment, within each business unit; (4)complete evaluation of such assets as ser-vice parts, production materials, and fin-ished goods in the global supply network;and (5) evaluation of cross-brand productand unit synergy to improve the manage-

ment of inventory and risk.We developed the Asset Management

Tool (AMT), a strategic decision-supporttool, specifically to address these issues.The integration of AMT with the otherasset-management reengineering initia-tives has resulted in the successful imple-mentation of extended-enterprise supply-chain management within IBM.The Asset Management Tool

An extended-enterprise supply chain isa network of interconnected facilitiesthrough which an enterprise procures,produces, distributes, and delivers prod-ucts and services to its customers. As pro-curement, distribution, and sales have be-come increasingly global, the supply

A company with an extendedsupply chain performs wellonly when it collaborates withits suppliers and resellers.

chains of large companies have becomedeeply intertwined and interdependent.Today’s extended-enterprise supply chainsare in fact networks of many supplychains representing the interests of manycompanies, from supplier’s suppliers tocustomer’s customers. Because of this in-terdependency, a company with an ex-tended supply chain performs well onlywhen it collaborates and cooperates ac-tively with its suppliers and resellers.

In high-technology industries, manage-ment of the extended-enterprise supplychain becomes very important. At its best,it keeps operating costs low and profitshigh. But a poorly managed supply chaincan reverse that relationship, eroding prof-its, compromising innovation, and ham-

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January–February 2000 9

pering business growth. Early in our ef-forts, we realized that there were twofundamental keys to overhauling IBM’ssupply chain. First, we had to reduce andmanage uncertainty to promote more ac-curate forecasts. Second, we had to im-prove supply-chain flexibility to facilitatequick adaptation to changes in the market-place. From the outset, we focused on theintrinsic interdependency of an extended-enterprise supply chain. We knew our sys-tem would perform as desired only if it re-flected the policies and processes used byour suppliers and channels, integratingtheir value chains with our own. This per-spective helped to shape our vision: an in-tegrated modeling and analysis tool forextended-enterprise supply chains. Itwould be a tool with new methodologiesto handle the uncertainties inherent in de-mand, lead time, supplier reliability, andother factors. It would be scalable, so thatit could handle the vast amounts of datadescribing product structure, supply-chainprocesses, and component stock informa-tion that typify the industry. Finally, thenew tool would be equally effective atmodeling basic types of supply-chain poli-cies and their interactions, because differ-ent companies may use different policies.

We designed AMT to address all ofthese issues. It is a modeling and analysissystem for strategic and tactical supply-chain planning that emerged from variousearlier internal IBM reengineering studies[Bagchi et al. 1998; Buckley 1996; Buckleyand Smith 1997; Feigin et al. 1996]. It sup-ports advanced modeling, simulation, andoptimization capabilities for quantitativeanalysis of multiechelon inventory sys-tems, along with such features as enter-

prise database connectivity and internet-based communication. AMT is built on sixfunctional modules: a data-modeling mod-ule, a graphical user interface, an experi-ment manager, an optimization engine, asimulation engine, and a report generator.

The data-modeling module provides arelational data interface, including productstructures, lead times, costs, demand fore-cast and the associated variability informa-tion. It has built-in explosion of bills ofmaterials and data-reduction capabilities,and automatic checks for data integrity. Itprovides access to IBM’s global and localoperational databases through databridges.

The graphical user interface (GUI) com-bines supply-chain modeling with dialog-based entry of supply-chain data. It allowsusers to build supply networks by drag-ging and dropping model components,such as manufacturing nodes, distributioncenters, and transportation nodes, onto thework space.

The experiment manager facilitates theorganization and management of data setsassociated with supply-chain experiments.It allows users to view and interactivelymodify parameters and policies. In addi-tion, it provides automated access to out-put data generated during experimentsand supports a variety of file-managementoperations.

The optimization engine performsAMT’s main function, quantifying thetrade-off between customer-service targetsand the inventory in the supply network.This module can be accessed from the GUIpull-down menu or called by the simula-tion engine.

The simulation engine simulates the

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performance of the supply chain undervarious parameters, policies, and networkconfigurations, including the number andlocation of suppliers, manufacturers, anddistribution centers; inventory and manu-facturing policies, such as base-stock con-trol, days of supply, build-to-stock, build-to-order, and continuous or periodicreplenishment policies. The simulation en-gine contains an animation module thathelps users to visualize the operation ofthe supply chain or vary parameters andpolicies while monitoring the simulationoutput reports.

The report generator offers a compre-hensive view of the performance of thesupply chain under study, including aver-age cycle times, customer-service levels,shipments, fill rates, and inventory. It alsogenerates financial results, including reve-nues, inventory capital, raw-material costs,transportation costs, and activity-basedcosts, such as material handling andmanufacturing.The Optimization Engine

The central function of the optimizationengine is to analyze the trade-off betweencustomer-service and inventory invest-ment in an extended-enterprise supplychain. The objective is to determine thesafety stock for each product at each loca-tion in the supply chain to minimize theinvestment in total inventory. We view thesupply chain as a multiechelon network inwhich we model each stocking location asa queuing system. In addition to the usualqueuing modeling, we incorporated intothe model an inventory-control policy: thebase-stock control, with the base-stocklevels being decision variables. To numeri-cally evaluate such a network, we devel-

oped an approach based on decomposi-tion. The key idea is to analyze eachstocking location in the network individu-ally and to capture the interactions amongdifferent stocking locations through theirso-called actual lead times.

We modeled each stocking location as aqueue with batch Poisson arrivals and infi-nite servers with service times followinggeneral distributions, denoted as MX/G/�

in queueing notation. To do this, we hadto specify the arrival and the service pro-cesses. We obtained the arrival process ateach location by applying the standardMRP demand explosion technique to theproduction structure. The batch Poisson

AMT embodies a creativecoupling of optimization,performance evaluation, andsimulation.

arrival process has three main parameters:the arrival rate, and the mean and the var-iance of the batch size. It thus accommo-dates many forms of demand data; for in-stance, demand in a certain period can becharacterized by its minimum, maximum,and most likely value. The service time isthe actual lead time at each stocking loca-tion. The actual lead time at a stocking lo-cation can be derived from its nominallead time (for example, the manufacturingor transportation time) along with the fillrate of its suppliers. In particular, when asupplier has a stock-out, we have to addthe resulting delay to the actual lead time.This delay is the time the supplier takes toproduce the next unit to supply the order.In our model, we derive the additional de-lay from Markov-chain analysis.

IBM


With the arrival and service processes inplace, we can analyze the queue and de-rive performance measures, such as inven-tory, back-orders, fill rates, and customer-service levels. The key quantity in theanalysis of a stocking location, i, is thenumber of jobs in the MX/G/� queue, de-noted Ni, which can be derived from stan-dard queueing results [Liu, Kashyap, andTempleton 1990]. To alleviate the compu-tational burden in large-scale applications,we approximated Ni by a normal distribu-tion. This way, we need to derive only themean and the variance of Ni, both ofwhich depend on the actual lead time,which is the service time in the queuingmodel. Figure 1 shows a snapshot of thedynamics at a stocking location.

The objective of the optimization modelis to minimize the total expected inventorycapital in the supply network. This total isa summation over all stocking locations,each of which carries two types of inven-

tory: finished goods (on-hand) inventory,and work-in-process (on-order) inventory.The constraints of the optimization modelare the required customer-service targets.They are represented as the probability,say 95 or 99 percent, that customer ordersare filled by a given due date. Our formu-lation allows users to specify customer-service targets separately for each demandstream. We first derive the fill rates foreach end product to meet the requiredcustomer-service target. These fill rates re-late to the actual lead times of all up-stream stocking locations, via the bills-of-materials structure of the network, and tothe actual lead times. The model thus cap-tures the interdependence of differentstocking locations, in particular the effectof base-stock levels and fill rates oncustomer-service. Related models insupply-chain and distribution networksinclude those of Lee and Billington [1993],Arntzen et al. [1995], Camm et al. [1997],

Figure 1: In this snapshot of the system dynamics at a stocking location, the base-stock level isnine, and when there are four units in stock, the other five units have been supplied to earlierorders, which translates into the five jobs in process.

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Kruger [1997], Graves, Kletter, and Hetzel[1998], and Andersson, Axsaeter, andMarklund [1998].

To allow fast execution of the optimiza-tion, we derived analytical gradient esti-mates in closed form and implemented agradient search algorithm to generate opti-mal solutions. Technical details of thiswork are presented by Ettl et al. [1998]and in the Appendix. In addition to thegradient search, we developed a heuristicoptimization procedure based on productclustering. To validate the solution ap-proach, we compared it against exhaustivesearches for test problems of moderatesize. For large-scale, industry-size applica-tions, the model has been extensivelytested at several IBM business units.The Simulation Engine

The simulation engine allows users tosimulate various supply-chain policies andin particular to verify and fine-tune theperformance of the solutions generated bythe optimization engine. We built thesimulation engine upon SimProcess[Swegles 1997], a general-purposebusiness-process simulator that was devel-oped jointly by IBM Research and CACIProducts Company. The simulation enginepreserves the capabilities of SimProcesswhile adding a supply-chain modelingfunctionality. Specifically, it providesmodeling functions for the followingsupply-chain processes:—The customer process represents outsidecustomers that issue orders to the supplychain, based on the modeled customer de-mand. It can also model information aboutthe desired customer-service target andpriority for the customer.—The manufacturing process models as-

sembly processes, buffer policies, and re-plenishment policies. It can also be used tomodel suppliers.—The distribution process models distri-bution centers and can also be used tomodel retail stores.—The transportation process modelstransportation time, vehicle loading, andtransportation costs.—The forecasting process represents prod-uct forecasts, including promotional andstochastic demand, for future periods.—The inventory-planning process modelsperiodic setting of inventory target levels.Underlying this process is the AMT opti-mization engine that computes recom-mended inventory levels at the variousstocking locations in a supply chain basedon desired customer-service target.

The simulation engine allows the user tovary a set of input parameters while moni-toring output reports to obtain the best setof output values. All input and output pa-rameters reside in the AMT modeling da-tabase. Users provide input parameters forthe simulation in the form of random vari-ables with stochastic distributions; theseinclude manufacturing lead times, trans-portation times, material-handling delaytimes, demand forecasts, product quantityrequired in a bill of material, and supplierreliability. The stochastic distribution func-tions supported include beta, Erlang, ex-ponential, gamma, normal, lognormal,Poisson, triangular, uniform, Weibull, anduser-defined distributions.

We designed the simulation engine toenable scenario-based analyses in whichsupply-chain parameters, such as thenumber and location of suppliers, manu-facturers, and distribution centers, inven-

IBM


tory levels, and manufacturing, replenish-ment, and transportation policies(build-to-plan, build-to-order, assemble-to-order, continuous replenishment, periodicreplenishment, full truckload, less-than-truckload, and so forth) are varied acrosssimulation runs. For each simulation run,the user can specify a planning horizon,the number of replicating scenarios (sam-ple runs), and a warm-up period duringwhich statistics are not retained. Thelength of the planning horizon depends onthe particular application in question andthe availability of historical demand fore-casts. We typically choose a horizon that isbetween six and 12 months.

The simulation-run outcome is in theform of measurement reports that can begenerated for turnaround times, customer-service, fill rates, stock-out rates, ship-ments, revenue, safety stock, and work-in-process. To analyze financial impacts,users can employ the following items, allof which are monitored during the simula-tion: cost of raw material; revenue fromgoods sold; activity-based costs, such asmaterial handling and manufacturing;inventory-holding costs; transportationcosts; penalties for incorrectly filled or lateorders delivered to customers; credits forincorrectly filled or late deliveries fromsuppliers; cost of goods returned by cus-tomers; and credits for goods returned tosuppliers.System Integration and TechnicalInnovations

We integrated the six functional mod-ules of AMT in a system architecture thatis flexible enough to accommodate users’varying computational needs. The archi-tecture is based on a client-server pro-

gramming model in which one can con-duct experiments using the resources of acomputer network (Figure 2). The AMTclient side provides a set of functions forviewing the graphical user interface anddialog-based data entry. The AMT serverside, which typically resides on a powerfulworkstation or midrange computer, pro-vides the full modeling and analysis func-tionality. For users with access to low-powered computers, such as laptops, wedeveloped an architecture in which theAMT client side is implemented as aplatform-independent Java application orapplet; web-enabled clients allow users toaccess AMT through a web browser.

To manage supply-chain operations,AMT requires data about the differentstages and processes that products gothrough. This data is accessible through arelational modeling database that is con-nected to the server through a relationalinterface. The database stores the informa-tion associated with the various modelingscenarios, including the supply-chainstructure, product structure, manufactur-ing data, and demand forecasts. The prod-uct structures are derived from a top-down bills-of-materials explosion that isprocessed for each end product. We ex-tracted all product data from corporatedatabases and from local site data sources.

To facilitate data extraction, we devel-oped a number of database connectivitymodules that provide automated databaseaccess, extract production data, and feedthem into the modeling database. All con-nectivity modules have built-in bills-of-materials explosion functionality. To de-tect inconsistencies in data recordingcaused by missing or incomplete informa-

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Figure 2: AMT is implemented using a client-server architecture in which the modeling func-tionality is separated from the graphical user interface. The modeling engines reside on aserver computer (eServer). The graphical user interfaces are piped to client computers that areimplemented as either Java applications (eManager) or Java applets (eClient). The AMT model-ing database can be accessed through a relational database interface. It contains such supplychain data as bills of materials, demand forecasts, lead times, costs, inventory policies, andcustomer-service requirements. Local and corporate data bridges provide automated access toenterprise data sources.

tion pertaining to the bills of materials, weadded database consistency checks thatgenerate missing data reports and reducethe data set to a consistent level that canbe downloaded to the modeling database.The data-collection process allows the userto supply missing data in relational tablesthat can be merged with the output of theexplosion. To keep the complexity of thebills-of-materials explosion manageable,we implemented data-reduction routinesthrough which one can eliminate noncriti-cal components automatically, based onthe item’s value class or annual require-

ments cost.AMT’s graphical user interface allows

modelers to build supply networks for avariety of supply chains by dragging anddropping generic supply-chain compo-nents on the workspace (Figure 3). Sophis-ticated algorithms are encapsulated in thecomponents. For instance, clicking the“PSG manufacturing” node will bring upscreens for the user to specify parametersand policies, such as delay time, manufac-turing lead times, bills of materials, andsuch manufacturing policies as build to or-der or build to plan. AMT also supports

IBM


Figure 3: AMT provides a graphical user interface that allows one to interactively constructsupply chain scenarios. In this example of an extended-enterprise supply chain, business part-ners (PSG Business Partners) send orders to a distribution center (PSG Distribution). The dis-tribution center processes the orders and sends products to a transportation node that ships theproducts to the business partners. The distribution center needs to replenish its stock from timeto time, so that it sends replenishment orders to the manufacturing site (PSG Manufacturing)that assembles finished products. The manufacturing site in turn replenishes its parts supplyby sending orders to its suppliers (PSG Suppliers). An inventory-planning node (PSG Inven-tory Planning) representing the AMT optimization engine computes optimal inventory levelsfor the distribution center based on forecasts of customer demand.

hierarchical process modeling. The usercan drill down to include other layers ofthe supply chain, adding scalability to themodeling approach. The customer nodecaptures demand, forecast, and customer-service requirements. We built in anima-tion to help users visualize the supply-chain activities of orders, goods, andtrucks moving between nodes. As thesimulation is running, users can view re-ports, such as service or inventory reports,

to see the current status of the simulation.In addition to these real-time reports,AMT also offers the financial and perfor-mance reports that we discussed earlier.

An important feature of AMT is thecomplementary functionality of the opti-mization and simulation engines. With theoptimization engine, the user can performfast yet very deep what-if analyses, whichare beyond the capability of any standardsimulation tool. With the simulation en-

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gine, the user can invoke the inventorymodule to perform periodic recalculationsof optimal inventory levels while simulat-ing dynamic supply-chain processes andpolicies. The user can run simulations onoptimized solutions, observing how differ-ent supply-chain policies at different loca-tions affect the supply chain’s perfor-mance. Simulation results can also be usedto adjust parameters of the simulation oroptimization runs. An automated interfacebetween the simulation engine and the op-timization engine allows users to invokeoptimization periodically during a simula-tion run, for example to recalculate targetinventory levels according to the latestforecast of demand. Users can also use theoptimization engine to periodically gener-ate build plans in a mixed push-pullmanufacturing environment, taking intoaccount service targets and systemuncertainty.

In summary, AMT embodies a creativecoupling of optimization, performanceevaluation, and simulation, integratedwith data connectivity and an Internet-enabled modeling framework. This makesit a powerful and versatile tool for captur-ing the stochastic and dynamic environ-ment in large-scale industrial supplychains. We model extended-enterprisesupply chains as networks of inventoryqueues, using a decomposition schemeand queuing analysis to capture the per-formance of each stocking location. Wedeveloped multiechelon, constrainedinventory-optimization algorithms thatuse conjugate gradient and heuristicsearches for efficient large-scale applica-tions. We developed a supply-chain simu-lation library consisting of an extensive set

of supply-chain processes and policies formodeling various supply-chain environ-ments with little programming effort. It of-fers performance measures, financial re-ports, and activity-based costing down tothe level of individual stock-keeping units.It also gives the user a way to validate andfine-tune supply-chain parameters basedon analytical results.Extended-Enterprise Supply ChainManagement at IBM Personal SystemsGroup

The IBM Personal Systems Group (PSG)is responsible for the development, manu-facture, sale, and service of personal com-puters (for example, commercial desktops,consumer desktops, mobile products,workstations, PC servers, network PCs,and related peripherals). PSG employsover 18,500 workers worldwide. Sales andmarketing groups are located in majormetropolitan areas, with manufacturingplants located in the United States, LatinAmerica, Europe, and Asia. In 1998, PSGsold approximately 7.7 million computersunder such brand names as IBM PC, Ap-tiva, ThinkPad, IntelliStation, Netfinity,and Network Station.

Increased competition from such PCmanufacturers as Dell and Gateway,which use a direct, build-to-order businessmodel, prompted PSG in 1997 to reevalu-ate its business practices and its relation-ships with its supply-chain partners. Thegoal was to design and implement a hy-brid business model, one that incorporatedthe best features of the direct model (buildto order, custom configuration, and inven-tory minimization) and the best features ofthe indirect model (final configuration,high customer service, and support), sell-

IBM


ing products through multiple channels.PSG formed a cross-functional team in

April 1997 with the task of quantifying therelationship between customer service andinventory throughout the extended supplychain under the existing business modeland under various proposed channel-assembly alternatives. We used productiondata from a subset of PSG’s commercialdesktop products to develop a baselinesupply-chain model in AMT. The modelwas triggered by end-user demand, re-seller ordering behavior, IBM manufactur-ing and inventory policies, supplier per-formance, and lead-time variability. Wecollected actual end-user sales data for 22reseller locations over five months. Resell-ers’ ordering behavior was influenced bymany factors, such as gaming strategies,marketing incentives, confidence in sup-plier reliability, and stocking for large cus-tomer purchases. Modeling each individ-ual activity would have been too complex.Our model captured the aggregate order-ing for each PSG reseller by substitutingalternative ordering policies, representingcurrent levels of sales activity in the chan-nel. For example, if a particular resellerheld an average of 60 days of inventory,the model established a target base-stocklevel representing 60 days of channel in-ventory for this reseller. To see whatwould happen if resellers changed theirordering policies, we changed the levels ofchannel inventory in the AMT model andran different what-if scenarios. For eachordering policy, we assumed that a re-seller would stock a product at a givenlevel of days of supply.

During the normal course of business,PSG forecasts its manufacturing volumes

over a rolling 13-week horizon. The cur-rent week’s forecast becomes the buildplan, which then pushes products built atPSG’s manufacturing sites to the distribu-tion warehouse where they are held untilthe products are eventually ordered, orpulled, by a reseller. This type of replen-ishment policy captured the logic of PSG’shybrid push-pull manufacturing and or-dering system in which PSG built prod-ucts to a forecast and held them as fin-ished goods in the warehouse until itreceived orders from its resellers. This sys-tem is not a true pull system because

PSG’s channel look-backexpenses dropped by morethan $100 million.

product availability influences reseller or-dering. Likewise, the system is not a truepush system because the backlog of resell-ers’ orders influences the schedules at PSGmanufacturing sites. To effectively capturevariability caused by component short-ages, capacity constraints, and require-ments for minimum lot sizes, we analyzedthe range of the 13-week forecasts.

PSG set a service target for customer de-liveries of three days, 95 percent of thetime, which translated directly into thecustomer-service constraint required bythe AMT optimization engine. Combiningthe simulation engine with the optimiza-tion engine, the model recalculated thebase-stock levels every week, according tothe latest available forecast of demand sothat customer orders could be filled withinthree days 95 percent of the time. This re-plenishment policy formed the basis forPSG’s supplier orders for components and

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subassemblies and for its subsequentmanufacturing activity. In Phase 1 of theproject, we used a reduced data set to con-struct a simplified prototype model ofPSG’s supply chain to test assumptions, toinvestigate alternative modeling algo-rithms, and to better understand possiblelimitations of the AMT application.

In Phase 2 of the project, we developedmore detailed modeling scenarios to varychannel inventory and to incorporate achannel-assembly policy at the resellers.PSG delivers to its resellers two types ofproducts, (1) standard machine-type mod-els (MTMs), which are fully configuredand tested computers, and (2) so-calledopen-bay machines, which are nonfunc-tional, basic computers without such pre-configured components as memory, hardfiles, and CD ROMs. These open bays al-low resellers to assemble machines accord-ing to specific customer requirements. Wefound that some resellers converted openbays into standard MTMs as needed andthen sold them to their customers. We re-fer to this as an example of flexibility be-cause resellers can use their current open-bay inventory to fill orders for standardMTMs, instead of stocking open bays ex-clusively to fill orders for nonstandardMTMs. Other resellers stockpiled open-bay inventory, and if they needed stan-dard MTMs to fill an order, they wouldreorder from PSG instead of configuringan open bay already in stock (an exampleof inflexibility). Both methods affect inven-tory and customer service. Because resellerflexibility could not be defined accurately,we designed different sets of simulationexperiments with the intent to bound, orframe, the true impact of channel assem-

bly within the two extreme cases of 100percent reseller flexibility and 100 percentreseller inflexibility.

We validated the accuracy of the AMTmodels by comparing the outputs of thesimulation runs to historical PSG data. Weadjusted our modeling assumptions andparameters as necessary and ran multiplesimulations using different parametersand policies. The key results of the studycan be summarized as follows:—Implementing channel assembly basedon PSG’s existing product structure, lowvolume environment, and present supply-chain policy reduces inventory very little(inflexible reseller channel behavior).—Allowing resellers to configure anyMTM from their stock of componentscould improve customer service by twopercent and simultaneously reduce inven-tories by 12 percent (flexible reseller chan-nel behavior).—Consolidating the demand at 22 config-uration sites into three large hubs couldimprove customer service by six percentand reduce inventories by five percent.—Based on the existing push-pull supply-chain policy, PSG can reduce channel in-ventory by 50 percent without affecting itscustomer-service level. The overall supply-chain inventory levels were far in excessof the optimum needed to maintain PSG’sservice target.

This and subsequent projects broughttogether four functional groups—market-ing and sales, manufacturing, distribution,and development—to seek a company-wide consensus on PSG’s strategic direc-tion and subsequent actions. Our studiescontributed directly to PSG’s advancedfulfillment initiative (AFI), an effort to in-

IBM


crease flexibility in the reseller channel byimproving parts commonality in PSG’sproduct structure [Narisetti 1998]. Also,PSG management endorsed the reductionof the number of configuration sites, as aresult of changing channel price-protectionterms and conditions. The specific termsand conditions were tied to the output ofthe AMT model, and they were imple-mented in November 1997 after a series ofrelated enhancements to the logisticsprocess.

PSG has based many of its decisions onhow to prioritize project deployment andmanage channel inventory on the resultsof subsequent AMT analyses. While theanalysis that drove PSG’s initial businesstransformation was conducted in 1997, the1998 business benefits were substantial.

The more accurate a reseller’sforecasts, the higher the levelof service.

PSG reduced its overall inventory by over50 percent from year-end 1997 to year-end1998. As a direct consequence of this in-ventory reduction, PSG’s channel look-back expenses dropped by more than $100million from 1997 levels. Look-back ex-penses account for payments to distribu-tors and business partners that compen-sate for price actions on the inventory theyare holding. In addition, by selling prod-ucts four to six weeks closer to when thecomponents are procured, PSG saved anadditional five to seven percent on prod-uct cost. This equates to more than $650million of annual savings.

In the months following the original as-sessment, we conducted further supply-

chain studies, including analyses that (1)incorporated the supply chains of businesspartners; (2) modeled additional geogra-phies; (3) assessed the impact on inventoryand customer service of delaying final as-sembly to the reseller’s distribution facili-ties; and (4) estimated the impact on in-ventory of reducing manufacturing cycletimes. These studies have helped PSG’sbusiness partners make more informed de-cisions on supply-chain policy. In particu-lar, they have led IBM and its major busi-ness partners to establish a colocationpolicy. In colocation, a business partner lo-cates its distribution space inside of IBM,eliminating the need for costly handlingand transportation among different sites.Finally, because we found that forecast ac-curacy greatly affected inventory and cus-tomer service, PSG used the AMT to de-termine the level of service it wouldpromise to its business partners, based ontheir ability to provide accurate forecasts.The more accurate a reseller’s forecasts,the higher the level of service PSG wouldprovide to that reseller. This policy is un-precedented in the industry and has beenfavorably received by PSG’s business part-ners. Overall, PSG believes that the AMThas been an invaluable asset in developingand implementing world-class supply-chain-management policies.Other AMT Applications Across IBM

AMT has also been applied and de-ployed in other IBM manufacturing divi-sions, including the printing systems divi-sion (PSC), the midrange computerdivision (AS/400), the office workstationdivision (RS/6000), the storage systems di-vision (SSD), the mainframe computer di-vision (S/390), and PSG’s European mar-

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INTERFACES 30:1 20

ket. A number of PSG’s business partnershave used AMT, including Pinacor, GECapital, and Best Buy. IBM’s Industry So-lution Unit uses the tool externally forconsulting engagements. Following arebrief descriptions of three recent AMTengagements:

The IBM Printing Systems Company(PSC) is a leading supplier of printer solu-tions for business enterprises. The productline includes printers for office printing tohigh-volume production printing. Thecompany employs approximately 4,550people, with total gross revenue for 1998of $1.95 billion. In 1996, PSC conducted anintensive testing process on the AMT overa five-month period. In its assessment re-port, the testing team concluded that AMTproduces accurate results, provides pro-ductivity improvements over existing

Financial savings amount tomore than $750 million at PSGin 1998.

supply-chain-management and inventorytools, and improves PSC’s precision in val-idating and creating inventory budgetsand turnover objectives. PSC then usedAMT to study the effects of forecast accu-racy, product structure, the introduction ofa new distribution center, and differentbusiness scenarios on the performance ofthe supply network for different productfamilies. In one of the cases alone, it re-ported inventory savings of $1.6 million,which represented 30 percent of the totalinventory holding cost.

IBM’s AS/400 division manufacturesmidrange business computers and servers,providing more than 150 models and up-

grades with up to 1,000 features. Assem-bling these systems requires several thou-sand unique part numbers, approximately1,000 of them used at the highest level ofassembly just prior to building a completesystem. Providing customers with the flex-ibility to customize the equipment they or-der by selecting features creates manufac-turing complexity and efficiencychallenges. The division used AMT to ana-lyze and quantify the impact on inventoryand on-time delivery of feature reduction,feature substitution, parts commonality,and delayed customization. The analysisshowed that eliminating low-volume partswould improve inventory turnover by 15percent and that substituting and postpon-ing their final assembly would improve in-ventory turnover by approximately 30 per-cent. The AS/400 division has reduced itsfeature count by approximately 30 percentsince 1998 with steady growth in totalrevenue.

In 1995, IBM established a quick-response service program to provide rapiddelivery for customers buying selectedmid-range computer memory, storage,and features. In September 1998, IBM in-stituted the quick-response program as afront end to provide real e-commerce forour large business partners. IBM usedAMT to analyze the trade-off between ser-vice and inventory in choosing an opti-mum performance point. It later used it toassess the impact of the quick-responseprogram on allocating inventory betweenmanufacturing and distribution centers.The results helped IBM to maximize busi-ness efficiency and contributed to dou-bling the growth of quick-response reve-nue in 1998.

IBM


ConclusionsThe AMT effort uses advanced OR tech-

niques and combines technical innovationswith practical and strategic implementa-tions to achieve significant business im-pacts. IBM has used AMT to address awide range of business issues, includinginventory management, supply-chain con-figuration, product structure, and replen-ishment policies. AMT has been imple-mented in a number of IBM business unitsand their business partners. Financial sav-ings through the AMT implementationsamount to more than $750 million at PSGin 1998 alone. Furthermore, AMT hashelped IBM’s business partners to meettheir customers’ requirements with muchlower inventory and has led to a co-location policy with many business part-ners. It has become the foundation for anumber of supply-chain-reengineering ini-tiatives. Several IBM business partnersview the AMT analyses as key milestonesin their collaboration with IBM in optimiz-ing the extended-enterprise supply chain.Acknowledgments

We gratefully acknowledge the contri-butions and support of the following peo-ple: Chae An, Ray Bessette, Rick Bloyd,Harold Blake, Richard Breitwieser, J. P.Briant, Bob Chen, Feng Cheng, ArthurCiccolo, Daniel Connors, Ian Crawford,Anthony Cyplik, John Eagen, Brian Eck,Gerry Feigin, Angela Gisonni, JohnKonopka, Tom Leiser, Tony Levas, NikornLimcharoen, Joe Magliula, Barbara Martin,Larry McLaughlin, Bob Moffat, GerryMurnin, Nitin Nayak, Jim Nugent, LynnOdean, Krystal Reynolds, Richard Shore,Mukundan Srinivasan, JayashankarSwaminathan, David Thomas, Bill Tulskie,

Burnie Walling, Wen-Li Wang, and JamesYeh.

APPENDIXOptimization of Multi-Echelon SupplyNetworks with Base-Stock Control

Here we provide a brief overview of thekey points of the mathematical model inthe optimization engine. Ettl, Feigin, Linand Yao [1998] give the full details, in-cluding topics that we do not touch uponhere, such as the treatment of nonstation-ary demands, the related rolling-horizonimplementation, the derivation of the gra-dients, and many preprocessing and post-processing steps.

We specify the configuration of the sup-ply network using the bills-of-materialsstructure of the products. Each site in thenetwork is either a plant or a distributioncenter. Associated with each site and eachproduct processed at the site is a multi-level bill of material. Each site has storageareas, which we refer to as stores, to holdboth components that appear on the billsof materials and finished products, whichcorrespond to input stores and outputstores. The subscripts i and j index thestores, and S denotes the set of all storesin the network. We assume a distributedinventory-control mechanism wherebyeach store follows a base-stock control pol-icy for managing inventory. The policyworks as follows: When the inventory po-sition (that is, on-hand plus on-order mi-nus backorder) at store i falls below somespecified base-stock level, Ri, a site placesa replenishment order. In our model, Ri isa decision variable.

For each store i, there is a nominal leadtime, Li, with a given distribution. Thenominal lead time corresponds to the pro-duction time or transshipment time at thesite where the store resides, assumingthere is no delay (due to stock-out) in anyupstream output stores. The actual leadtime, Li, in contrast, takes into accountpossible additional delays due to stock-

LIN ET AL.

INTERFACES 30:1 22

out. Whereas Li’s are given data, Li’s arederived performance measures.

To analyze the performance of eachstore i, we use an inventory-queue model,for example, Buzacott and Shanthikumar’s[1993]. Specifically, we combine the base-stock control policy with an MX/G/�queue model, where arrivals follow a Pois-son process with rate ki, and each arrivalbrings in a batch of Xi units, or orders. Thebatch Poisson arrival process is a goodtrade-off between generality and tractabil-ity. In particular, it offers at least three pa-rameters to model the demand data: thearrival rate and the first two moments ofthe batch size (whereas a simple Poissonarrival process has only one parameter).

To derive the performance measures ateach store i, we need to first generate theinput process to the MX/G/� queue. Todo this, we take the demand stream (fore-cast or real) associated with each class,translate it into the demand process ateach store by going through the bills-of-materials structure level by level, and shiftthe time index by the lead times at eachlevel. This process is quite similar to theexplosion and offsetting steps in standardMRP analysis. A second piece of dataneeded for the MX/G/� queue is the ser-vice time, which we model as the actuallead time.

Let Ni be the total number of jobs in thequeue MX/G/� in equilibrium. Followingstandard queueing results [Liu, Kashyap,and Templeton 1990], we can derive themean and the variance of Ni, denoted as li

and . We then approximate Ni with a2ri

normal distribution:

N � l � rZ, (1)i i i

where Z denotes the standard normal var-iate. Accordingly, we write the base-stocklevel as follows:

R � l � k r , (2)i i i i

where ki is the so-called safety factor. As Ri

and ki relate to each other via the aboverelation, either can serve as the decisionvariable. Let Ii be the level of on-hand in-ventory, and Bi the number of back-ordersat store i. These relate to Ni and Ri asfollows:

�I � [R � N ]i i i

�and B � [N � R ] , (3)i i i

where [x]� � max(x,0). We can then de-rive the expectations:

E[I ] � rH(k ), and E[B ] � rG(k ) (4)i i i i i i

where�

H(k ) � (k � z)�(z)dz andi i�ki

�

G(k ) � (z � k )�(z)dz, (5)i i�ki

and �(z) � exp(�z2/2)/ is the density2p�function of Z. Furthermore, writing U(x)� �(z)dz, the distribution function of Z,x�0

and � 1 � U(x), we can derive theU(x)stock-out probability pi and the fill rate fiat store i as follows:

¯p � U(k ), andi i

¯f � 1 � r�(k )/l � U(k ). (6)i i i i i

All of the above performance measuresinvolve the actual lead time at store i,which can be expressed as follows:

L � L � max(s ), (7)i i jj�S�i

where S�i denotes the set of stores thatsupply the components needed to buildthe units in store i, and sj denotes the ad-ditional delay at store j � S�i. As sj isquite intractable in general, with queueinganalysis, we have derived the followingapproximation:

E[B ]j˜s � L r where r : � . (8)j j j j p (R � 1)j j

Intuitively, E[Bj]/pj is the average numberof back-orders at location j conditioned

IBM


upon a stock-out there, and each of theseback-orders requires an average time ofLj/(Rj � 1) to fill, that is, during the stock-out, on the average, there are (Rj � 1) out-standing orders in process.

Customer demands are supplied from aset of end stores, S0, stores at the bound-ary of the network. Consider a particularcustomer class, and suppose its demand issupplied by one of the end stores, i � S0.Let Wi denote the waiting time to receivean order. The required customer-servicetarget is

P[W � b ] � � , (9)i i i

where bi and �i are given data. When thedemand is supplied from on-hand inven-tory, the delay is simply the transportationtime Ti, time to deliver the finished prod-ucts to customers, which is given; other-wise, there is an additional delay of sj.Hence,

P[W � b ] � f P[T � b ]i i i i i

� (1 � f )P[T � s � b ].i i i i

For the above to be at least �i, we need toset fi, the fill rate, to the following level:

� � P[T � s � b ]i i i if � . (10)i P[T � b ] � P[T � s � b ]i i i i i

The quantity si involved in the right-hand side of the above equation can be ex-pressed as si � Li ri, following (8). Since ri

involves Bi and Ri, both of which are func-tions of ki, and so is fi, we need to solve afixed-point problem defined by the equa-tion in (8) to get fi (or ki). In the iterationsinvolved in the optimization procedure,however, this fixed-point problem can beavoided by simply using the ri value ob-tained from the previous iteration. Oncewe derive fi and ki, the base-stock level (2)and the stock-out probability (6) thenfollow.

The objective of our optimization modelis to minimize the total expected inventorycapital throughout the supply network

while satisfying customer-service require-ments. Each store has two types of inven-tory: on-hand inventory and work-in-process (WIP) inventory. (The WIPincludes the orders in transition, that is,orders being transported from one store toanother.) From the above discussion, theexpected on-hand inventory at store i isE[Ii] � riH(ki), and the expected WIP isE[Ni] � li. Therefore, the objective func-tion takes the following form:

C(k) � [c�l � c rH(k )], (11)� i i i i ii�S

where c�i and ci denote the inventory capi-tal per unit of the on-hand and WIP in-ventory, respectively, with ci assumedgiven, and c�i derived from the ci’s alongwith the BOM. We want to minimize C(k),subject to meeting the fill-rate require-ments in (10), for all the end stores: i � S0

� S. This is a constrained nonlinear opti-mization problem. We derive the partialderivatives �/�kj C(k), all in explicit analyt-ical forms based on the relations derivedabove (and others). We use these in aconjugate-gradient search routine, for ex-ample that of Press et al. [1994]. As thesurface of the objective function is quiterugged, to avoid local optima, we also im-plemented several heuristic search proce-dures. For instance, evaluate a set of ran-domly generated initial points and pickthe best one (in terms of the objectivevalue) to start the gradient search.ReferencesAndersson, J.; Axsaeter, S.; and Marklund, J.

1998, “Decentralized multi-echelon inventorycontrol,” Production and Operations Manage-ment, Vol. 7, No. 4, pp. 370–386.

Arntzen, B. C.; Brown, G. G.; Harrison, T. P.;and Trafton, L. L. 1995, “Global supply chainmanagement at Digital Equipment Corpora-tion,” Interfaces, Vol. 25, No. 1, pp. 69–93.

Bagchi, S.; Buckley, S.; Ettl, M.; and Lin, G.1998, “Experience using the supply chainsimulator,” Proceedings of the Winter Simula-tion Conference, Washington, DC, December,pp. 1387–1394.

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Buckley, S. 1996, “Supply chain modeling,”Proceedings of the Autofact Conference, Detroit,Michigan, pp. 749–756.

Buckley, S. and Smith, J. 1997, “Supply ChainSimulation,” Georgia Tech Logistics ShortCourse, Atlanta, Georgia, pp. 1–17.

Buzacott, J. A. and Shanthikumar, J. G. 1993,Stochastic Models of Manufacturing Systems,Prentice-Hall, Englewood Cliffs, New Jersey.

Camm, J. D.; Chorman, T. E.; Dill, F. A.; Evans,J. R.; Sweeney, D. J.; and Wegryn, G. W. 1997,“Blending OR/MS, judgment, and GIS: Re-structuring P&G’s supply chain,” Interfaces,Vol. 27, No. 1, pp. 128–142.

Ettl, M.; Feigin, G.; Lin, G.; and Yao, D. D.1998, “A supply network model with base-stock control and service requirements,”Operations Research, forthcoming.

Feigin, G.; An, C.; Connors, D.; and Crawford,I. 1996, “Shape up, ship out,” ORMS Today,Vol. 23, No. 2, pp. 1–5.

Graves, S.; Kletter, D. B.; and Hetzel, W. B.1998, “A dynamic model for requirementsplanning with application to supply chainoptimization,” Operations Research, Vol. 46,No. 3, pp. S35–S49.

Kruger, G. A. 1997. “The supply chain ap-proach to planning and procurement man-agement,” Hewlett-Packard Journal, Vol. 48,No. 1, pp. 1–9.

Lee, H. L. and Billington, C. 1993, “Materialmanagement in decentralized supplychains,” Operations Research, Vol. 41, No. 5,pp. 835–847.

Liu, L.; Kashyap, B. R. K.; and Templeton,J. G. C. 1990, “On the GIX/G/� system,”Journal of Applied Probability, Vol. 27, No. 3,pp. 671–683.

Narisetti, R. 1998, “How IBM turned around itsailing PC Division,” Wall Street Journal, Octo-ber 3, p. B1.

Press, W. H.; Teukolsky, S. A.; Vettering, W. T.;and Flannery, B. P. 1994, Numerical Recipes inC, second edition, Cambridge UniversityPress, New York.

Swegles, S. 1997, “Business process modelingwith SIMPROCESS,” Proceedings of the WinterSimulation Conference, Piscataway, New Jer-sey, pp. 606–610.

Bob Moffat, general manager for manu-

facturing, procurement, and fulfillment atIBM Personal Systems Group, said duringthe presentation of the paper at the Edel-man competition: “We reduced our chan-nel inventory from over three months toapproximately one month. As a direct con-sequence of this inventory reduction, ourdivision has reduced 1998 price protectionexpenses by over $100M from the previ-ous year. Price protection expenses arewhat we reimburse business partnerswhenever we take a price action on prod-ucts they are holding. We had reduced ourend-to-end inventory from four and a halfmonths to less than two months by theend of 1998. By closing the gap betweencomponent procurement and product saleby four to six weeks, there is a savings onproduct cost of at least five percent. Thisequates to more than $650 million of an-nual savings. AMT has improved our rela-tionships with business partners, makingthem more efficient, more productive, andultimately more powerful in the market-place. I believe this will lead to a funda-mental change in our business culture, aunification of basic value among suppliers,manufacturers, and resellers.”

Jean-Pierre Briant, IBM vice presidentfor integrated supply chain, further ex-plained: “The AMT tool has found appli-cation in almost every supply chain withinIBM. It helps us understand our extendedsupply chain—from our suppliers’ sup-pliers to our customers’ customers. Wehave deployed the AMT tool to assist ex-ternal companies in managing their sup-ply chains, with very effective results.”

Jim Manton, president and COO ofPinacor, said: “The results that the [AMT]team delivered on the supply chain analy-

IBM


sis helped Pinacor identify opportunitiesfor optimizing the product flow betweenour companies. . . . I am pleased to see thatboth IBM and Pinacor are focusing on therecommendations to make the necessaryimprovements. . .”.

Mac McNeill, senior vice president ofglobal operations for GE Capital IT Solu-tions, who sponsored a four-month projectusing the AMT to model GE Capital’spersonal-computer supply chain com-mented: “The modeling allowed us to de-velop a base case using actual end-usercustomer sales and then to quickly modeland optimize many alternatives based onvarious levels of GE forecast accuracy,IBM fill rates, transit times, in-bound andout-bound delays, and commonality ofparts. The optimization results will allowus to develop action plans to balance im-proved levels of serviceability with lowerlevels of inventory.”

Copyright � 2000 INFORMS0092-2102/00/3004/0065/$05.001526–551X electronic ISSN

INVENTORY—PRODUCTION—MULTI-ITEM, ECHELON, STAGEINDUSTRIES—COMPUTER—ELECTRONICS

This paper was refereed.

INTERFACES 30: 4 July–August 2000 (pp. 65–80)

Xilinx Improves Its Semiconductor SupplyChain Using Product and ProcessPostponementAlexander O. Brown Owen Graduate School of Management

Vanderbilt University401 21st Avenue SouthNashville, Tennessee 37203

Hau L. Lee Graduate School of Business and theDepartment of Management Science andEngineering, Stanford UniversityStanford, California 94305

Raja Petrakian Xilinx, Inc.2100 Logic DriveSan Jose, California 95124

The semiconductor firm Xilinx uses two different postpone-ment strategies: product postponement and process postpone-ment. In product postponement, the products are designed sothat the product’s specific functionality is not set until after thecustomer receives it. Xilinx designed its products to be pro-grammable, allowing customers to fully configure the functionof the integrated circuit using software. In process postpone-ment, a generic part is created in the initial stages of the manu-facturing process. In the later stages, this generic part is cus-tomized to create the finished product. Xilinx manufactures asmall number of generic parts and holds them in inventory.The use of these generic parts allows Xilinx to hold less inven-tory in those finished products that it builds to stock. And forsome finished products, Xilinx can perform the customizationsteps quickly enough to allow it to build to order.

High technology industries, such assemiconductors and computers, are

characterized by short product life cyclesand proliferating product variety. Facedwith such challenges, companies in theseindustries have found that delaying the

point of product differentiation can be aneffective technique to cut supply-chaincosts and improve customer service. Thispostponement technique is a powerfulway to enable cost-effective mass customi-zation [Feitzinger and Lee 1997]. To use

BROWN, LEE, PETRAKIAN

INTERFACES 30:4 66

postponement effectively, companies mustcarefully design their products and pro-cesses. Through careful design of theproduct and the process, many electronicsand computer companies have been ableto delay the point of product differentia-tion, either by standardizing some compo-nents or processes or by moving the cus-tomization steps to downstream sites, suchas distribution centers or retail channels.Lee [1993, 1996]; Lee, Billington, andCarter [1993]; Lee, Feitzinger, andBillington [1997]; and Lee and Sasser[1995] give examples.

Postponement concepts have also beenapplied in other industries, such as the au-tomobile industry [Whitney 1995] whereproduct modularity enables delayed cus-tomization of auto parts. Indeed, Ulrich[1995] showed that a high degree of prod-uct modularity coupled with component-process flexibility could render postpone-ment possible and effective. Lee,Padmanabhan, and Whang [1997] alsosaid that both product and process modu-larity support postponement. Modular de-signs for products or modular processes (amanufacturing process that can be brokendown into subprocesses that can be per-formed concurrently or in different se-quential order) are techniques that enablepostponement.

The semiconductor industry has beenplagued by a proliferation of product vari-ety because of the overlapping product lifecycles—companies introduce new or en-hanced versions of products before exist-ing products reach the ends of their lifecycles. In the programmable-logic segmentof the industry, new customers will usethe enhanced versions in their products,

but some existing customers may delayadopting the new versions despite theirimproved performance and price. Periodsof appreciable demand for a version of aproduct may range from six months totwo years, with products sometimes hav-ing an extended period of very low end-of-life demand. Thus, semiconductorcompanies must offer many products si-multaneously. The product-variety prob-lem is compounded by unpredictable de-mands and long manufacturing leadtimes.

Semiconductor firms face unpredictabledemand, in large part, because of their up-stream position in the supply chain. Anintegrated circuit (IC) made by a semicon-ductor firm is a component of other subas-semblies or final products. Thus, it mustpass through other companies, such ascontract manufacturers, distributors, andresellers, before the final product reachesthe end consumer. Lee, Padmanabhan,and Whang [1997] describe the “bullwhipeffect” in which demand fluctuations in-crease as you travel upstream in the sup-ply chain. Since semiconductor firms arelocated far upstream in the supply chain,they often face such large fluctuations.

Manufacturing cycle times in the semi-conductor industry are still very long de-spite advances in the process technology.The manufacturing process, consisting ofwafer fabrication, packaging, and testing,takes about three months. With suchlong manufacturing lead times, the semi-conductor companies must hold largeinventories of finished goods or theircustomers—computer assemblers, telecom-munication manufacturers, or other elec-tronics manufacturers—must hold large

XILINX

July–August 2000 67

inventories to hedge against demanduncertainties.

Product variety, long production leadtimes, and demand unpredictability nega-tively affect the manufacturing efficiencyand performance of both semiconductorcompanies and their customers. Thesecharacteristics also affect the customer’sproduct-development processes. For ex-ample, one part of a telecommunications-equipment manufacturer’s product-development process might be the custom

Semiconductor companiesoffer many productssimultaneously.

design of application-specific integratedcircuits (ASICs). The design process oftenincludes creating a number of prototypesbefore settling on a final working design.Because of the long production times,there is often a significant delay betweendesigning and receiving prototypes. Sincetime to market is a key factor in the suc-cess of high-tech products, this delay maybe very costly for the manufacturer. Tocompress the cycle, such manufacturersmay request many prototypes towards thebeginning of the design process, resultingin additional design and developmentcosts.

Product variety, long lead times, anddemand unpredictability are all unavoid-able and problematic characteristics of thesemiconductor industry. However, somecompanies are finding new ways to copewith them. Xilinx, Inc., uses innovative de-sign principles of postponement to avoidexcessive inventory while providing greatservice to its customers. It uses both prod-

uct and process postponement exten-sively.

In product postponement, the firm de-signs the product so that it can delay itscustomization, often by using standard-ized components. Xilinx relies on a moreextreme form of product postponement.Instead of the firm performing the finalconfiguration during manufacture or evendistribution, it designs the ICs so that itscustomers perform the final configurationusing software. Consequently, Xilinxgreatly shortens the product-developmentcycles of its customers, as the customersdo not have to specify the full featuresand functionalities of the ICs beforeproduction.

Using proprietary design technologies,Xilinx creates many types of ICs, differen-tiated by such general features as speed,number of logic gates, package type, pincount, and grade. Although the customersperform the final configuration of thelogic, they must order products with theappropriate general features. For example,a customer with a large and complex de-sign requiring high speed must select aphysical device type with a large numberof logic gates and a high speed. Later thecustomer can configure the logic of the de-vice using software, creating an enormousnumber of possible designs. Product post-ponement is very suitable for programma-ble devices because a near-infinite numberof designs can be created from a few thou-sand physical-product permutations.

In process postponement, the firm de-signs the manufacturing and distributionprocesses so that it can delay product dif-ferentiation, often by moving the push-pull boundary or decoupling point toward


INTERFACES 30:4 68

the final customer. A push-pull boundaryis the point in the manufacturing-and-distribution process at which productioncontrol changes from push to pull. Earlyin the process, prior to the push-pullboundary, the firm builds to forecast.Later in the process, after the push-pullboundary, it builds to order. Often, pro-cess designs allow manufacturers tochange their push-pull boundaries. Ahighly celebrated example of process post-ponement is the case of Benetton, whichused to make sweaters by first dyeing theyarns and then knitting them into finishedgarments of different colors. Its push-pullboundary used to be at finished sweat-ers—all production was built to forecast.Benetton resequenced its production pro-cess so that it first knits undyed garments,and then dyes them (and thereby custom-izes them to the different color versions)on demand. Hence, its new push-pullboundary is between knitting and dyeing[Dapiran 1992].

To improve its manufacturing process,Xilinx focused on creating a new push-pull boundary, working with its suppliers.Rather than going through all the steps tocreate an IC in its finished form from araw silicon wafer, Xilinx divides the pro-cess in two stages. In the first step, itswafer-fabrication supply partners manu-facture unfinished products, called dies,and hold inventory of this material. Thisinventory point is the push-pull boundary.Based on actual orders from the custom-ers, another set of supply partners pulldies from inventory and customize theminto finished ICs.The Xilinx Supply Chain

Digital semiconductor devices can be

broadly grouped into three categories:memory, microprocessors, and logic.While the general-purpose microproces-sors can execute almost any logical ormathematical operation, logic devices pro-vide specific functionality at lower costand greater speed. However, the tradi-tional method of defining the functions of

Xilinx was one of the first touse a virtual business model.

a logic device is to configure it during thefabrication process. Recently, with the in-troduction of programmable logic devices,it has become possible to customize a ge-neric but more expensive logic device us-ing software after the logic device hasbeen completely manufactured andpackaged.

Founded in 1984, Xilinx developed thefield-programmable gate array (FPGA), aprogrammable logic device, and it has be-come one of the two largest suppliers ofprogrammable logic solutions in theworld. The company’s revenues in 1997were $611 million and the gross marginwas around 62 percent. Xilinx was one ofthe first semiconductor companies to use avirtual business model: it subcontracts outlogistics, sales, distribution, and mostmanufacturing to long-term partners. Xil-inx’s only manufacturing facilities are itsCalifornia and Ireland facilities that justperform some final testing. It meets about74 percent of its total demand through dis-tributors, whose expertise has evolved be-yond traditional warehousing and inven-tory management to include engineeringfunctions, such as helping customers de-sign Xilinx parts into their systems. Xilinx

XILINX


keeps certain core functions in-house, suchas technology research, circuit design,marketing, manufacturing engineering,customer service, demand management,and supply-chain management. This vir-tual business model provides Xilinx with ahigh degree of flexibility at low cost. Itspartners benefit because Xilinx uses stan-dard manufacturing and business pro-cesses and aggressively drives process im-provements through technical innovationand re-engineering. Although the virtualmodel has strategic risks (the core compe-tencies becoming commodity-like) and op-erational risks (unexpected lack of avail-able capacity at suppliers), it has provenhighly successful in the industry [Lineback1997].

Today, most of Xilinx’s competitorshave access to the same fabrication processtechnology through their own wafer-fabrication partners. The technology andmanufacturing gap between members ofthe industry is closing. Consequently, Xil-inx sees management of the demand-and-supply chain as providing it with a com-petitive advantage in the market. In 1996,Xilinx executive management initiated a

major initiative to overhaul the company’spractices and processes for managing sup-ply and demand.

In the Xilinx supply chain, the flow ofmaterials begins with the fabrication pro-cess (front end), where raw silicon wafersare started and manufactured using hun-dreds of complex steps that typically taketwo months (Figure 1). Anywhere from 20to 500 integrated circuits come from eachfabricated wafer. In the last process stepsof the front end, the wafers are sorted andtested for basic electrical characteristics.Although precise information is not avail-able until the final test after assembly, thisstep provides some useful indications ofthe proportion of good integrated circuitson the wafer and the speed mix that theyare likely to yield. After sorting and pre-liminary testing, wafers are stored in in-ventory—the die-bank. Planning waferstarts to ensure proper die-bank inventoryis a major challenge, requiring such infor-mation as demand forecasts, projectedyields, and work in process to determinethe volume and mix of wafers needed tomeet the demand and inventory tar-gets.

Figure 1: In the Xilinx supply chain, supply partners perform the wafer fabrication and assem-bly, while Xilinx manages production levels and the inventory levels in die bank and finishedgoods. After production, a distributor buys the integrated circuits and supplies them to originalequipment manufacturers that incorporate the integrated circuit into their products. Consumerspurchase the products through retailers. Triangles represent inventory stocking locations, andsquares represent manufacturing processes.


INTERFACES 30:4 70

The next link in the supply chain is theback end, a term that refers to both the as-sembly and test processes. In the back-end, wafers are first cut into dies, or indi-vidual “raw” integrated circuits. There areapproximately 100 different types of dies.To be usable, the integrated circuits mustbe placed in a package, a plastic casingwith electric lead pins, that allows them tobe later mounted in a circuit board. Thereare usually about 10 to 20 package typesfrom which a customer can select for agiven die. The dies are wire bonded toform a permanent electrical contact withthe package. The packaged dies are thentested electrically to determine if theymeet stringent design and quality require-ments and to determine their speed. Thereare usually about five to 10 different possi-ble speed grades. The packaged devicesthat pass the quality tests are then storedin finished-goods inventory. With leadtimes of three weeks for assembly andtest, planning of back-end starts is diffi-cult, requiring information on both thebacklog of orders and demand forecasts.One complexity involves the issue of de-vice speed. Although Xilinx understandsthe expected fraction of dies that will yieldto each speed level, the actual fraction forany given die is different. Thus, planningusing the expected fraction of dies at eachspeed level will often result in a mismatchof supply and demand. To meet the de-mand, Xilinx will start more material inthe back end and pick wafers intelligentlyusing measurements collected in thefabrication-and-sort step.

Most Xilinx customers are servicedthrough distributors who maintain inven-tories of Xilinx finished-goods parts. The

advantages distributors provide to Xilinxare that they have cost-effective means forhandling large numbers of small tomedium-size customer orders and they of-fer such value-added services as inventoryconsolidation, inventory management, andprocurement-program support. The cost ofXilinx is that they add an extra link to thesupply chain, causing a potential distor-tion in demand information. The lack ofend-demand visibility can be partially off-set when distributors provide Xilinx withsystematic data regarding point of sale(POS), bookings, backlog, and inventory.Most Xilinx customers are original equip-ment manufacturers (OEMs) that put oneor more Xilinx parts on a circuit board andthen assemble a large system using theboard and other components. The OEMsthen sell these systems to other customersusing various marketing and distributionchannels. The Xilinx supply chain is fur-ther complicated by the practice of manyOEMs of subcontracting the board assem-blies to specialized vendors.

On-time delivery is emphasized at Xil-inx. As a result, Xilinx has often resolvedthe trade-off between inventory and on-time delivery by adding inventory. One ofthe key goals of the supply-chain-management initiative is to achieve thesame levels of customer service with lowerinventory costs throughout the supplychain.Product Postponement: TheProgrammable Logic Devices

Before recent developments in program-mable logic, logic devices were primarilyASICs in which the logic was built in dur-ing wafer fabrication. Typically, the OEMcustomer would design an ASIC as part of

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a larger design of the system board onwhich the ASIC would be mounted. TheOEM customer submitted a design for theASIC to a semiconductor manufacturer,who fabricated a prototype of the deviceaccording to design specifications. Thecharacteristics of ASICs were fully deter-mined during fabrication, and hence theOEM customer receiving an ASIC coulduse it only for the intended design. Yet,because of changes in the system specifica-tions or design flaws, design iterationswere very common in such product-development projects in the high-technology industries (Figure 2). Anychange in the design of an ASIC requiredboth modifying the semiconductor-fabrication process and manufacturing ad-ditional prototype ASICs using the modi-fied process. A change in the fabricationprocess could cost hundreds of thousandsof dollars and manufacturing prototypeASICs could take over three months. As aresult, design iterations in systems usingASICs were very time consuming[Trimberger 1994].

With programmable logic devices, theOEM customer receives a “generic” de-vice. These devices are not completely ge-neric—each type has features that cannot

be customized. Thus, once a customerchooses a generic die type, the customercan customize within a certain range ofparameters. The features that create thesehard design limits include die packaging,speed grade, maximum number of logicgates, voltage, power, maximum die inputand output, and software programmingmethodology:—The customer chooses from a set of pos-sible package types and lead-pin counts.Different packages have different thermaland protective properties and have differ-ent maximum electrical input and outputcharacteristics.—The customer chooses from a set ofspeed grades, each of which produces adifferent clock rate. Higher speeds may berequired for some applications.—The customer chooses from a set of pos-sible device sizes, specified by the numberof logic gates. The number of logic gatesdetermines the size and complexity of thelogic design that can be implemented.—The customer selects from a variety ofvoltages used to power the device (usually2.5 V, 3.3 V, or 5V).—Each generic device type has differentpower constraints.—Each generic device has different maxi-

Figure 2: When building a system using an ASIC, the manufacturer incorporates the logic whenthe integrated circuit is manufactured. Thus, the designer must wait for a new integrated circuitto be manufactured to make design changes.


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mum input and output electrical charac-teristics, for example, the maximum levelof current that the device can put out.—The customer may select a device thatuses a familiar programmingmethodology.

Although the customer must decide onsome characteristics in advance, the essen-tial characteristic of the final device, thelogic function of the device, is not definedin physical processing. Instead, the OEMcustomer programs, in minutes or hours,the programmable logic device using soft-ware running on a personal computer. Theuser downloads the information into thegeneric die and thus completes a fully cus-tomized logic device. With such a pro-grammable logic device, the process fordesigning an end system is now dramati-cally different (Figure 3). Each design iter-ation takes less time as does the overalldesign and development process.

Besides shortening the design-processtime, product postponement can improvethe operational efficiency of the supplychain by reducing the procurement leadtimes. ASIC suppliers often operate undera build-to-order system, not maintaining

finished-goods inventory (but they mayhave some in-process inventory). As a re-sult, the procurement lead times for OEMcustomers are sometimes two to threemonths long. Since accurate forecasting ofdemand at the specific ASIC device levelover such a long horizon is difficult, OEMcustomers using ASICs often keep largeinventories of the ASICs. Programmablelogic suppliers can afford to keep inven-tory in finished-goods form or in the diebank because programmable logic devicesare more generic with more predictabledemand. Thus, lead times for procuringprogrammable logic devices are in days orweeks so OEM customers who use themneed less inventory.

In-system programming (ISP) allowseven greater product postponement. Withthis capability, customers can easily pro-gram or reprogram the logic even after thedevice is installed in the system (Figure 4).For example, electronic systems such asmulti-use set-top boxes, wireless-telephonecellular base stations, communications sat-ellites, and network-management systems,can now be fixed, modified, or upgradedafter they have been installed.

Figure 3: When building a system with a programmable logic device, the customer incorporatesthe logic using software after the integrated circuit is manufactured. Thus, design changes canbe made quickly using software. In contrast to Figure 2, the steps “manufacture logic IC” and“design system” are reversed.

XILINX


Figure 4: In a system built with a programmable logic device with in-system programmingcapability, the logic can be incorporated after the system is set up with the customer.

Process Postponement: The Die Bank asthe Push-Pull Boundary

The use of product postponement al-lows Xilinx’s customers to create a near-infinite number of different products (dif-ferent logic designs) from a few thousandtypes of physical products (Xilinx finishedgoods). However, since demand for eachfinished good is usually very uncertainand manufacturing takes around threemonths, achieving excellent service withreasonable overall inventory levels was achallenge with this many different fin-ished goods. Since many of the finishedgoods use the same type of die, Xilinx rec-ognized an opportunity to implement pro-cess postponement to simultaneously re-duce inventory and increase serviceresponsiveness.

Its revised process using postponementworks as follows. Instead of using the pro-jected demands for individual finishedgoods to determine production at the frontend, Xilinx aggregates the demands forfinished goods into die demands and usesthe projected die demands to determinethe front-end production starts. After com-

pleting the front-end stage, it decides howto customize the dies into different fin-ished goods in the back-end stage. It thuspostpones product differentiation, movingit from the beginning to the end of thefront-end stage. It still bases customizationin the back-end stage on demand forecast(push), with inventory being held infinished-goods form. Thus, the push-pullboundary remains at the end of the pro-cess. Since the point of product differentia-tion moves forward but the push-pullboundary is still at the end of the process,we refer to this approach as partial post-ponement. Eppen and Schrage [1981] ini-tially proposed this approach in a multi-level distribution setting; it is equallyapplicable to this manufacturing setting.

Although partial postponement pro-vides benefits, moving the push-pullboundary to an earlier point in the processcan increase them. In full die-bank push-pull postponement, the generic dies areheld in inventory (the die bank) immedi-ately after the front-end stage, and this diebank becomes the new push-pull bound-ary. No inventory is held in finished-


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goods form; instead, the dies are custom-ized according to customer orders.

We compared die-bank push-pull post-ponement and the no-postponement ap-proach by analyzing the inventory andservice trade-off for each approach usingdata from a family of finished goods de-rived from the one die type. We assumedindependent and normally distributed de-mands and a weekly periodic review base-stock policy. For the no-postponement ap-proach, we modeled the system asindependent inventory nodes, each repre-senting a finished goods part. We calcu-lated the minimum inventory required tomeet a service constraint (maximum ex-pected back orders) for each node andsummed the inventory across nodes. For agiven level of safety stock, we estimatedthe expected back orders for each node us-ing the demand uncertainty and the plan-ning lead time [Nahmias 1993]. For thedie-bank push-pull postponement ap-proach, we modeled the system as a singleinventory node at the die bank. We esti-mated expected back orders at this nodeusing the demand uncertainty of the ag-gregated die demand. We showed that thedie-bank push-pull strategy offers signifi-cant improvements (Figure 5).

Although this die-bank push-pull post-ponement strategy offers performance im-provements, it is not acceptable for cus-tomers that require fast deliveries. Thus, ifthe back-end lead time is two weeks andthe customer needs delivery in one, Xilinxcould not meet the customer’s require-ment. Xilinx wanted to move from a par-tial postponement approach to the die-bank push-pull approach and still satisfysuch customer requirements. Thus, it has

adopted a hybrid approach. Xilinx hasbeen reducing back-end lead times, andthe times for the majority of products arenow shorter than customers usually re-quire. It builds these products for the diebank according to customer orders (thedie-bank push-pull strategy). It builds fin-ished goods with longer back-end leadtimes and shorter delivery time to forecast(the partial-postponement strategy).

To determine the distribution of inven-tory between finished goods and die bank,we used the same number of finishedgoods as in the previous analysis. We as-sumed each finished-goods part had oneof two back-end lead times: a time equalto the customer-response time and onelonger than the customer-response time(set at the average for the parts with leadtimes greater than the customer-responsetime). We increased the percent of partswith the short lead time from 0 to 100 per-cent to generate the results. To avoid con-cerns about the order in which we selected

Figure 5: The graph shows the expected num-ber of back orders as a function of the totalinventory for two approaches: the no-postponement approach and the die-bankpush-pull approach. For the same level of in-ventory investment, the expected number ofback orders is much lower under the die-bankpush-pull approach.

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Figure 6: The figure illustrates the inventorydistribution between die bank (white) andfinished goods (black) when adopting a hy-brid strategy. The horizontal axis is the pro-portion of finished goods that have back-endlead times within the customer-response-timewindow. As this proportion grows, more ofthe products can be built to order. Thus, thetotal inventory decreases significantly and themix of inventory becomes more heavilyweighted to die bank. Results are shown for aconstant service level (as measured by ex-pected back orders).

finished goods for back-end lead time re-duction, we assumed equal demands forall finished goods. So that we could useEppen and Schrage’s [1981] model to ana-lyze the partial postponement approach,we assumed all parts had the same coeffi-cient of variation.

For parts with the short back-end leadtime, we used the die-bank push-pull ap-proach and determined the minimum die-bank inventory to maintain the desiredlevel of service. For the parts with thelonger back-end lead time, we used thepartial-postponement approach. For theseparts, we determined the inventory levelsrequired for the given service level usingEppen and Schrage’s results [1981]. Theirresults are for just such a partial-postponement structure (under a differentname), and they allow us to calculate theeffective demand uncertainty as a functionof the individual finished-goods uncer-tainty levels and the front-end and back-end lead times. Using these results, wecalculate the total safety stock in finishedgoods for a maximum level of expectedbackorders.

When few parts have short lead times,we must manage most parts using thepartial-postponement approach, keepingmost inventory in finished goods. As thenumber of products with short back-endlead times increases, we can build moreparts from the die bank to meet customerorders, decreasing inventory in finishedgoods and increasing that at the die bank.The decrease in finished-goods inventoryis much more rapid than the increase indie-bank inventory. Thus, moving towardsthe pure die-bank push-pull approach re-duces inventory and dramatically reduces

cost since the cost of finished goods is 40percent more that the die cost.

Table 1 summarizes the four process-postponement approaches. The primarydriver of the benefits of process postpone-ment is the risk pooling or statistical pool-ing that occurs when aggregating de-mands for many finished goods intodemand for fewer dies. The aggregate de-mand is less uncertain, and thus the firmcan hold less inventory to provide thesame level of service. The risk-pooling ef-fect is large when the number of finishedgoods for each die type is large and thecorrelation between finished-goods de-mands is small. A large correlation be-tween two finished goods means that ifdemand is larger than expected for onefinished good, it will likely also be largerthan expected for the second finishedgood. Fortunately, at Xilinx, there are a


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StrategyPostponement ofproduct decision

Inventory atdie bank

Inventory atfinished goods

No postponement �

Partial postponement � �

Die bank push-pull � �

Hybrid � � �

Table 1: For each of four approaches to managing Xilinx’s process, the table indicates whetheror not postponement is used and where inventory is held—in the intermediate and genericform at die bank or in the final form at finished goods. Xilinx adopted the hybrid model, allow-ing it to reduce inventories and maintain a high level of customer service.

large number of finished goods for eachdie (50 to 150) and the average correlationbetween the finished goods was found tobe only 0.018.

Using postponement and holding mostinventory at die bank has a number of ad-ditional benefits. Inventory held at the diebank is less costly than that at finishedgoods. About 30 to 50 percent of eachproduct’s total value is added in the back-end stage. Inventory held at the die bankalso has a lower risk of obsolescence.Many finished goods have just a few cus-tomers. If demand drops unexpectedly,Xilinx may be left with inventory of thesegoods that it cannot sell to anyone else.Die inventory, however, has not yet beencustomized, and its flexibility greatly re-duces the risk of obsolescence. Obsoles-cence costs in the industry are often aboutfive percent of gross inventory per year,nearly all for finished goods. Postpone-ment makes inventory management easier.In practice, inventory cannot be managedsolely by a model-based system. Its deci-sions must be adjusted for issues beyondthe model’s scope. With process postpone-ment, management can focus on managingthe inventory of the 100 dies rather thantrying to make decisions for 10,000 fin-ished goods.

Implementing Process PostponementImplementing process postponement of-

ten requires redesigning current productswhile trying to keep the changes transpar-ent to the customer. Fortunately this canbe done fairly easily in high-technologymanufacturing because of the short life ofproducts. To redesign a product to enableprocess postponement, a manufacturer cansimply wait the short time until the nextproduct-generation release when manycustomers will convert their designs totake advantage of speed and pricebenefits.

Xilinx designs products to allow for theuse of process postponement, keeping thedegree of customization low through thefront-end stage. For a few general productcategories, the die options (for example,many options for logic cell count) are nu-merous but packaging options are few.Thus, process postponement providesminimal advantage, and little can be donefrom a design perspective because somefeatures (such as logic cell count) can becreated only during the front-end stage.

Xilinx has pursued three process-relatedinitiatives to make process postponementmore effective—inventory modeling,supply-mix prediction, and back-endcycle-time reduction. It uses inventory

XILINX


modeling to determine the appropriatepush-pull boundaries for finished goodsand to determine inventory levels at vari-ous stocking locations. For parts infinished-goods stock, it is optimal to keepinventory in the die bank for quick replen-ishment instead of using pure partial post-ponement. Xilinx uses inventory models toimprove the hybrid strategy and to deter-mine the optimal level of inventory tohold in the die bank to replenish finishedgoods and to fill orders for build-to-orderparts. It currently uses a multi-echelonmodel developed jointly with IBM [Ettl etal. forthcoming; Brown et al. 1999].

In the supply-mix-prediction initiative,Xilinx uses statistical models to predict thespeed mix of the die-bank inventory. Cus-tomer orders specify the desired speed. To

Xilinx reduced its inventorylevels without harmingoverall customer service.

customize dies from the die bank to meetcustomer orders or to replenish finished-goods stock, Xilinx must know how manydies are in each speed yield in the die-bank inventory. Xilinx can easily predictthe average fraction of die per wafer thatwill be of each speed. However, due toslight perturbation in the wafer-fabricationprocess, the actual fraction for each indi-vidual wafer will be different. The objec-tive of the supply-mix initiative is to pre-dict this fraction. Although the true speedof a device is not known until it completesthe assembly and test stages, Xilinx canget initial data using a test on die-bank in-ventory called wafer sort. Using this data,Xilinx applies regression and other statisti-

cal methods to estimate speed yield distri-butions quite accurately [Ehteshami andPetrakian 1998]. This knowledge enables aplanner to choose wafers from the die-bank inventory that closely match the or-der requirements, thus reducing thewasted dies and improving responsetimes.

The third initiative to improve processpostponement is a continuing process toreduce back-end lead times. Xilinx hasworked with its manufacturing partners toreduce the wafer-fabrication time fromthree to one-and-a-half months. For Xilinxto make the die bank the push-pullboundary, the back-end lead time must beshort. With a shorter back-end lead time,Xilinx can satisfy a larger proportion ofcustomer orders using the die bank as thepush-pull boundary instead of finishedgoods. Much of the back-end lead time isadministrative time. Thus, Xilinx has beenable to streamline the process and reducethe lead time through information technol-ogy and closer supplier (for assembly andtesting) involvement. Internal planningand order fulfillment systems have beenmade more responsive and electronic datainterchange or Extranet web-based toolshave been used to expedite the exchangeand processing of information betweenXilinx and its worldwide vendors.Conclusion

Xilinx has created tremendous valuesthrough product and process postpone-ment. In the case of product postpone-ment, it has found the value of ISP andIRL to be tremendous. For example,Hewlett-Packard Company used a Xilinxfield-programmable gate array, a powerfulvariety of programmable logic devices,


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when it designed the LaserJet Companion,reducing its design cycle by an estimatedsix to 12 months [Rao 1997]. For the elec-tronics industry, Reinertsen [1983] esti-mated that a six-month delay in the devel-opment time of a product reduces theprofits generated over the product’s lifecycle by a third.

Firms are only beginning to realize thepotential of product postponement. Rao[1997] describes how IBM designed asyn-chronous transfer mode (ATM) network-ing switches when the industry had notyet fully developed standards and proto-cols. Using programmable logic deviceswith ISP capabilities, it was able to deliversystems to its customers that could easilybe upgraded to the latest standards withno hardware changes. With more recenttechnological advances, firms can evenprovide these upgrades through the Inter-net for systems that are online. Villasenorand Mangione-Smith [1997] describe howFPGAs are changing the field of comput-ing, possibly resulting in major technologi-cal breakthroughs. They envision comput-ing devices that adapt their hardwarealmost continuously in response to chang-ing input. They also predict that configur-able computing is likely to play a growingrole in the development of high-performance computing systems, resultingin faster and more versatile machines thanare possible with either microprocessors orASICs. With such technology, firms canpostpone the definition of a product with-out limit, an ultimate form of productpostponement.

Process postponement has also signifi-cantly improved financial performance atXilinx. Although Xilinx has not kept per-

formance metrics since it first introducedprocess postponement, its refinement ofthe process-postponement hybrid from thethird quarter of 1996 to the third quarterof 1997 helped it to reduce corporate in-ventory from 113 dollar days to 87 dollardays (dollar days is the net inventory di-vided by the cost of goods sold for thequarter times 90 days per quarter). Thistranslates directly into cost savings andimprovements in the company’s return onassets. At the same time, customer service,measured by the percentage of times that

Gaining acceptance of themodels took time and effort.

customer orders are filled on time, has re-mained the same. This is particularly im-pressive because during that period, Xilinxreleased an unusually large number ofnew products. Despite the proliferation ofproduct variety and the increase in serviceback orders associated with technicalproblems with the new products, Xilinxreduced its inventory levels without harm-ing overall customer service. During thistime period, the inventory levels at thekey competitors increased to well over 140dollar days.

Currently, Xilinx is working closely withits partners to further reduce lead times atboth the front-end and back-end stages.Clearly, reducing front-end lead times willresult in even less safety stock needed inthe die bank; while reducing the back-endlead times will enable Xilinx to satisfymore customer orders by using the diebank as the push-pull boundary.

Implementing postponement at Xilinxrequires tremendous organizational sup-

XILINX


port. The change from stocking primarilyin finished goods to stocking primarily indie bank initially created some nervous-ness among the sales and logistics person-nel who dealt with customers’ orders. Al-though the company realized that itneeded to use scientific inventory modelsto manage inventory levels effectively,gaining acceptance of the actual modelstook time and effort. We ran extensivecomputer simulations to demonstrate theeffectiveness of the model and conductedintensive training and education programswith various functions within the com-pany to create confidence in the modeland acceptance of this new approach. Theresults showed that all these efforts wereworthwhile, and postponement is now akey part of Xilinx’s overall supply-chainstrategy.

AcknowledgmentsWe thank Chris Wire, a key figure in

driving the demand-and-supply-chaininitiative at Xilinx, for his general input.We also thank John McCarthy and DeanStrausl for their support and vision in theprojects and Donald St. Pierre for provid-ing the engineering details of in-systemprogramming for logic devices.

ReferencesBrown, A. O.; Ettl, M.; Lin, G. Y.; and

Petrakian, R. 1999, “Implementing a multi-echelon inventory system at a semiconductorcompany: Modeling and results,” IBM Wat-son Labs technical report, Yorktown, NewYork.

Dapiran, P. 1992, “Benetton—Global logistics inaction,” Asian Pacific International Journal ofBusiness Logistics, Vol. 5, No. 3, pp. 7–11.

Ehteshami, B. and Petrakian, R. 1998, “Speedyield prediction,” Working paper, Xilinx,Inc., San Jose, California.

Eppen, G. D. and Schrage, L. 1981, “Central-

ized ordering policies in a multi-warehousesystem with lead times and random de-mand,” in Multi-Level Production/InventorySystems: Theory and Practice, ed. L. B.Schwarz, North-Holland, Amsterdam andNew York.

Ettl, M.; Feigin, G. E.; Lin, G. Y.; and Yao, D. D.forthcoming, “A supply network model withbase-stock control and service requirements,”Operations Research.

Feitzinger, E. and Lee, H. L. 1997, “Mass cus-tomization at Hewlett-Packard: The power ofpostponement,” Harvard Business Review, Vol.75, No. 1, pp. 116–121.

Lee, H. L. 1993, “Design for supply chain man-agement: Concepts and examples,” in Per-spectives in Operations Management: Essays inHonor of Elwood S. Buffa, ed. R. Sarin, KluwerAcademic Publishers, Boston, Massachusetts,pp. 45–65.

Lee, H. L. 1996, “Effective inventory and ser-vice management through product and pro-cess redesign,” Operations Research, Vol. 44,No. 1, pp. 151–159.

Lee, H. L.; Billington, C.; and Carter, B. 1993,“Hewlett-Packard gains control of inventoryand service through design for localization,”Interfaces, Vol. 23, No. 4, pp. 1–11.

Lee, H. L.; Feitzinger, E.; and Billington, C.1997, “Getting ahead of your competitionthrough design for mass customization,”Target, Vol. 13, No. 2, pp. 8–17.

Lee, H. L.; Padmanabhan, V.; and Whang, S.1997, “The bullwhip effect in supply chains,”Sloan Management Review, Vol. 38, No. 3, pp.93–102.

Lee, H. L. and Sasser, M. 1995, “Product uni-versality and design for supply chain man-agement,” Production Planning and Control:Special Issue on Supply Chain Management,Vol. 6, No. 3, pp. 270–277.

Lineback, R. J. 1997, “The foundry/fablessmodel could become dominant,” Semiconduc-tor Business News, Vol. 1, No. 5, p. 1.

Nahmias, S. 1993, Production and OperationsAnalysis, second edition, Richard D. Irwin,Inc., Homewood, Illinois.

Rao, S. S. 1997, “Chips that change their spots,”Forbes, Vol. 160, No. 1, pp. 294–296.

Reinertsen, D. G. 1983, “Whodunit? The searchfor new-product killers,” Electronic Business,Vol. 9, No. 7, pp. 34–39.


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Trimberger, S. M. 1994, Field-Programmable GateArray Technology, Kluwer Academic Publish-ers, Boston, Massachusetts.

Ulrich, K. 1995, “The role of product architec-ture in the manufacturing firm,” Research Pol-icy, Vol. 24, No. 3, pp. 419–440.

Villasenor, J. and Mangione-Smith, W. H. 1997,“Configurable computing,” Scientific Ameri-can, Vol. 276, No. 6, pp. 66–71.

Whitney, D. E. 1995, “Nippondenso Co. Ltd.: Acase study of strategic product design,”Working paper, Massachusetts Institute ofTechnology, Cambridge, Massachusetts.

Randy Ong, Vice-President, Operations,Xilinx Inc., 2180 Logic Drive, San Jose,California 95124–3400, writes: “This is tocertify that the supply-chain efforts at Xil-inx as described by the authors . . . haveindeed been carried out. We have ob-served tremendous payoffs via such ef-forts, improving the efficiencies and effec-tiveness of our supply-chain andorder-fulfillment processes. As a fablesssemiconductor company, Xilinx has to relyon tight integration with our supply part-ners, distributors, and customers to re-main competitive. Demand and supply-chain management is a cornerstone of ourmanufacuring strategy, and we arepleased to see such efforts creating greatvalues for the company. I am also pleasedto report that we are continuing our ef-forts to build supply-chain excellency sothat Xilinx can become the leading edgesupply-chain company in the semiconduc-tor industry.”

1523-4614/99/0101/0077$05.00Copyright q 1999, Institute for Operations Researchand the Management Sciences

Manufacturing & Service Operations ManagementVol. 1, No. 1, 1999 77

Stock Positioning and PerformanceEstimation in Serial Production-

Transportation Systems

Guillermo Gallego • Paul ZipkinDepartment of Industrial Engineering & Operations Research, Columbia University, New York, New York 10027

The Fuqua School of Business, Duke University, Durham, North Carolina 27708

This paper considers serial production-transportation systems. In recent years, researchershave developed a fairly simple functional equation that characterizes optimal system

behavior, under the assumption of constant leadtimes. We show that the equation covers avariety of stochastic-leadtime systems as well. Still, many basic managerial issues remain ob-scure: When should stock be held at upstream stages? Which system attributes drive overallperformance, and how? To address these questions, we develop and analyze several heuristicmethods, inspired by observation of common practice and numerical experiments. One ofthese heuristics yields a bound on the optimal average cost. We also study a set of numericalexamples, to gain insight into the nature of the optimal solution and to evaluate the heuristics.(Inventory/Production; Multistage; Solutions and Heuristics)

1. IntroductionConsider a serial production-transportation system.

• There are several stages, or stocking points, ar-ranged in series. The first stage receives supplies froman external source. Demand occurs only at the laststage. Demands that cannot be filled immediately arebacklogged.

• There is one product, or more precisely, one perstage.

• To move units to a stage from its predecessor, thegoods must pass through a supply system, representingproduction or transportation activities. The cost for ashipment to each stage is linear in the shipmentquantity.

• There is an inventory-holding cost at each stageand a backorder-penalty cost at the last stage. The ho-rizon is infinite, all data are stationary, and the objec-tive is to minimize total average cost. Information andcontrol are centralized.

We focus on a basic system, where time is continu-ous, demand is a Poisson process, and each stage’s

supply system generates a constant leadtime. How-ever, virtually all the results remain valid for adiscrete-time system with i.i.d. demands, forcompound-Poisson demand in continuous time, andfor more complex supply systems with stochastic lead-times. Also, since an assembly system can be reducedto an equivalent series system (Rosling 1989), the re-sults apply there too.

Clark and Scarf (1960) initiated the analysis of thissystem, assuming discrete time with a finite horizonand nonstationary data. They showed that the optimalpolicy has a simple, structured form (an echelon base-stock policy) and developed a tractable scheme to com-pute it. Federgruen and Zipkin (1984) adapted the re-sults to the stationary, infinite-horizon setting andpointed out that the algorithm becomes simpler there.Rosling (1989) and Langenhoff and Zijm (1990) pro-vided streamlined statements of the results. Chen andZheng (1994) further streamlined the results and ex-tended them to continuous time. The outcome is afairly simple functional equation, Equation (5) in §2,

GALLEGO AND ZIPKINStock Positioning and Performance Estimation

Manufacturing & Service Operations Management78 Vol. 1, No. 1, 1999

that characterizes the optimal policy. See Federgruen(1993) for a review of this literature.

There is another, very different stream of researchon multistage systems, one that emphasizes policyevaluation. It assumes a particular policy type, usuallya base-stock policy, and estimates key performancemeasures, especially average inventories and backor-ders. Those measures are used to construct an opti-mization model, whose solution yields the best suchpolicy. The supply systems can be fairly complex, in-deed some generate stochastic leadtimes. In most casesthe performance estimates are approximations. Thesystem structure too can be more complex; in additionto series systems, the approach applies to distributionand assembly systems. This literature begins with theMETRIC model of Sherbrooke (1968). Recent contri-butions include Graves (1985), Sherbrooke (1986), andSvoronos and Zipkin (1991). Reviews can be found inNahmias (1981) and Axsater (1993). We explain in §3that, despite these differences, the solution to Equation(5) also yields the best base-stock policy for such a sys-tem, up to the approximation.

Still, many basic managerial issues concerning suchsystems remain obscure: When should stock be held atupstream stages? Which system attributes drive over-all performance, and how? To address these questions,we study several heuristic methods (§4), inspired byobservation of common practice and numerical exper-iments, including one that yields a bound on the op-timal average cost. Sensitivity analysis of this resultreveals interesting features of system behavior. Wealso study a set of numerical examples (§5), both togain insight into the nature of the optimal solution andto evaluate the heuristics.

Section 6 presents our conclusions. A key finding isthat system performance is fairly insensitive to stockpositioning, provided the overall system inventory isnear optimal. In particular, certain heuristic policieswhich concentrate stock at a few locations performquite well.

We also discuss a broader system design problem, asin Gross et al. (1981). Here, the stages are potential stor-age locations, but none have yet been built. The designproblem is to select a subset among them and then todetermine a control policy for the resulting network.There is a cost to open each facility, and such costs

appear in either the objective function or a constraint.There may be several products sharing the same fa-cilities. This is a hard problem, but several of our heu-ristics apply to it as well.

2. Base-Stock Policy Evaluation andOptimization

This section reviews the basic facts concerning policyevaluation and optimization.

2.1. StagesFor now, assume Poisson demand and constant lead-times. Denote

J 4 number of stagesj 4 stage index, j 4 1, . . . , Jk 4 demand rateL 4 supply leadtime to stage jj

L 4 total system leadtime 4 R L .j j

The numbering of stages follows the flow of goods;stage 1 is the first, and stage J is the last, where demandoccurs. The external source, which supplies stage 1, hasample stock; it responds immediately to orders.

2.2. Base-Stock PoliciesIn a single-stage system, a base-stock policy aims to keepthe inventory position constant. The target inventoryposition is a policy variable, the base-stock level, de-noted s. When the inventory position falls below s, thepolicy orders enough to raise the inventory position tos; otherwise, it does not order. Thus, once the inventoryposition hits s, orders precisely equal demands.

In a multi-stage system, there are two classes of base-stock policy, local and echelon. Although they seemquite different, the two classes are equivalent (Axsaterand Rosling 1993).

A local base-stock policy is a decentralized controlscheme, where each stage monitors its own local in-ventory position and places orders with its predeces-sor. Each stage j follows a standard, single-stage base-stock policy with parameter

s8 4 local base-stock level for stage j,j

a nonnegative integer. The overall policy is character-ized by the vector .Js8 4 (s8)j j41

An echelon base-stock policy is a centralized control



scheme. It monitors each stage’s echelon inventory (thestage’s own stock and everything downstream), anddetermines external orders and inter-stage shipmentsaccording to a base-stock policy. The policy parame-ters are

s 4 echelon base-stock level for stage j,j

also a nonnegative integer. Let . As shownJs 4 (s )j j41

by Chen and Zheng, given stationary parameters, sucha policy is optimal in either a periodic-review or acontinuous-review setting.

Given a local base-stock policy s8, an equivalent ech-elon base-stock policy has parameters .s 4 ( s8j i$j i

Conversely, starting with an echelon base-stock policys, one can construct an equivalent local policy, setting

and , where1 1 1 1s 4 min {s } s8 4 s 1 s s 4j i#j i j j j`1 J`1

. (Also, the echelon base-stock policy is1 1 J0 s 4 (s )j j41

equivalent to s.)

2.3. CostDenote

E[•] 4 expectation`[x] 4 max{0, x}

D(t) 4 cumulative demand in the interval (0, t].V[•] 4 variance

1[x] 4 max{0, 1x}

The following are state random variables inequilibrium:

I8 4 local on-hand inventory at stage jj

B8 4 local backorders at stage jj

B 4 customer backorders 4 B8J

IT 4 inventory in transit to stage j (units in j’sj

supply system)I 4 echelon inventory at stage j 4 I8 ` Rj j i.j

(IT ` I8)i i

IN 4 echelon net inventory at stage j 4 I 1 B.j j

Also, let

Dj 4

leadtime demand for stage j, a generic random variablehaving the distribution of D(Lj). The Dj areindependent.

The cost factors are

b 4 backorder penalty-cost rate4 local inventory holding-cost rate at stage jh8j

hj 4 echelon inventory holding-cost rate at stage j 4

1h8 h8 ,j j11

where .h8 4 00

The usual accounting scheme for in-transit invento-ries charges on ITj`1 as well as . We exclude suchh8 I8j j

costs, in order to facilitate comparison among policiesand systems. Thus, the total average cost, expressed inlocal terms, is

JC(s8) 4 E[R h8I8 ` bB]. (1)j41 j j

The equivalent expression in echelon terms isJC(s) 4 E[R h IN ` (b ` h8)B]j41 j j J

J1 E[R h8D ]. (2)j41 j j`1

(Here, DJ`1 4 0. The second term is necessary, becausethe first includes the usual in-transit holding cost, andE[ITj] 4 E[Dj].)

2.4. Local Policy EvaluationFor any policy s8, the equilibrium local backorder vari-ables satisfy the following recursion:

B8 4 00 (3)`B8 4 [B8 ` D 1 s8] .j j11 j j

And,

I8 4 s8 1 (B8 ` D ) ` B8. (4)j j j11 j j

(See, e.g., Graves 1985.) From these, we can computeE[B] and E[ ] and thus the average cost [Equation (1)].I8j

2.5. Echelon Policy OptimizationWe now present a method to determine an optimalechelon base-stock policy, denoted s*. This is the Clark-Scarf algorithm, essentially as stated by Chen andZheng:

Set . For j 4 J,J 1 1, . . . , 1,1C (x) 4 (b ` h8)[x]J`1 J

given Cj`1, compute

C (x) 4 h x ` C (x)j j j`1

ˆC (y) 4 E[C (y 1 D )]j j j

s* 4 argmin {C (y)}j j

C (x) 4 C (min{s*, x}). (5)j j j

At termination, set .JC* 4 C (s*) 1 E[( h8D ]1 1 j41 j j`1

This is the optimal cost.



A similar calculation can be used to evaluate anypolicy s. Just omit the optimization step, and use sj inplace of in the last step. One can show that thiss*jmethod is equivalent to Equations (2) through (4). Con-versely, one can show directly that Equation (5) opti-mizes over policies evaluated by Equations (2) through(4). This point underlies the extensions of §3. (To ourknowledge, these observations are new here.)

Recursion (5) deserves to be called the fundamentalequation of supply-chain theory. It captures the basic dy-namics and economics of serial systems. It omits much,but any more comprehensive theory must build on it.We know little about its solution, however. The re-mainder of the paper begins to investigate it.

2.6. Decreasing Holding CostsExamination of Equation (5) reveals that, for j , J, ifhj`1 # 0, then , which implies . In thiss* 4 ` s8* 4 0j`1 j

case, we can eliminate stage j, replacing Lj`1 by Lj`1

` Lj and hj`1 by hj`1 ` hj. (Rosling observes this.)Continue to eliminate stages in this way, until all theremaining hj . 0. Thus, a stage holds stock only when itis cheaper to hold it there than anywhere downstream. Thismakes sense intuitively; downstream inventory pro-vides more direct, effective protection against cus-tomer backorders than upstream inventory. The onlypossible advantage of upstream inventory is lowerinventory-holding cost.

3. Other Demand and SupplyProcesses

The same methods can be used to evaluate and opti-mize, exactly or approximately, under a variety ofother model assumptions.

3.1. Compound-Poisson DemandSuppose that demand is a compound-Poisson process,and each increment of demand can be filled separately.All the results above remain valid. Here, each Dj has acompound-Poisson distribution, but that is the onlydifference.

3.2. Exogenous, Sequential Supply SystemsConsider a system like that of Svoronos and Zipkin(1991), specialized to a series structure: Each stage’ssupply system is stochastic. Stage j’s system generates

a virtual leadtime Lj(t); a shipment to j initiated at timet arrives at t ` Lj(t). The system processes orders se-quentially, so shipments arrive in the same sequence asthe corresponding orders; that is, t ` Lj(t) is nonde-creasing in t. Each supply system is exogenous, i.e., itsinternal state and Lj(t) are stochastic processes, butthey are unaffected by shipments. Each system is er-godic, i.e., Lj(t) approaches a steady-state random vari-able Lj, regardless of initial conditions. Finally, thesesystems, and hence the Lj(t) and Lj, are independentover j.

Svoronos and Zipkin show that Equations (3)through (4) evaluates a base-stock policy. Here, Dj hasthe distribution of D(Lj), the demand over the (sto-chastic) virtual leadtime Lj, so E[Dj] 4 kE[Lj] and V[Dj]4 kE[Lj] ` k2V[Lj]. These Dj are again independent.Consequently, as explained in §2.5, Equation (5) findsthe best base-stock policy.

3.3. Independent LeadtimesReturn to the Poisson-demand case. Suppose that eachstage’s leadtimes are i.i.d. random variables; in effect,each supply system consists of multiple identical pro-cessors in parallel. Let Lj be the generic leadtime ran-dom variable for stage j. In this context, Equations (3)through (4) remain valid with ITj in place of Dj.

It is difficult to characterize the ITj, in general. Thereis one case where it is easy, namely, when s 4 s8 4 0.There, the system is equivalent to a tandem networkof queues with Poisson input, where each node j hasan infinite number of servers with service times Lj. So,ITj has the Poisson distribution with mean kE[Lj], andthe ITj are independent. (See, e.g., Kelly 1979.). For gen-eral s8 $ 0 we can use this same distribution to approx-imate the ITj. This is, in fact, the key approximationunderlying the METRIC procedure (see Sherbrooke1968, 1986 and Graves 1985), specialized to series sys-tems. It is quite accurate.

With this approximation, using Dj to stand for theapproximate ITj, Equations (3) through (4) evaluate alocal policy. Therefore, Equation (5) computes the bestbase-stock policy, up to the approximation.

3.4. Limited-Capacity Supply SystemsNow, suppose each supply system consists of a singleprocessor and its queue. The processing times at stage



j are i.i.d., distributed exponentially with rate lj. As-sume k , l [ minj {lj}. Recursion (3), with ITj in placeof Dj, applies here too. Again, it is difficult to charac-terize the ITj in general, but easy in the case s8 4 0.Here, ITj has the geometric distribution with parame-ter qj 4 k/lj, and the ITj are independent (Kelly). Thisworks well as an approximation for the general case,as shown by Buzacott et al. (1992), Lee and Zipkin(1992), and Zipkin (1995). So, Equation (5) again findsthe (approximately) best base-stock policy.

4. Bounds and Heuristics

4.1. The Restriction-DecompositionApproximation

This section presents a fairly simple way to determinea useful heuristic policy and an upper bound on theoptimal cost. The approach involves restriction of thepolicy space and decomposition of the resulting modelinto single-stage submodels. Accordingly, we call it therestriction-decomposition or RD approximation. Thisapproach, or something like it, is widely used in prac-tice. It is striking that this simple idea actually boundsthe original system.

Let J` be any subset of stages that includes J. Weconstruct an approximation for any choice of J` andthen select the best J`. Index these stages in order byj(m), m 4 1, . . . , M. So, j(M) 4 J. Also, denote j(0) 4

0. Let

D 4 D ` D ` . . . ` D , 0 # i , j # J,(i,j] i11 i`2 jmD 4 D , m 4 1, . . . , M.(j(m11),j(m)]

First, restrict 4 0, j Ó J`, so that only stages in J`s8jare allowed to hold stock. Using Equation (3), one canreadily show that

m `B8 4 [B8 ` D 1 s8 ] , m 4 1, . . . , M.j(m) j(m11) j(m)

The next steps effectively decompose the system atstages J`. It is easy to show that

m `B8 # B8 ` [D 1 s8 ] , m 4 1, . . . , Mj(m) j(m11) j(m)

M m `B 4 B8 # [D 1 s8 ]j(M) o j(m)m41

m Ì8 # [s8 1 D ] , m 4 1, , . . . , M.j(m) j(m)

Consequently,

M m ` m `C(s8) # E[h8 [s8 1 D ] ` b[D 1 s8 ] ].o j(m) j(m) j(m)m41

Equivalently, let

` 1C8(x) 4 h8[x] ` b[x]j j

ˆC (y) 4 E[C8(y 1 D )], i , j.(i,j] j (i,j]

Then,MC(s8) # C (s8 ).o (j(m11),j(m)] j(mm41

Each term in this sum is the cost of a single-stagesystem. It charges the full penalty cost b to local bac-korders at each stage j(m), while ignoring the effects ofthose backorders on downstream stages. In this senseit splits the system into separate subsystems.

Now, let s(i,j] minimize C(i,j](y), and denote the min-imal cost by . Then,C*(i,j]

MC* # C* .o (j(m11),j(m)]m41

This relation holds for any J`. To find the best suchbound over all possible J`, consider the following net-work: The nodes are {0,1, . . . ,J}, the arcs are (i,j), i , j,and the arc lengths are . The best bound, then, isC*(i,j]the length of the shortest path from 0 to J. This problemhas precisely the same structure as the dynamic eco-nomic lot-size problem of Wagner and Whitin (1958),and can be solved using the same algorithm.

From the best J` one can construct a plausible heu-ristic policy: Set 4 0, j Ó J`, and for j 4 j(m) [ J`,s8jset 4 s(j(m11),j(m)]. The actual cost of this policy is nos8jmore than the upper bound. (Alternatively, use Equa-tion (5) to find the optimal policy for the system re-stricted to J`. We have not tested this more refinedapproach.)

The RD approximation extends directly to the de-sign problem: If there is a fixed cost kj to build stage j,just add kj to each . Also, if several products shareC*(i,j]the network, compute the for each product, andC*(i,j]then sum them over the products. The algorithm abovethen provides a heuristic solution and an upper bound.

We remark that the complexity of the RD heuristicappears to be O(J2), compared to O(J) for the optimiz-ing algorithm Equation (5). Indeed, we have observedthat, for very large J, the heuristic can take longer thanEquation (5). For smaller, plausibly-sized systems,however, the heuristic is usually much faster. And, itis a tractable method for the design problem.



Here is a further useful approximation: Scarf (1958)and Gallego and Moon (1993) show that

1/2 `C* # (bh8) r [ C ,(i,j] j (i,j] (i,j]

where r(i,j] is the standard deviation of D(i,j]. Using thein the calculations above yields a distribution-free`C(i,j]

bound, one that depends only on two moments ofleadtimes and demands, not their actual distributions.Call this the maximal RD approximation. The same anal-ysis yields a heuristic solution of the form 4`s(i,j]

E[D(i,j]] ` r(i,j], where is a safety factor dependingz8 z8j j

on b and , whose cost is no more than the upperh8jbound. (This approach is much faster than the originalRD heuristic, since is easier to compute than .)`C C*(i,j] (i,j]

This simple form facilitates sensitivity analysis: Ob-serve that, in the Poisson-demand, constant-leadtimecase, each depends on k through a factor . Thus,`C k!(i,j]

the shortest path is independent of k, and the costbound is proportional to . That is, the heuristic’s choicek!of stocking points is independent of the demand volume, andthe true optimal cost is bounded above by a function pro-portional to . A similar analysis of suggests that`k s! (i,j]

the overall safety stock is proportional to . The samek!is true for stochastic, independent leadtimes (§3.3). Forexogenous, sequential leadtimes (§3.2), however, r(i,j]

4 (kE[L(i,j]] ` k2V[L(i,j]])1/2, so the optimal cost isbounded by a linear function of k, as is the safety stock.

Likewise, the shortest path is independent of b, andthe cost bound is proportional to .b!

The leadtime Lk affects r(i,j] for all (i,j] with i , k #

j. It has the biggest impact on the r(i,j] for short intervals(i,j] around k. Thus, for small k, Lk has a major impactonly on terms with small j and hence low . Con-`C h8(i,j] j

versely, Lk for large k affects terms with large . Thish8jsuggests that downstream leadtimes have a greater impacton system performance than upstream ones.

The familiar normal approximation yields an ap-proximation to of the same form as , namely,`C* C(i,j] (i,j]

a factor depending on the cost parameters, times r(i,j].It also yields a solution of the same form as . Call`s(i,j]

this the normal RD approximation. So, the observationsabove about k and the Lk remain valid. The cost factors,however, grow more slowly in b than .b!

Some additional bounds for two-stage systems canbe found in Gallego and Zipkin (1994).

4.2. The Zero-Safety-Stock HeuristicThis approach (the ZS heuristic, for short) sets 4s8jE[Dj], j , J, and then optimizes over . More precisely,s8Jto cover the case of non-integral E[Dj], the heuristic sets

, j , J. Then, using Equations8 4 ( E[D ] 1 s8j i#j i j11

(3), it computes the distribution of . Finally, itB8J11

chooses to minimize the stage-J holding and penaltys8Jcosts, a single-stage problem. (This method was in-spired by some preliminary numerical results, inwhich the optimal was near E[Dj], j , J.) Evidently,s8jthis is an O(J) calculation, and it is very fast in practice.

4.3. The Two-Stage HeuristicThis approach (the TS heuristic) restricts inventory totwo stages, the last one J and some single j , J. Givenj , J, it finds the optimal policy for the resulting two-stage system. It then selects the best such policy over j, J. (This method too was based on empirical obser-vations, namely, that restricting the number of loca-tions sometimes has little cost impact.)

This technique requires solving J 1 1 two-stageproblems, nearly as much work as the full optimiza-tion algorithm Equation (5). The purpose of the heu-ristic is not speed. Rather, it is a tool to investigatestock-positioning issues: Where is stock most useful?And, how costly is the restriction to two stages? Thisapproach also extends easily to the design problem; inthat context it is a plausible heuristic for systems withlarge fixed facility costs.

5. Numerical ResultsThis section presents some numerical examples, to pro-vide insight into the behavior of the optimal policy andthe performance of the heuristics.

5.1. Specification

5.1.1. System Structure and Parameters. We as-sume Poisson demand and constant leadtimes. With-out loss of generality, we fix the time scale so that thetotal leadtime is L 4 1, and the monetary unit so thatthe last stage’s holding cost is 4 1. The stages areh8Jspaced symmetrically, so each stage j’s leadtime is Lj

4 1/J. We consider four numbers of stages, J 4 1, 4,16, 64; two demand rates, k 4 16, 64; and two penaltycosts, b 4 9, 39 (corresponding to fill rates of 90%,97.5%).



Figure 2 Optimal Policy: Linear Holding Costs

Figure 1 Holding Cost Forms

5.1.2. Holding Cost Forms. We consider severalforms of holding costs , depicted in Figure 1. Theh8jsimplest form has constant holding costs, where all h8j4 1. Here, there is no cost added from source to cus-tomer. This is a rather unrealistic scenario, but it is auseful starting point to help understand other forms.The linear holding-cost form has 4 j/J, or hj 4 1/J.h8jHere, cost is incurred at a constant rate as the productmoves from source to customer. This is quite realistic.Affine holding costs, where 4 a ` (1 1 a)j/J forh8jsome a [ (0,1), are even more realistic. Here, the ma-terial at the source has some positive cost, and the sys-tem then adds cost at a constant rate. This form is acombination of the constant and linear forms. In Figure1 and the calculations below, a 4 0.75.

The last two forms represent deviations from linear-ity. The kink form is piecewise linear with two pieces.The system incurs cost at a constant rate for a while,but at some point shifts to a different rate, which re-mains constant from then on. Here, the kink occurshalfway through the process, at stage J/2. So, for somea [ (11,1), hj 4 (1 1 a)/J, j # J/2, and hj 4 (1 ` a)/J, j . J/2. Again, we set a 4 0.75. Finally, in the jumpform, cost is incurred at a constant rate, except for onestage with a large cost. Here, the jump occurs just afterstage J/2. So, hj 4 a ` (1 1 a)/J, j 4 J/2 ` 1, and hj

4 (1 1 a)/J otherwise, for some a [ (0,1). We canview this as linear cost before J/2 and affine cost after.Here again, a 4 0.75.

5.2. Optimal Policy

5.2.1. Constant Holding Costs. The optimal policyin this case is simple: For j , J, 4 0; only the lasts8*j

stage carries inventory. Stage J, in effect, becomes asingle-stage system with leadtime L. The optimal pol-icy is the same for all J. This is also the optimal policyfor J 4 1 under any other holding-cost form.

5.2.2. Linear Holding Costs. Figure 2 shows theoptimal policy s* for J 4 64 and two values each of k

and b. Several observations are worth noting: Thecurves are smooth and nearly linear; the optimal policydoes not lump inventory in a few stages, but ratherspreads it quite evenly. The departures from linearityare interesting too: The curves are concave. Thus, thepolicy focuses safety stock at stages nearest thecustomer.

5.2.3. Affine Holding Costs. Figure 3 shows theoptimal policy. For j . 1, the curves follow the samepattern as in Figure 2. (Indeed, the curves for b 4 9here are identical to those for linear costs and b 4 39,because these two cases have identical ratios hj/(b `

), j . 1.) However, the curves break down sharply ath8jj 4 1 (because h1 is large). Therefore, the equivalentpolicy s1 is flat for small j, and so the policy holds noinventory at early stages. This solution is intermediatebetween those for constant and linear costs. As a in-creases and the costs move upwards, stocks shift to-ward the customer. The total system stock decreases



Figure 6 Optimal Policy: Effects of J

Figure 5 Optimal Policy: Jump Holding Costs

Figure 4 Optimal Policy: Kink Holding Costs

Figure 3 Optimal Policy: Affine Holding Costs

slightly. But, perhaps surprisingly, stocks near the cus-tomer actually increase.

5.2.4. Kink Holding Costs. Figure 4 displays s*.Downstream from the kink (before Algorithm (5) en-counters it), the curves exhibit the same pattern as inthe linear case. Upstream from the kink, the policyagain follows the linear pattern, almost as if the kinkwere the last stage. The net result is substantial stockat and just before the kink, where holding costs are lowrelative to later stages.

5.2.5. Jump Holding Costs. Figure 5 displays s*.From the jump on, the policy behaves much as in the

affine case: smooth, concave decrease beyond thejump, but a sharp break downwards at the jump. Up-stream from the jump, the policy again follows the pat-tern of the linear case. Thus, there is substantial stockjust before the jump and none just after it.

5.3. Sensitivity Analysis

5.3.1. Number of Stages. Figure 6 compares the s*for different Js, each with linear holding costs, k 4 64,and b 4 39. The curves follow the same patterns asbefore, as closely as the restricted numbers of stagesallow. Indeed, the actual echelon stock at a stockingpoint is nearly identical to the J 4 64 case. Closer in-spection shows that the total system stock is slightly



Figure 8 Optimal Cost: Kink Holding Costs

Figure 7 Optimal Cost: Linear Holding Costs

higher for larger J. Likewise, the optimal cost decreasesin J, but quite slowly, as shown in Figure 7.

Similar results hold for affine holding costs. Indeed,the optimal cost is even less sensitive to J. For kinkholding costs (Figure 8), the optimal cost is signifi-cantly lower at J 4 4 than at J 4 1, due to the avail-ability of the low-cost stocking point at the kink. LargerJs yield relatively minor improvements. The jumpform displays a similar pattern. Thus, for these twoforms, it is important to position stock at the kink (orjump). Otherwise, the cost is quite insensitive to J.

These results suggest that the system cost is relatively

insensitive to stock positioning, provided the overallstock level is about right, and obvious low-cost stock-ing points are exploited. We shall see further evidencefor this below.

5.3.2. Demand Rate. In Figures 7 and 8 the opti-mal cost for k 4 64 is about twice that for k 4 16 inevery case. This is consistent with the notion that theoptimal cost is nearly proportional to , as suggestedk!in §4.1. We have also plotted, but omit here, the cu-mulative safety stocks . The curves for( s8* 1 kj/Ji#j j

k 4 64 are about twice those for k 4 16. So, the safetystocks too are nearly proportional to k.!

5.3.3. Backorder Cost. The figures above indicatethat the base-stock levels and optimal cost are increas-ing in b. The policy, however, is not very sensitive tob. The cost, though rather more sensitive, grows con-siderably slower than , as suggested by the normalb!RD approximation.

5.3.4. Leadtimes. Figures 7 and 8 provide someevidence for the notion that downstream leadtimes aremore important than upstream ones. Starting with lin-ear holding costs, contract the downstream leadtimesand expand the upstream ones, keeping L and the h8jfixed. The result looks much like the kink form with a

{ (0,1). And, the kink form has lower optimal cost forJ . 1.

5.4. Performance of Bounds and Heuristics

5.4.1. The RD Approximation. Figure 9 shows thepolicies chosen by the RD heuristic in one case (J 4 64,k 4 64, b 4 39) for all four holding-cost forms. (Thesame policy is chosen for the kink and jump forms.)These policies are quite different from the correspond-ing optimal ones; they concentrate stock in just a fewstages. For the linear form, the policy places a smallinventory near the source (9 units at stage 3) and alarge one (77) at the last stage. For the affine form, thepolicy is even more extreme, placing all its stock (80)at the end. For the kink and jump forms, the policyplaces substantial inventory (46) at stage 32, just beforethe cost increase, a little near the source (9 at stage 2),and the rest (44) at the end. Also, the total systemstocks are slightly larger than optimal. The results forother J, k, and b are similar.



Figure 9 RD Heuristic’s Policies Table 1 Heuristics’ Percentage Costs over Optimal

Form RD ZS TS

Linear 10–20% 2–8% 4–11%Affine 1–3% 3–14% 0–2%Kink 9–22% 11–25% 5–17%Jump 5–7% 11–15% 1–3%

Even so, the RD heuristic and the cost bound per-form fairly well. Table 1 shows the percentage errorsfor all three heuristics. For example, for the linear form,the RD policy’s cost exceeds the optimal by 10%–20%.(The errors tend to increase slowly in J, k, and b.) Theseerrors are far smaller than the cost differences betweensystems. The cost bound is usually just a bit more thanthe actual heuristic policy’s cost.

Thus, the RD approximation provides crude but ro-bust estimates of system performance. It is certainlyaccurate enough for rough-cut design studies. Thisfact, coupled with the gross differences between theRD and optimal policies, is further evidence of the in-sensitivity of performance to stock positioning.

5.4.2. The ZS Heuristic. The ZS heuristic, by def-inition, sets to the average leadtime demand up tos8jthe last stage. It sets larger than the optimal policys8jdoes, to compensate for the lower stocks at earlierstages. It generates the same policy for all four costforms. It works very well for linear holding costs,rather less well for affine costs, and not so well for thekink and jump forms.

5.4.3. The TS Heuristic. For J 4 64, for linearcosts, the TS heuristic places stock just past the middleof the system, in addition to stage J. Specifically, for k

4 64 and b 4 39, it chooses j 4 36. For affine costs,the heuristic places stock further downstream, at j 4

48. (The locations are just slightly different for theother k and b.) For the kink and jump forms, it selectsj 4 32, just before the cost increase, in all cases. Theresults are similar for smaller J. As Table 1 indicates,this method performs quite well; it is the best amongthe three heuristics. This is yet more evidence of theinsensitivity of performance to stock positioning.

6. ConclusionsWe have seen that the optimal policy depends on thegrowth of holding costs between source and customer.For constant costs, the policy puts all stock at the laststage. For linear costs, the policy distributes stock quiteevenly, though favoring downstream sites. In othercases the policy can be understood as a systematiccombination and variation of these patterns.

On the other hand, although it is important to opti-mize the system-wide inventory and to exploit espe-cially low holding costs, system performance is oth-erwise fairly insensitive to stock positioning. One candeviate substantially from the optimal policy for arather small cost penalty, as in the restriction to smallerJ and the heuristics. In particular, the RD and TS heu-ristics work fairly well; they capture the gross behaviorof the optimal policy, though differing substantially indetail. Consequently, they are reasonable heuristics forthe design problem.

The sensitivity of the system to its parameters issimilar in many ways to the familiar single-stage sys-tem. For instance, with constant leadtimes, the optimalcost and safety stocks increase as the square root of thedemand rate. Multistage systems have certain addi-tional characteristics, however. For example, down-stream leadtimes have greater impacts on performancethan upstream ones.

We have presented these results to several groups of



managers in different industries. Their reactions areworth reporting. They showed considerable interest inthe forms of the figures as diagnostic devices. For ex-ample, they wanted to plot their own holding costs inthe style of Figure 1, to see where cost accrues quicklyand where slowly. (This type of diagram is called atime-cost profile by Fooks (1993) and Schraner (1994).Observe that the in-transit holding cost is essentiallythe area under each curve.) Likewise, a plot of actualinventories in the manner of Figures 2 through 5 is aconvenient way to see just where stock is concentrated.

Many managers at first resisted the notion that stockshould be concentrated close to customers. After all,the downstream sites are the most expensive ones. But,following discussion of the sites’ different degrees ofstockout protection, as in §2.6, most agreed that theoptimal policy was at least plausible. Several notedthat their own firms’ stock-positioning policies werequite different, and planned to investigate the alter-native suggested by the model. Similarly, many hadembraced the idea of reducing total leadtime, and weredubious that downstream leadtimes could be more im-portant than upstream ones. Once the logic was ex-plained, however, they accepted it.

Finally, none of the managers found it hard to be-lieve that the heuristics perform well. Indeed, they pre-ferred solutions that concentrate stock in only a fewlocations, and they appreciated the simplicity of theheuristics. Their experience suggested that all real sys-tems incur some fixed costs, as in the design problem.

Several questions remain: Are there better heuris-tics? Do the results extend to more complex systems,such as distribution systems and systems with fixedorder costs? These are subjects of ongoing research.1

ReferencesArrow, K., S. Karlin, H. Scarf, eds. 1958. Studies in the Mathematical

Theory of Inventory and Production. Stanford University, Stan-ford, CA.

Axsater, S. 1993. Continuous review policies for multi-level inven-tory systems with stochastic demand. S. Graves, A. RinnooyKan, P. Zipkin, eds. Logistics of Production and Inventory. Elsevier(North-Holland), Amsterdam, The Netherlands. Chapter 4.

——, K. Rosling. 1993. Installation vs. echelon stock policies for mul-tilevel inventory control. Management Sci. 39 1274–1280.

1We are grateful to Jing-Sheng Song for helpful comments on earlierversions of this paper.

Buzacott, J., S. Price, J. Shanthikumar. 1992. Service level in multi-stage MRP and base stock controlled production systems. G.Fandel, T. Gulledge, A. Jones, eds. New Directions for OperationsResearch in Manufacturing. Springer, Berlin, Germany.

Chen, F., Y. Zheng. 1994. Lower bounds for multi-echelon stochasticinventory systems. Management Sci. 40 1426–1443.

Clark, A., H. Scarf. 1960. Optimal policies for a multi-echelon inven-tory problem. Management Sci. 6 475–490.

Federgruen, A. 1993. Centralized planning models for multi-echeloninventory systems under uncertainty. S. Graves, A. RinnooyKan, P. Zipkin, eds. Logistics of Production and Inventory. Elsevier(North-Holland), Amsterdam, The Netherlands. Chapter 3.

——, P. Zipkin. 1984. Computational issues in an infinite-horizon,multiechelon inventory model. Oper. Res. 32 818–836.

Fooks, J. 1993. Profiles for Performance. Addison-Wesley, New York.Gallego, G., I. Moon. 1993. The distribution-free newsboy problem:

Review and extensions. J. Oper. Res. Soc. 44 825–834.——, P. Zipkin. 1994. Qualitative analysis of multi-stage production-

transportation systems: Stock positioning and performance es-timation. Working paper, Columbia University, New York.

Graves, S. 1985. A multi-echelon inventory model for a repairableitem with one-for-one replenishment. Management Sci. 31 1247–1256.

——, A. Rinnooy Kan, P. Zipkin, eds. 1993. Logistics of Production andInventory. Handbooks in Operations Research and Manage-ment Science, Volume 4, Elsevier (North-Holland), Amster-dam, The Netherlands.

Gross, D., R. Soland, C. Pinkus. 1981. Designing a multi-product,multi-echelon inventory system. L. Schwarz ed. Multi-Level Pro-duction/Inventory Control Systems: Theory and Practice. North-Holland, Amsterdam, The Netherlands. Chapter 1.

Kelly, F. 1979. Reversibility and Stochastic Networks. Wiley, New York.Langenhoff, L., W. Zijm. 1990. An analytical theory of multi-echelon

production/distribution systems. Statist. Neerlandica 44 3, 149–174.

Lee, Y., P. Zipkin. 1992. Tandem queues with planned inventories.Oper. Res. 40 936–947.

Nahmias, S. 1981. Managing reparable item inventory systems: Areview. L. Schwarz ed. Multi-Level Production/Inventory ControlSystems: Theory and Practice. North-Holland, Amsterdam, TheNetherlands. Chapter 13.

Rosling, K. 1989. Optimal inventory policies for assembly systemsunder random demands. Oper. Res. 37 565–579.

Scarf, H. 1958. A min-max solution of an inventory problem. K.Arrow, S. Karlin, H. Scarf, eds. Studies in the Mathematical Theoryof Inventory and Production. Stanford University, Stanford, CA.Chapter 12.

Schraner, E. 1994. Optimal production operations sequencing. Work-ing paper, Stanford University, Stanford, CA.

Schwarz, L., ed. 1981. Multi-Level Production/Inventory Control Sys-tems: Theory and Practice. North-Holland, Amsterdam, TheNetherlands.

Sherbrooke, C. 1968. METRIC: A multi-echelon technique for recov-erable item control. Oper. Res. 16 122–141.



——. 1986. VARI-METRIC: Improved approximations for multi-indenture, multi-echelon availability models. Oper. Res. 34 311–319.

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Accepted by Stephen Graves; received November 18, 1996. This paper has been with the authors 8 months for 4 revisions.1o2

1523-4614/99/0102/0089$05.00Copyright � 1999, Institute for Operations Researchand the Management Sciences

Manufacturing & Service Operations ManagementVol. 1, No. 2, 1999, pp. 89–111 89

Quantity Flexibility Contracts and SupplyChain Performance

A. A. Tsay • W. S. LovejoyDepartment of Operations & Management Information Systems, Leavey School of Business, Santa Clara University,

Santa Clara, California 95053-0382School of Business Administration, University of Michigan, Ann Arbor, Michigan 48109-1234

The Quantity Flexibility (QF) contract is a method for coordinating materials and infor-mation flows in supply chains operating under rolling-horizon planning. It stipulates a

maximum percentage revision each element of the period-by-period replenishment scheduleis allowed per planning iteration. The supplier is obligated to cover any requests that remainwithin the upside limits. The bounds on reductions are a form of minimum purchase com-mitment which discourages the customer from overstating its needs. While QF contracts arebeing implemented in industrial practice, the academic literature has thus far had little guid-ance to offer a firm interested in structuring its supply relationships in this way. This paperseeks to address this need, by developing rigorous conclusions about the behavioral conse-quences of QF contracts, and hence about the implications for the performance and design ofsupply chains with linkages possessing this structure. Issues explored include the impact ofsystem flexibility on inventory characteristics and the patterns by which forecast and ordervariability propagate along the supply chain. The ultimate goal is to provide insights as towhere to position flexibility for the greatest benefit, and how much to pay for it.(Supply Chain Management; Supply Contracts; Quantity Flexibility; Forecast Revision; MaterialsPlanning; Bullwhip Effect)

1. IntroductionMany modern supply chains operate under decentral-ized control for a variety of reasons. For example, out-sourcing of various aspects of production is currentlya popular business model in many industries (cf.Farlow et al. 1995, Iyer and Bergen 1997), which au-tomatically distributes decision-making authority.Even for highly vertically integrated firms, today’scharacteristically global business environments oftenresult in multiple sites worldwide working together todeliver product, while reporting to different organi-zational functions or units within the corporation. Op-erational control of these sites may be intentionallydecentralized for informational or incentive consider-ations. However, decentralization is not without risks.For expository purposes, we describe some of these in

the context of the single-product, serial supply chaindepicted in Figure 1. Each node represents an inde-pendently managed organization, and each pair ofconsecutive nodes is a distinct supplier-buyerrelationship.To reconcile manufacturing/procurement time-lags

with a need for timely response, agents within suchsupply chains often commit resources to productionquantities based on forecasted, rather than realized de-mand. A period-by-period replenishment schedule(e.g., six months’ worth of monthly volume estimates)is a common format by which many firms communi-cate information about future purchases to their sup-ply partners. Rolling horizon updating is a standardoperational means of incorporating new informationas it accrues over time. For example, each period the

TSAY AND LOVEJOYQuantity Flexibility Contracts

Manufacturing & Service Operations Management90 Vol. 1, No. 2, 1999, pp. 89–111

Figure 1 Decentralized Supply Chain

retailer creates a forecast of the uncertain and poten-tially non-stationary market demand e.g., [100, 120,110, . . .] where the 100 denotes the current period’s de-mand, 120 is an estimate of the next period’s demand,and so on. Based on this, the retailer provides to themanufacturer a schedule of desired replenishments,e.g., [50, 150, 90, . . .], where the numbers may differfrom the market forecast due to whatever inventorypolicy the retailer may use, and any stock carried overfrom the previous period. The manufacturer treats thisschedule as its own “demand forecast” and in turn cre-ates a replenishment schedule for the parts supplier tofill, and so on. This information flow is represented bythe dotted lines in Figure 1. We assume that each partyknows only the schedule provided by its immediatecustomer, and is only concerned with its own costperformance.Such estimates are intended to assist an upstream

supplier’s capacity and materials planning. However,buyers commonly view them as a courtesy only, andindeed craft the supply contracts to preserve this po-sition. To some buyers this presents an opportunity toinflate these figures as a form of insurance, only to laterdisavow any undesired product (cf. Lee et al. 1997). Acareful supplier must then deflate the numbers toavoid over-capacity and inventory. This game of mu-tual deception may be individually rational given the

circumstances, but increases the uncertainties andcosts in the system (cf. Magee and Boodman 1967,Lovejoy 1998).Various remedies to this well-known inefficiency

have been attempted, a number of which are noted in§2. One approach that has become popular in manyindustries is the Quantity Flexibility (QF) contract,which attaches a degree of commitment to the forecastsby installing constraints on the buyer’s ability to revisethem over time. The extent of revision flexibility is de-fined in percentages that vary as a function of the num-ber of periods away from delivery. This is made con-crete in Figure 2.Since individual nodes share common structure and

wemay wish to consider chains of considerable length,we use common variable names for node attributeswherever possible, and associate them with specificparties via superscripts (P,M, and R in the example inFigure 2).At each time period, indexed by t, the period-by-

period stochastic market demand is described by {l(t)}� [l0(t), l1(t), l2(t), . . .], where

l (t) � actual market demand occurring in0

period t (1)

l (t) � estimate of period (t � j) demand,j

for each j � 1.



Figure 2 Decentralized Supply Chain with QF Contracts

The statistical structure of this process is known to theretailer, who incorporates it into supply planning. Theretailer in turn provides the manufacturer with a re-plenishment schedule vector {r(t)}R � [r0(t), r1(t),r2(t), . . .]R, where

r (t) � actual purchase made in period t (2)0

r (t) � estimate of purchase to be made inj

period (t � j), for each j � 1.

This becomes the upstream supplier’s release schedulevector, denoted { f(t)}M � [ f0(t), f1(t), f2(t), . . .]M, where

f (t) � quantity sold in period t (3)0

f (t) � estimate of quantity to be sold inj

period (t � j), for each j � 1.

Thus far we have simply formalized the informationflow described in Figure 1. Next, we consider the QFcontract between each pair of nodes. Themanufacturer-retailer QF contract is parametrized by(�, x), where � � [�1, �2, . . .] and x � [x1, x2, . . .].This places bounds on how the retailer may revise{r(t)}R going forward in time. Specifically, for each tand j � 1:

[1 � x ]r (t) � r (t � 1) � [1 � � ]r (t). (4)j j j�1 j j

That is, the estimate for future period (t � j) cannotbe revised upward by a fraction of more than �j ordownward by more than xj. Contingent on this, thecontract stipulates that the retailer’s eventual orderswill all be filled with certainty.1

1It is natural to expect that any reasonable flexibility agreementshould be such that the interval bounding a given future period’spurchase becomes progressively smaller as that period approaches.Although not readily apparent from Equation (4), the QF arrange-ment has this feature. For instance, according to Equation (4), inplanning for period (t � 2) the retailer’s period t estimate r2(t) con-strains the period (t � 1) estimate by

[1 � x ]r (t) � r (t � 1) � [1 � � ]r (t).2 2 1 2 2

In turn, by another application of Equation (4), r1(t � 1) is knownto constrain the eventual purchase r0(t � 2) by

[1 � x ]r (t � 1) � r (t � 2) � [1 � � ]r (t � 1).1 1 0 1 1

Together these define from the period t perspective the windowwithin which the eventual purchase must fall:

[1 � x ][1 � x ]r (t) � r (t � 2) � [1 � � ][1 � � ]r (t).1 2 2 0 1 2 2

Hence, the window bounding the actual purchase evolves from [(1� x1)(1 � x2)r2(t), (1 � �1)(1 � �2)r2(t)] to [(1 � x1)r1(t � 1), (1 �

�1)r1(t � 1)]. Assuming Equation (4) is observed, the latter window



Because { f(t)}M � {r(t)}R, Equation (4) means themanufacturer can be sure that revisions to estimates ofits “demand” will obey

[1 � x ] f (t) � f (t � 1) � [1 � � ] f (t) (5)j j j�1 j j

and is contractually obligated to support the resultingsequence of purchases. The manufacturer in turnpasses a replenishment schedule, denoted {r(t)}M, to itsown supplier. This will obey constraints analogous toEquation (4) above, except with flexibility parameters

Thus the parts supplier knows that revisions to(�, x).{ f(t)}P will stay within the bounds, and in turn(�, x)passes upstream the replenishment schedule {r(t)}P

(staying within the bounds), and so on. This ex-(�, x)ercise is repeated each period, with all estimates up-dated in rolling-horizon fashion.QF contracts are intended to provide a benefit to

each party. The supplier formally guarantees the buyera specific safety cushion in excess of estimated require-ments. In return, the buyer agrees to limit its orderreductions, essentially a form of minimum purchaseagreement. In this way the buyer accepts some of thedownside demand risk which, were forecasts com-pletely divorced of commitment, would be left to thesupplier. Mutual agreement on the significance of fore-casts improves the planning capabilities of both par-ties. Any favoritism expressed by this arrangement canbe mitigated in setting the flexibility limits, as we willdemonstrate.The emergence of QF contracts as a response to cer-

tain supply chain inefficiencies is described in Lee etal. (1997). Sun Microsystems uses QF contracts in itspurchase of monitors, keyboards, and various other

(one period prior to purchase) is contained entirely in the former(two periods prior). More generally, requiring Equation (4) at everyrevision generates a sequence of nested intervals that ultimately con-verge to the actual purchase. This will become clear when, in §3, weformalize this “cumulative” perspective on the flexibility terms ofthe contract, taking an alternative view of the per-period incrementalflexibilities in Equation (4). Both representations have been observedin industry. The incremental form would be preferred by a buyer,since this constrains the successive updating of its replenishmentschedules. The cumulative form would be used by a supplier, sincethis renders future capacity needs more transparent. But as theseforms are mathematically equivalent, our results apply equally wellto each.

workstation components (cf. Farlow et al. 1995).Nippon Otis, a manufacturer of elevator equipment,implicitly maintains such contracts with Tsuchiya, itssupplier of parts and switches (cf. Lovejoy 1998). So-lectron, a leading contract manufacturer for many elec-tronics firms, has recently installed such agreementswith both its customers and its rawmaterials suppliers(Ng 1997), implying that benefits may accrue to eitherend of such a contract. QF-type contracts have alsobeen used by Toyota Motor Corporation (Lovejoy1998), IBM (Connors et al. 1995), Hewlett Packard, andCompaq (Faust 1996). A similar structure, called a“Take-or-Pay” provision, is often embedded in long-term supply contracts for natural resources (cf. Mastenand Crocker 1985, Mondschein 1993, National EnergyBoard 1993). In addition to being used to govern re-lations between separate companies, QF structureshave also appeared at the interface between the manu-facturing and marketing/sales functions (taking therole of supplier and buyer, respectively) within singlefirms (cf. Magee and Boodman 1967).While QF contracts are being implemented in in-

dustrial practice, the academic literature has thus farhad little guidance to offer a firm interested in struc-turing its supply relationships in this way. This paperseeks to address this need, by pursuing the followingobjectives: (a) to provide a formal framework for theanalysis of such contracts, with explicit considerationof the non-stationarity in demand that drives the desirefor flexibility; (b) to propose behavioral models, i.e.,forecasting and ordering policies, for buyers who aresubject to such constraints in their procurement plan-ning, and for suppliers who promise such flexibility totheir customers; and (c) to link these behaviors to localand systemwide performance (e.g., inventory levelsand order variability), and therefore guide the nego-tiation of contracts. In the following discussion, ourintent is not necessarily to advocate the QF contract,but to provide conclusions about the implications ofits usage.Section 2 positions this paper in the literature. Sec-

tions 3 and 4 introduce the modeling primitives. Wewill analyze complex systems such as the one in Figure2 by decomposing the supply chain into modules ofsimpler structure. All interior nodes, meaning those



which have QF contracts on both their input and out-put sides, can be represented by one node type. Herewe will derive a reasonable inventory policy that rec-onciles the constraints and the commitments impliedby the input and output flexibility profiles. Anothernode type represents the node at the market interface,which has a QF contract on its input side only, but hasstatistical knowledge about demand on its output side.Here we will suggest an ordering policy that takes intoaccount the market demand dynamics, the relativecosts of holding and shortage, and the input-side flex-ibility parameters. The decision problems of each nodetype are formidable due to the large number of deci-sion variables and the statistical complexity of cus-tomer ordering, so we will utilize heuristic policies.This enables us to explore in §5 the performance prop-erties of supply chains controlled with QF contracts.We investigate the implications of flexibility character-istics for both inventory and service, as well as howorder variability propagates along the supply chain.Once these relationships are established, the issue ofcontract design, i.e., the choice of flexibility parame-ters, may be pursued. In particular, §6 examines thevalue of flexibility in the supply chain. We conclude in§7 with discussion of these results and implementationissues. For clarity of exposition, all proofs are deferredto Appendix 1.

2. Literature ReviewIt is not generally the case that a supply chain com-posed of independent agents acting in their own bestinterests will achieve systemwide efficiency, often dueto some incongruence between the incentives faced lo-cally and the global optimization problem. In oursingle-product setting in which the only uncertainty isin the market demand and the only decision is productquantity, this is because overstock and understockrisks are visited differently upon the individualparties.One response is to reconsider the nature of the sup-

ply contracts along the chain. (See Tsay et al. (1999) fora recent review.) The general goal is to install rules formaterials accountability and/or pricing that will guideautonomous entities towards the globally desirableoutcome (cf. Whang 1995, Lariviere 1999). This type of

approach recurs in a broad range of settings, for ex-ample the economic literature on “vertical restraints”(cf. Mathewson and Winter 1984, Tirole 1988, Katz1989), the marketing literature of “channel coordina-tion” (e.g., Jeuland and Shugan 1983, Moorthy 1987),and agency theory (cf. Bergen et al. 1992, Van Ackere1993). Recent examples in the multi-echelon inventoryliterature include Lee and Whang (1997), Chen (1997),and Iyer and Bergen (1997). When recourse in light ofinformation changes is admitted, results are limited tosingle-period settings. Contractual structures that havebeen shown to replicate the efficiency of centralizedcontrol in that context include buyback/return ar-rangements (cf. Pasternack 1985, Donohue 1996,Kandel 1996, Ha 1997, Emmons and Gilbert 1998) andthe QF contract (cf. Tsay 1996). In all the above works,information about market demand is common to allparties.Some flexible supply contracts with risk-sharing in-

tent have been studied in more realistic settings.Bassok and Anupindi (1995) consider forecasting andpurchasing behavior when the buyer initially forecastsmonth-by-month demand over an entire year and thenmay revise each month’s purchase once within speci-fied percentage bounds. Bassok and Anupindi (1997a)analyze a contract which specifies that cumulative pur-chases over a multi-period horizon exceed a previ-ously (and exogenously) specified quantity, a form ofminimum-purchase agreement. Bassok and Anupindi(1997b) study a rolling-horizon flexibility contractsimilar to our QF structure, focusing on the retailer’sordering behavior when facing an independent andstationary market demand process. Eppen and Iyer(1997) analyze “backup agreements” in which thebuyer is allowed a certain backup quantity in excess ofits initial forecast at no premium, but pays a penaltyfor any of these units not purchased. These models donot attempt to demonstrate efficiency of the contract,instead focusing on the operational implications of thespecified prices and constraints for the buyer. No con-sideration is made for how the supplier might bestsupport its obligations, as the upstream decision prob-lem is rendered difficult by the statistical complexityof the demand that is transmitted through. Moreover,the information structure is kept simplified, with the



forecast for a given period’s demand updated at mostonce, if at all.What little is known about ongoing relationships

with information updating is limited to a single nodesetting with very stylized demand models. For exam-ple, Azoury (1985), Miller (1986) and Lovejoy (1990,1992) consider demand whose structure is known ex-cept for a single uncertain parameter that is updatedeach period in a very specific way (e.g., Bayesian up-dating, or exponentially smoothed mean). Base stockpolicies with moving targets turn out to be optimal ornear-optimal. While these are quite powerful results,they apply only when delivery is immediate. Whenlead times are non-zero, a properly made current-period decision would need to account for the behav-ior of demand over several subsequent periods. Evenwith these relatively straightforward demand models,the statistics required for the policy calculations be-come computationally formidable. This is the caseeven absent supply side flexibility.Industrially, rolling horizon planning is the most

common approach to non-stationary problems withpositive lead times, a prominent application beingMa-terial Requirements Planning (MRP). As in our setting,MRP seeks a supply schedule that attends to a period-by-period schedule of materials needs. Baker (1993)provides a recent review of lot-sizing studies, for bothsingle and multiple level models. Numerical simula-tion is the predominant means of evaluating algorithmperformance, largely due to the complexity of thesetting.Our primary interest is in the way these studies

model demand and how demand information is in-corporated into the planning process. In general, theinstalled policies rarely explicitly account for the tem-poral dynamics of the underlying demand. The accu-racy of the forecasts may be specified as a forecast errorthat gets incorporated into safety stock factors for eachperiod (cf. Miller 1979, Guererro et al. 1986). However,there is no consideration for how each forecast mightchange from one period to the next. Typically, eitherdeterministic end demand is assumed (in which caseforecast updating is not an issue) or the forecast is fro-zen over the planning horizon. Either way, the re-sponse is reactive. Finding that the “stochastic, se-quential, and multi-dimensional nature” of this class

of problem defies an optimization-based approach,Heath and Jackson (1994) suggests that this approxi-mates “reasonable” decision-making. We share thisview in our pursuit of insights for industrialapplication.One limitation of the MRP framework and other

conventional models is the notion of a fixed, or whatwe call “rigid”, lead time. In many real systems, thelead times that are loaded into the materials planningmodel are exaggerated to hedge against uncertaintiesin the supply process (e.g., queuing or raw materialsshortages) (cf. Karmarkar 1989). The QF contract for-malizes the reality that a single lead time alone is aninadequate representation of many supply relation-ships, as evinced by the ability of buyers to negotiatequantity changes even within quoted lead times.This paper seeks insights for a setting including all

of the above features: resources which require advancecommitments, non-stationary demand about which in-formation evolves over time, and the possibility of re-vising the commitments within bounds in reaction toinformation changes. Because this work evolved fromcollaboration with an industrial partner competing ina volatile industry, we have avoided as much as pos-sible any dependence on specific statistical assump-tions about market demand. In this context, optimalpolicies are unknown, so we seek behavioral modelsthat mimic rational but potentially suboptimal policy-makers. We also consider the perspectives of both par-ties to each contract. In addition to specifying thebuyer’s behavior, we recommend how a suppliermight economically deliver the promised flexibility,and characterize how the costs of both parties varywith the contract parameters.

3. Analysis of an Interior NodeWe first specify the structure and behavior of a flexnode,which we use to represent an agent which has QFcontracts with both its supplier and customer (e.g., themanufacturer or the parts supplier in Figure 2). In §4we will introduce the semi-flex node to handle the casewhen the customer-side interface is unstructured. Wewill model multi-stage supply chains by linking thesemodular units.At each period t, the node receives { f(t)}� [ f0(t), f1(t),



f2(t), . . .] as defined in Equation (3), the release scheduledelineating the downstream node’s needs. The nodewill in turn provide its upstream supplier with a re-plenishment schedule {r(t)} � [r0(t), r1(t), r2(t), . . .] as de-fined in Equation (2). Note that one node’s releaseschedule is simultaneously the downstream node’s re-plenishment schedule. I(t) is the node’s period t endingstock, calculated as I(t) � I(t � 1) � r0(t) � f0(t). Allquantities are measured in end-item equivalents.The input and output QF parameters are denoted as

(�in, xin) and (�out, xout) respectively, superscripted tosignify the node’s point of reference. Restating Equa-tions (4) and (5) with this notation gives the followingground rules for schedule revisions, termed Incremen-tal Revision (IR) constraints:

out out[1 � x ] f (t) � f (t � 1) � [1 � � ] f (t),j j j�1 j j

for all t, each j � 1 (6)in in[1 � x ] r (t) � r (t � 1) � [1 � � ] r (t),j j j�1 j j (7)

for all t, each j � 1.

Naturally, we assume � 0 and 0 �in out in out� , � x , xj j j j

� 1. Since these IR constraints are assumed to hold inall future iterations, the current period’s fj(t) suggestsbounds on f0(t � j), the actual customer purchase inperiod (t � j). Specifically, Equation (6) implies

out out[1 � X ] f (t) � f (t � j) � [1 � A ] f (t),j j 0 j j

for all t, each j � 1, where (8)j

out out•1 � X � (1 � x ) andj � qq�1

jout out•1 � A � (1 � � ). (9)j � q

q�1

Similarly, on the replenishment side, Equation (7)implies

in in[1 � X ] r (t) � r (t � j) � [1 � A ] r (t),j j 0 j j

for all t, each j � 1, where (10)j

in in•1 � X � (1 � x ) andj � qq�1

jin in•1 � A � (1 � � ). (11)j � q

q�1

Equations (8) and (10) are termed Cumulative Flexibility(CF) constraints. Clearly and arein in out outA , X , A Xj j j j

non-negative and increasing in j, indicating thatgreater cumulative flexibility is available for periodsfurther out, which is helpful since longer-term projec-tions are generally less informative. As noted in §1, theIR and CF systems of constraints are mathematicallyequivalent, so that QF contracts may be stated eitherway. Each perspective has certain advantages, andthroughout this paper we will use whatever form ismore convenient for the given context.

Replenishment Planning at a Flex NodeThe flex node decision problem is to construct the {r(t)}to be passed upstream, given the {f(t)} faced and thelocal inventory level. The only policies we deem “ad-missible” are those that uphold the release-side con-tract without violating the replenishment-side con-tract. That is, an admissible policy is one for which,given any arbitrary sequence of {f(t)} whose updatesobey Equation (6), (a) updates to {r(t)} obey (7), and (b)coverage is provided (i.e., I(t � 1) � r0(t) � f0(t) for allt).The stochastic optimization problem to be solved at

period t, called program (F), is:H

min E[G(I(t � j))|{f(t)}]{r(t)},(r (t�1), . . ,r (t�H)) �0 0j�0

subject to (12)

I(t � j) � I(t � j � 1) � r (t � j) � f (t � j)0 0

for j � 0, . . , H (13)

I(t � j) � 0 for j � 0, . . , H (14)

in(1 � x )r (t � 1) � r (t)j�1 j�1 j

in� (1 � � )r (t � 1) for j � 0, . . H � 1j�1 j�1

(15)

in in(1 � X )r (t) � r (t � j) � (1 � A )r (t)j j 0 j j

for j � 0, . . , H. (16)

G() is some convex cost function (minimized at zero)that is charged against future ending stock levels, sothe objective is to minimize expected total cost over Hperiods for some fixed H. This problem is stochasticbecause, as suggested by balance Equation (13), G(I(t



� j)) depends on the random variables ( f0(t � 1), . . . ,f0(t � j)) conditional on { f(t)}. The decision variablesare {r(t)} (the current replenishment schedule, which isall that must be formally stated to the supplier) and,for internal planning purposes, (r0(t � 1), . . . , r0(t �

H)) (the sequence of intended future purchases, whichstill enjoys some opportunity for revision).2 Equation(14) enforces the coverage commitment, Equation (15)states what {r(t)} is allowed given {r(t � 1)} and theinput side IR constraint3 and Equation (16) then com-putes the CF bounds on the node’s future purchasesbased on the {r(t)} chosen.Exact solution to (F) is difficult for two primary rea-

sons. First, dimensionality of the decision space is verylarge, with each decision variable subject to con-straints. In particular, Equation (16) acts like a capacityconstraint, which precludes closed-form solution in astochastic setting (cf. Federgruen and Zipkin 1986,Tayur 1992). Here, the added wrinkle is that futurecapacity limits can not only vary by period, but areactually decision variables that can be dynamically ad-justed. Second, and more problematically, the statisti-cal properties of the random variables ( f0(t � 1), f0(t� 2). . .) are in general very complex, since not onlyare they ultimately derived from a non-stationary andmultivariate market demand/forecast process, theyare filtered through the inventory policies of one ormore intermediaries (see Figure 2) and all interveningQF constraints. Hence, while the expectation in the ob-jective function may be well-defined in theory, in prac-tice it is intractable, rendering the search for an optimalpolicy problematic. However, we can identify an open-loop feedback control (OLFC) policy (cf. Bertsekas1976) that has some satisfying mathematical and in-tuitive properties. In an OLFC policy, at each period asequence of actions is computed looking forward andassuming perfect information, and the first action is in-voked. The information is then updated the followingperiod and another forward-looking sequence of ac-tions is computed, and so forth. In this way, a complex

2{r(t � 1)}, {r(t � 2)}, etc. need not be specified at this point sinceany influence they may have are reflected implicitly through Equa-tion (16). Values consistent with any feasible solution can be inferredif desired.3{r(t � 1)} is data resulting from the period (t � 1) planning iteration.

stochastic dynamic program is approximated by a se-ries of deterministic models. Such policies are com-monplace in problems with complex or incompletelyspecified process dynamics. The conventional wisdomis that OLFC is a fairly satisfactory mode of control formany problems. This, in fact, is the approach taken byindustry practitioners in their adoption of the MRPparadigm.To construct an OLFC policy for the control of a flex

node, we suppress explicit consideration of future up-dates to { f(t)}. Instead, the contractual coverage obli-gation suggests fixed targets to which the flex node canposition. In particular, this node must fill any ordersprovided that the customer’s revisions do not exceedthe defined bounds.4 The resulting deterministic prob-lem, which we denote program (F-OLFC) is:

h

min G(I(t � j)) subject to{r(t)},(r (t�1), . . ,r (t�h)) �0 0j�0

I(t � j) � I(t � j � 1) � r (t � j)0

out� (1 � A )f (t) for j � 0, . . , h (17)j j

I(t � j) � 0 for j � 0, . . , h (18)in(1 � x )r (t � 1) � r (t) �j�1 j�1 j

in(1 � � )r (t � 1) for j � 0, . . h � 1 (19)j�1 j�1


for j � 0, . . , h. (20)

f0(t � j) has been replaced with (1 � ) fj(t) foroutAj

reasons discussed above. This program also considersa potentially shorter time window, of length h � H, asa practical consideration. Naturally, this assumes thatall flexibility parameters are well-defined for an h-pe-riod outlook.

Proposition 1. The following {r(t)} is optimal for pro-gram (F-OLFC), and is admissible:

in•r (t) � max[T (t), (1 � x )r (t � 1)]j j j�1 j�1

for j � 0, . . , h, where (21)

4This is not the same as guaranteeing to meet all customer demand,since the allowable order is groomed in advance by the flexibilityconstraints, i.e., it is a truncated version of what the customer mightdesire otherwise.



out(1 � A ) f (t) � l (t)j j j•• T (t) � (22)j in1 � Aj

I(t � 1) for j � 0in•• l (t) � [l (t) � (1 � X )r (t) (23)j j�1 j�1 j�1� out �� (1 � A )f (t)] for j � 1.j�1 j�1

This is named the Minimum Commitment (MC) policyas the present decisions minimize commitment to fu-ture costs subject to supporting service obligations.(r0(t � 1), . . . , r0(t � h)) is not stated explicitly sinceonly {r(t)} needs to be provided to the supplier (seeAppendix 1 for the complete optimal solution). lj(t) isthe period t projection of inventory assured to be avail-able at period (t � j), anticipating the future actions ofthe OLFC-optimal decision rule. From here on, we as-sume that flex nodes use the MC policy. The next sec-tion investigates the relationships among flexibility, in-ventory, and information subject to this behavioralassumption.

The Effect of Flexibility Disparities Across a FlexNodeThis section makes rigorous the notion that inventoryresults from a disparity between input and output flex-ibility. The intuition is as follows. The goal is for sup-ply to track customer orders as closely as possible. Be-cause of forecast updating, those orders are movingtargets and the output flexibility defines the range ofpotential movement. Meanwhile, the input flexibilityrepresents the node’s tracking ability. A node with dif-ficulty in matching upside movement compensates byincreasing its general positioning. Inventory accrueswhen the node is unable to pare down its replenish-ments as quickly as the customer is allowed to reduceits own requirements.Proposition 2 demonstrates that a flex node which

possesses more flexibility (in CF form) in its supplyprocess than it offers its customer can meet all obli-gations with zero inventory.

Proposition 2. If (a) updates to { f(t)} obey IR con-straints, (b) the MC policy is used, (c) I(0) � 0, and (d)(Ain,Xin)� (Aout,Xout), then I(t)� 0 for all t. In the specialcase that (Ain, Xin) � (Aout, Xout), then rj(t) � fj(t) for allj � 0, t � 1.

Note that (�in, xin) � (�out, xout) is sufficient, but not

necessary, to guarantee that (Ain, Xin) � (Aout, Xout).The result holds under the latter, less restrictivecondition.This proposition provides insight into one aspect of

flexibility contracting. Once the input profile matchesthe output profile, additional supply side flexibility iswasted and represents an irrational configuration.(Formally, this would be the case if, in addition to con-dition (d), or for at least one j.)out in out inA � A X � Xj j j j

Such a node “absorbs” flexibility with no benefit to thesystem, and would be able to provide better service(more flexibility) at no cost to itself (no increase in in-ventory) by passing its excess flexibility downstreamuntil (Ain, Xin) � (Aout, Xout). This will result in a per-fect non-distortive conduit of information and mate-rials. Orders are filled exactly, no inventory accumu-lates, and every schedule received is transmittedstraight upstream unaltered (a pure lot-for-lot policy).In all other scenarios, the node serves as an “amplifier”of flexibility, offeringmore to the customer than it itselfreceives. Such nodes must carry inventory to meettheir contracted goals. The specific inventory require-ment will be driven not only by the flexibility profiles,but also the nature of the {f(t)} process facing the node.Analytical results predicting inventory from the in-

stalled flexibilities are currently limited. While thisquestion will be addressed for the general setting vianumerical simulation in §5, to obtain insight into howinventory builds we consider here the simplest con-ceivable sequence of {f(t)}: deterministic and stable re-lease schedules, i.e., fj(t) � for all j � 0, where theˆ ˆf fj j

are constants which satisfy Equation (6) ([ ]out ˆ1 � x fj j

� � [1� ] for j � 1). These “stable forecasts”outˆ ˆf � fj�1 j j

are perfect in that the actual release is exactly everyf0time period. Naturally, if this were known in advance,the output flexibility could be eliminated since the cus-tomer has no real need for revision capability. How-ever, to investigate the inventory impact of non-zeroflexibilities we consider how the MC policy will per-form if applied to this predictable process. Inventorywill still arise due to the need to cover the possibilityof increases.An equilibrium for a flex node facing stable forecasts

consists of an inventory level and replenishmentschedule that, once in place as the state variables, per-sist for all subsequent periods. Proposition 3 providesexplicit characterization of the equilibrium behavior.



Proposition 3. An equilibrium for a flex node facing

stable forecasts is {r, I} where:ˆ{ f }

f0 for 0 � j � j*in1 � Xjr � (24)j 1� max {z } for j* � j � hk�j kin1 � Xj

j*outˆ ˆ ând I � [(1 � A )f � f ]� k k 0

k�1

in inA � Xj* j*ˆ� f where (25)0 � in �1 � Xj*

out ˆ(1 � A )fj j in•z � [1 � X ] andj j� in �1 � Aj

ˆ ˆmax {j: z � f } if ∃ j s.t. z � fj 0 j 0•j* � �0 otherwise.

The above expressions may be interpreted in the fol-lowing way. As it is increasing in the output flexibilityand decreasing in the input flexibility, zj reports therelative inadequacy of the input side flexibility over aj-period-away outlook. Based on the zjs, j* defines theflexibility shortfall horizon, the shortest horizon lengthwithin which input flexibility constraints bind. Beyondj*, the zks are “small,” which may be interpreted as asurplus of input flexibility. Indeed, for these indices,Equation (24) indicates that maximal replenishmentflexibility is not exercised. j* plays a key role in thecomputation shown in Equation (25), which accumu-lates period-by-period the amount by which the cov-erage target exceeds the actual demand over the flex-ibility shortfall horizon (the last term is a boundaryeffect adjustment). Inventory results from a non-zeroj*, i.e., the existence of a window within which flexi-bility is lacking, an insight that extends beyond the“stable forecasts” setting. Comparative statics for theinventory level are cataloged in Proposition 4.

Proposition 4. Under the conditions of Proposition 3,the following properties apply: (a) Release Schedule: (i)

� 0, (ii) � 0 for j � 1 (the inequality isˆ ˆ ˆ ˆDI/Df DI/Df0 j

strict for j � j*); (b) Upside Output Flexibility: outˆDI/DAj

� 0 for j � 1 (the inequality is strict for j � j*); (c) DownsideOutput Flexibility: � 0 for all j; (d) Upside InputoutˆDI/DXj

Flexibility: � 0 for j � j*, � 0 otherwise;in inˆ ˆDI/DA DI/DAj j

(e) Downside Input Flexibility: � 0 for j � j*,inˆDI/DXj

� 0 otherwise.inˆDI/DXj

Proposition 4 may be interpreted as follows. First,the inventory level is determined by the size of theactual release relative to the upside coverage targets.In (a.i), increasing suggests that the demand out-f0comematerializes higher relative to forecast, which de-creases inventory. Increasing the forward-lookingcomponents of the release schedule as in (a.ii) neces-sitates inflation of corresponding replenishments,hence potentially more inventory. Comparing (b) to(a.ii) suggests that and have similar effects,outf Aj j

which follows since only the product (1� playsout Â )fj j

into the MC logic. As appears nowhere in Prop-outXj

osition 3, , which may seem counterin-outˆDI/DX � 0j

tuitive. However, (c) assumes that remains con-ˆ{ f }stant. In reality, a rational downstream customershould increase its {r} (which becomes this flex node’s

in response to an increase in its downside inputˆ{ f })flexibility (this flex node’s Xout). Hence the net effectwould actually be more consistent with that describedin (a), a network phenomenon not captured in thissingle-node analysis. Items (d) and (e) show that im-provements in input flexibility reduce inventory, butonly on the boundary of the flexibility shortfall hori-zon. Adding within the horizon does not help, sincethe constraint that defines the boundary continues tobind. Beyond the boundary additional flexibility onlycontributes to an existing surplus. Of course, withmore realistic release schedule dynamics, j* will moveabout, so that increasing any component of the inputflexibility would likely be beneficial. This and all otherinsights reported above have been corroborated by nu-merous simulation experiments.

4. The Market InterfaceA QF contract delineates conditions under which allorders will be filled. However, at the market interfacethis may be an inappropriate representation of the sup-ply relationship. For example, consider a retailer thatserves the external market, which is not a single entitywith which a contract of this sort may bewritten. Thereis no rationale for limiting a customer’s entitlement toproduct, nor is there a customer-provided forecast to



which to tie a minimum purchase requirement. Werepresent this situation with a “semi-flex node”.Like a flex node, the semi-flex node has replenish-

ment governed by a QF contract. However, there is nosuch structure on the release side. {l(t)} � [l0(t), l1(t),l2(t), . . .] represents information at period t regardingthe period-by-period demand, as defined in Equation(1). The construction of {l(t)} is exogenous to the nodebut will certainly impact performance. As with the flexnode, the decision is {r(t)}, with updates governed bythe IR constraints in Equation (7). Ending inventory isupdated by I(t) � I(t � 1) � r0(t) � l0(t), which as-sumes complete backordering.The optimization problem faced by a semi-flex node

is analogous to program (F) faced by a flex node, ex-cept that the expectation in the objective functionEquation (12) would be conditional on {l(t)} ratherthan {f(t)}, and l0(t � j) should appear in Equation (13)in place of f0(t � j). The same issues that complicatethe solution of (F) and motivate an OLFC approach(dimensionality and statistical complexity) also applyhere. Hence, following the logic applied at the flexnode, we formulate program (S-OLFC) as the open-loop version of the semi-flex node’s decision problem:

h

min E[G(I(t � j))|{l(t)}]{r(t)},(r (t�1), . . ,r (t�h)) �0 0j�0

subject to

I(t � j) � I(t � j � 1) � r (t � j) � l (t � j)0 0

for j � 0, . . , h(26)

in in(1 � x )r (t � 1) � r (t) � (1 � � )r (t � 1)j�1 j�1 j j�1 j�1

for j � 0, . , h � 1(27)


for j � 0, . . , h.(28)

Whereas for the flex node the release-side contractualobligation induced a deterministic schedule of futurereleases on which to focus, here there is no such com-

mitment, reflected in the lack of an analog to Equation(18). Hence, in contrast to (F-OLFC), this open-loop ob-jective function still involves an expectation, whichwill be based on the distribution of (l0(t � 1), . . , l0(t� h)) conditional on {l(t)}. The open-loop approach isto suppress consideration of how {l(t)} might be up-dated over time.Even with IID market demand and a G() of simple

structure, (S-OLFC) is difficult to solve analytically dueto the dimensionality and the constraint structure. In-stead, we have considered a number of computation-ally attractive, heuristic approaches based on relaxa-tions of (S-OLFC), and performed a series of numericalsimulation tests, assuming a specific market demandprocess. In particular, since flexibility is most mean-ingful when tracking a non-stationary process, for allstudies in this paper we have used an ExponentiallyWeighted Moving Average (EWMA) process (cf. Box etal 1994). In an EWMA process, period t demand is l0(t)� � 1) � nt. nt � N(0, r2) is an IID normal fore-l (t1casting noise with known variance, and � 1) is thel (t1mean of period t’s demand, which follows exponentialsmoothing dynamics: � (1 � d) • � 1) � d •l (t) l (t1 1

l0(t). 0 � d � 1, with d � 0 corresponding to IID de-mand and larger values of d indicating more volatiledemand environments. The demand and forecast pro-cess then has two parameters of volatility, d and r, andtests were conducted for numerous parameter combi-nations. Based on the discussion and simulation anal-ysis detailed in Appendix 2, we propose the followingheuristic.

The “Sequential Fractile” (SF) policy constructs {r(t)} as

follows. Define (t) � l0(t) and E[G(Sj•S* S*(t) � argmin0 j Sj

� Dj(t))|{l(t)}], where Dj(t) l0(t � j) is the cu-j•� �q�0

mulative demand for periods t through (t � j). Letting y �

[a, b] denote the point in the interval [a, b] closest to y, forj � 0, . . , h, select:

r (t � j)0r (t) � �j in in(2 � A � X )/2j j

in[(1 � x )r (t � 1),j�1 j�1

in(1 � � )r (t � 1)], where (29)j�1 j�1



Figure 3 Supply Chain for System Performance Analysis

j�1

r (t � j) � S*(t) � I(t � 1) � r (t � q)0 j � 0� �q�0

in� [(1 � X )r (t � 1),j�1 j�1

in(1 � A )r (t � 1)].j�1 j�1

It is straightforward to verify that in a conventionalscenario of a fixed lead-time with no flexibility, thisreduces to the classical policy of maintaining stock on-hand plus on-order at a critical fractile of cumulativedemand over the lead-time. In fact, the SF policy maybe viewed as a generalization of multi-period news-vendor logic, known to be optimal with IID demand,to rolling horizon planning in the presence of flexibil-ity. Replenishment policies based on IID logic but ap-plied to real (almost certainly not IID) demand pro-cesses have been demonstrated both in research andpractice to be very effective, if not optimal (cf. Lovejoy1990, 1992). We make no claim that the SF policy isoptimal in more general settings, only that it includeslogic approximating the behavior of a reasonable prac-titioner and has intuitive appeal. Bassok andAnupindi(1997b) propose alternative OLFC semi-flex node pol-icies under slightly different assumptions, which allowfor the development of certain performance bounds.The computationally intensive nature of their policiesunderscores the need for simplifying heuristics.

5. Performance Properties of QFSupply Chains

We are now prepared to explore the performanceproperties of multi-level supply chains controlledwithQF contracts, which can be modeled by linking to-gether the individual node building blocks presentedin §2 and §3. Below we characterize the following met-rics: (i) system-wide inventory patterns, (ii) variability

of orders placed at each node, and (iii) service pro-vided at the market interface. In particular, the com-parative statics of each of these with respect to themar-ket demand volatility and system flexibilitycharacteristics will be provided.

Modeling Supply ChainsInventory points whose replenishments and releasesare both controlled by QF contracts are represented byflex nodes (cf. §2). Only the single node furthest down-stream in the chain may deviate from this structure,and semi-flex structure (cf. §3) accommodates its dis-tinctive features.The link between two nodes is described by the flex-

ibility profile of the QF contract and, if desired, a logis-tical delay (LD). The LD allows the representation ofdelay that is truly unavoidable (e.g., for ocean transit).As in MRP explosion calculus, a buyer node’s replen-ishment schedule becomes its supplier’s release fore-cast, differing by the intervening LD time offset:

(t) → (t) for j � LD. A non-zero LD alsosupplier buyerf rj�LD j

leads the parties to perceive the QF contract differ-ently. Along with the time offset, i.e. ( out� ,j�LD

↔ ( )buyer, the immutability of ordersout supplier in inx ) � , xj�LD j j

within the incoming logistical pipeline is representedby ( )buyer � ( )buyer � 0 for j � LD. Hence, a logis-in in� xj j

tical delay may be regarded as an extreme form ofinflexibility.

Supply Chain PerformanceFor the following experiments we consider the serialchain depicted in Figure 3. Nodes 1–3 are flex nodesand node 0 is a semi-flex node. Logistical delays are aslabeled.Figure 4 presents the assumed system flexibility

characteristics, stated in CF form since the computa-tional algorithms were easier to implement this way.Conversion back to IR form is easy, via Equations (9)and (11). Parameter values were chosen to provide



Figure 4 Base-Case System Flexibilities

j 1 2 3 4 5 6 7 8 9 10Node 1 out outA and Xj j 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

in inA and Xj j 0.00 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32Node 2 out outA and Xj j 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32

in inA and Xj j 0.00 0.00 0.03 0.06 0.10 0.13 0.16 0.19Node 3 out outA and Xj j 0.03 0.06 0.10 0.13 0.16 0.19

in inA and Xj j 0.00 0.00 0.03 0.05 0.08 0.10

Figure 5 Summary of Experiments and Observations

Observations and ConclusionsSystem Parameter UnderConsideration

Inventory Variability of Orders Node 0 Cost & Service Level

1. Demand forecast error. r isincreased incrementally.

increases at every node (Fig. 6) over all r considered, upstreamvariability � market demandvariability (Fig. 10)

both cost and fill rate worsen withr (Fig. 14)

2. Parameter governing movementof mean demand. d is increasedincrementally.

increases at every node (Fig. 7) for low d, upstream variability �

market demand variability; as d

increases, bullwhip effect eventuallyoccurs (Fig. 11)

both cost and fill rate worsen withd (Fig. 15)

3. Flexibility between flex nodes.Components of (Aout, Xout)Node2 areincreased incrementally. {d, r} �

{0.3, 20}

decreases at Node 1, increases atNode 2; impact on Node 3 is minor(Fig. 8)

upstream variability is fairly robust tosmall perturbations of internalflexibility parameters (Fig. 12)

NOT APPLICABLE

4. Flexibility between flex node andsemi-flex node. Components of (Aout,Dout)Node1 are increased incrementally.{d, r} � {0.3, 20}

decreases at Node 0, increases atNodes 1 and 2; impact on Node 3is minor (Fig. 9)

order variability is apparently fairlyrobust to small perturbations ofinternal flexibility parameters (Fig. 13)

more supply-side flexibilityimproves both cost and fill rate(Fig. 16)

flexibility amplification (cf. Proposition 2) at each flexnode, with upside-downside symmetry in each profile.This network configuration will be referred to as theBase-Case. We again use the EWMA demand and fore-cast process detailed in Appendix 2, with � 100l (0)1

and (co, cu) � (30, 150).In a series of simulation experiments, we consider

the relationship between key parameters and perfor-mance outcomes. The parameters studied are: (1) r, thedemand forecast error, (2) d, the parameter governingmovement of the mean demand, (3) the flexibility pro-file between two flex nodes (Nodes 1 and 2), and (4)the flexibility profile between a flex node and a semi-flex node (Nodes 1 and 0, respectively). The outcomesreported for each node are: (1) average inventory, and(2) variability of orders (i.e., StdDev(r0())). The investi-gation of variability is motivated by concern for the

“bullwhip” effect, an empirically common phenome-non in which the variability of replenishment ordersplaced by a node exceeds the variability of customerorders encountered. That is, order variability exceedsmarket demand variability, and increases on movingupstream. Lee et al. (1997) reports that the QF contracthas appeared in industry as a counter-measure to thebullwhip effect.For stated combinations of the system parameters

we report the performance metrics over 100 separate500-period simulation runs. The four experiments andobservations are summarized in Figure 5, and illus-trated in Figures 6–16.Note that increasing the flexibility between flex

nodes (Experiment 3 in Figure 5) has no bearing onNode 0 performance. This is because Node 0 continuesto receive the same flexibility from Node 1, regardless



Figure 8 Inventory vs. (Aout, Xout)Node 2Figure 7 Inventory vs. d, with r � 20

Figure 6 Inventory vs. r, with d � 0

of what happens further upstream. Of course, wewould expect that in a real supply chain an increase inupstream flexibility should potentially benefit evendownstream parties further removed. This would oc-cur if, for instance, Node 1 were to be willing to passto Node 0 some of the inventory savings enabled bythe improved flexibility provided by Node 2. Thiscould be in some combination of increased flexibilityand lower unit cost. Such behaviors are not consideredwithin the scope of these experiments.Figures 6 and 7 validate our intuitions regarding de-

mand variability and inventory. Figure 8 is consistentwith the intuitions developed in Proposition 4. Node

1 is receiving improved service (higher input flexibil-ity), therefore can meet its commitments with less in-ventory. Node 2 is in turn promising a higher level ofservice, and carries more inventory as a result. Fromthis we note that all else equal, increasing the param-eters of the QF contract reduces the customer’s costs atthe expense of the supplier. This conflict of preferencesprovides the tension in the contract negotiation pro-cess. Even though Node 3’s flexibility status is unal-tered, its inventory situation does change. The effectsare carried upstream via changes in the dynamics ofthe information vector. Each flexibility profile trans-forms the information flow, so changes in any profilewill have ramifications for all nodes upstream no mat-ter how far removed. As with Figure 8, Figure 9 showsthat increasing the flexibility between two nodes (thistime a flex node and a semi-flex node) shifts inventoryupstream. Slight upward pressure is also expressed atNode 2, which apparently gets damped out beforereaching Node 3. At this point it is still unclear whereinventory, and by implication flexibility, should bestbe positioned from a system-optimizing perspective.This design question requires additional structure de-scribing the relative economic implications of holdinginventory at the various locations, which we do notpursue in this paper. A methodology for addressingthis issue is provided in Tsay (1995).The next several figures investigate the prevalence

of the bullwhip effect in QF environments. In Figure10, which has IID market demand, no bullwhip occurs.



Figure 12 System Variability vs. (Aout, Xout)Node 2, with {d, r} � {0.3,20}

Figure 11 System Variability vs. d, with r � 20

Figure 10 System Variability vs. r, with d � 0

Figure 9 Inventory vs. (Aout, Xout)Node 1

This was not unexpected since the phenomenon is usu-ally associated with non-stationary demand. However,dampening of variability is achieved. When demand isnon-stationary (Figure 11), increasing volatility in themarket demand and forecasts eventually overwhelmsthe variability-diffusing capability of the installed flex-ibility. However, a true bullwhip, which would cor-respond to an upward-sloping curve, is not alwayspresent. Figures 10 and 11 confirm that at each nodeStdDev(r0()) increases with either demand variabilityparameter. Figures 12 and 13 suggest that the patternsof variability are fairly robust to small perturbations offlexibility parameters.

We conclude that the presence of flexibility candampen the transmission of order variability up thechain. This is because an entire replenishment schedulecan move in response to changes in the demand en-vironment. For example, suppose demand forecastsare revised upwards in a given period, which wouldlead a node to generally increase the elements of itsreplenishment schedule. If the demand forecasts arerevised back down in the next period, the node has theopportunity to undo some of the previous increases inthe replenishment schedule. The ability to dynamicallyadjust the estimates is what enables a node to recoverfrom some of the overreacting that becomes a bullwhip



Figure 16 Node O Performance vs. (Ain, Xin)Node 0, with {d, r} � {0.3,20}

Figure 15 Node O Performance vs. d, r � 20

Figure 14 Node O Performance vs. r, d � 0

Figure 13 System Variability vs. (Aout, Xout)Node 1, with {d, r} � {0.3,20}

effect in rigid lead-time settings. As market demandbecomes more volatile, the dampening capabilities ofthe installed flexibilities are eventually overwhelmed,and a bullwhip-type of effect may then be expressed.As the semi-flex node (Node 0) has distinct structure

due to its interface with the external market, additionalperformance metrics are appropriate. Figures 14through 16 report this node’s average holding and bac-korder cost per period and service performance (de-fined as a fill rate) for the relevant experiments. As wewould expect, increasing market demand uncertaintyand forecast volatility (Figures 14 and 15) cause both

the cost and fill rate to worsen, and increased inputflexibility (Figure 16) enables an improvement in both.Natural performance benchmarks are apparent only

for the semi-flex node. These include a single-locationmodel with immediate replenishment (extreme flexi-bility) and one with a fixed lead time of H � 0 (zeroflexibility), which are well understood in IID demandsettings (this approach is taken in Bassok andAnupindi 1997b). However, what remains lacking issome basis for evaluating the absolute magnitudes ofthe performance outcomes observed at individual flexnodes and across the system. Are there ways to control



Figure 17 Tandem Supply Chain for Contract Evaluation

the same supply chain which will result in lower in-ventory levels across the board? Would these methodsincrease or decrease the order variability? Models ofbehavior and performance under alternative controlschemes are necessary. To the best of our knowledge,these remain open research areas.

6. Contract DesignThus far we have provided primitives for modelingsupply chains controlled by QF contracts and charac-terized system performance for fixed flexibility param-eters. We now consider these as decision variables,since this will be a manager’s ultimate interest.5 Ourgoal is to provide the “willingness-to-pay” for incre-ments of flexibility, which a materials manager canthen compare against the menu of flexibility vs. unitprocurement cost combinations offered by a vendor orpool of vendors, as well as other cost considerationsnot included in this analysis.To illustrate ourmethodologywe use the simple tan-

dem chain depicted in Figure 17, in which a single flexnode (Node 1) feeds into a semi-flex node (Node 0)located at the market interface. Given a contract be-tween Node 0 and Node 1 of (A, X), we wish to placea value on Node 1’s supply-side flexibility, denoted as

(A, Both contracts have h � 4. While we use aX).multi-level system for greater realism in the dynamicsof the materials and information flows, the results andintuitions that follow are not materially different fromthose obtained for a single node model.

5In general, the planning horizon H should also be open to negoti-ation, and the method we present could easily handle this simply byincreasing the dimensionality of the experiment design (i.e., repeat-ing the process for alternative values of H).

The general methodology is straightforward, in thatwe incrementally increase and record the correspond-ing reductions in Node 1’s inventory cost given a hold-ing cost per period of 15, using the method of §4 tocompute average inventory levels in each case. Ratherthan varying (A, along its eight degrees of freedomX)independently, here we limit consideration to a spe-cific parametric form: A � � {0.04s, 0.08s, 0.12s,X

0.16s} with s � 0, . . , 5. Using � 100 and r � 20,l (0)1

this procedure was repeated for d values of {0.3, 0.5,0.7}. The cost outcomes are reported in Figure 18 asNode 1’s average inventory cost per unit of demand,which is appropriate for comparison against unit pro-curement cost.The left figure reports how inventory costs varywith

the external contract, while on the right is the samedata in terms of savings relative to the zero-flexibilitycase (s � 0). This describes the buyer’s “willingness topay” (WTP) for positive increments of flexibility rela-tive to a rigid supply lead time. The cost curves indi-cate that for any external contract the costs are increas-ing with the market’s d. Each cost curve is decreasingin s, as would be expected. As s becomes arbitrarilylarge the cost approaches zero since demand can betracked perfectly with infinite flexibility. The WTPcurves suggest, for example, that in a market with d �

0.7 the materials manager of Node 1 should to be will-ing to pay the external vendor an additional $7.60/unitto go from a no-flexibility contract (s � 0) to an s � 5supply contract. The curves shift upwardwith d, whichwe expect since flexibility, the ability to track a movingtarget, should increase in value with the extent ofmovement to be tracked. More generally, flexibilitycannot be valued without an environmental context.For example, the WTP curve will be uniformly zero ina world of completely deterministic demand as long



Figure 18 Node 1 Inventory Cost, Willingness-to-Pay (WTP) (per unit) vs. Supply Flexibility

as the internal contracts are specified properly. In eachdemand environment there appears to be a point ofdiminishing returns beyond which additional flexibil-ity becomes practically worthless, suggesting thatthere is already sufficient flexibility on hand to suitablyrespond to the degree of schedule volatility encoun-tered. A buyer always prefers more flexibility, butshould be happy to settle for less if the price is right.

7. Concluding RemarksThis paper proposes a framework for performanceanalysis and design of QF supply chains. We have pro-vided local policies that, in addition to suggesting arational way to make use of flexible supply, dictatewhat actions must be taken to support flexibility prom-ised to a customer. While these are not necessarily op-timal in the traditional sense, we feel they provide a

reasonable compromise in light of their computationalproperties and the complexity of the general problem.We have developed the notion of inventory as a con-

sequence of disparities in flexibility. In particular, in-ventory is the cost incurred in overcoming the inflex-ibility of a supplier so as to meet a customer’s desirefor flexible response, which we call flexibility amplifi-cation. All else equal, increasing a node’s input flexi-bility reduces its costs. And all else equal, promisingmore output flexibility comes at the expense of greaterinventory costs. We therefore recommend that inven-tory management should be viewed as the manage-ment of process flexibilities.The modular design of our local nodal models en-

ables multi-echelon analysis, which has been lackingin the literature of flexible supply contracts. Our ex-periences have revealed that the distribution of the in-ventory burden across QF supply chains is determined



by the system flexibility characteristics and the volatil-ity in the market demand and forecast process. Wehave found in addition that QF contracts can dampenthe transmission of order variability throughout thechain, thus potentially retarding thewell-known “bull-whip effect”.We provide a methodology for computing a mate-

rials manager’s “willingness-to-pay” for flexibilityfrom an external vendor, which has certain properties.These include the notions that flexibility increases invalue as the market environment becomes more vola-tile, and that flexibility observes a principle of dimin-ishing returns. The buyer always prefers more flexi-bility, but should be careful to make the appropriatecost-benefit assessment in negotiating the contract.As firms have experimented with QF contracts, cer-

tain implementation issues have come to light. The QFcontract represents a radical change in procurementpractice for some firms, and change rarely comeswith-out organizational resistance.Materials buyers may present one source of oppo-

sition. Some are accustomed to manipulating orderswithout perceived consequence, and are reluctant tosurrender this position. For others it is the formality ofthe flexibility limits, rather than the particular latitudesspecified, that inspires discontent. Some of these indi-viduals thrive on the thrill and challenge of the dy-namic bargaining process, and have confidence in theirability to extract greater concessions in an ad-hoc sys-tem than any supplier would actually commit to for-mally. A large part of this problem is in the difficultyof understanding just how much flexibility is actuallyneeded and how much is available in the relationship.More fundamentally, it can be problematic for a ma-terials organization to recalibrate its intuitions andbusiness practices around specifying flexibilities in-stead of inventories. The intent of this paper has beento inform these issues.Depending on what behavior is being replaced, it is

unclear whether the move to a QF arrangement willdrive procurement prices down or up. Even if theseincrease, this may still be the best solution in terms oftotal costs. Yet this can be obstructed by a conflict ofinterest within the buyer organization. TheQF contractis precisely about trading off procurement price for in-ventory cost, yet in many firms different groups are

held accountable for each of these. In Sun Microsys-tems, for example, the Supplier Management organi-zation is responsible for the unit price, while the Ma-terials organization owns the inventory (cf. Farlow etal. 1995). Will the group concerned with procurementprice pay for the supply flexibility that will help thefactory operate with less inventory?A similar conflict can occur within the supplier or-

ganization. The supplier benefits from themore honestforecasts that the buyer may provide due to the QFcontract, but in exchange may need to lower its sellingprice and carry additional inventory to meet its prom-ise of coverage. Resistance may result if inventory andprice (which now affects revenue) are concerns of dif-ferent groups.These, and other cultural and organizational consid-

erations, will join efficiency and valuation issues in de-termining the popularity of QF contracts over time.6

Appendix 1. Proofs of Propositions

PROOF OF PROPOSITION 1.We solve (F-OLFC) in several steps, outlined as follows. First, wemomentarily relax the upper bounds in Constraints (19) and (20) toavoid potential infeasibility. The relaxed solution is not unique in{r(t)}, so we pick the option that has the lowest values component-wise. Finally, we show that if updates to {f(t)} satisfy the required IRconstraints, our solution to the relaxed program automatically sat-isfies the upper bounds of Equations (19) and (20), and hence isadmissible as well as being optimal for (F-OLFC). We now proceedin this fashion.(F-OLFC) is potentially infeasible since the upper bounds in Equa-

tions (19) and (20), which act like capacity constraints, may precludecoverage. The problem is that in converting to a deterministic prob-lem, the information indicating that updates to {f(t)} are alsobounded is lost. So for the moment we relax these upper bounds, inwhich case Equations (19) and (20) can be combined into (1 �

, and the optimal (r0(t � 1), . . , r0(t �inX )r (t � 1) � r (t � j)j�1 j�1 0

h)) can be stated as:

6The authors would like to thank a number of individuals. TimothyEckert and Richard Goldstein of Sun Microsystems engaged us inmany meaningful conversations in the model design stage. Profes-sors J. Michael Harrison, Warren Hausman, Martin Lariviere, HauLee, James Patell, Evan Porteus, Seungjin Whang and RobertWilsonhave provided many insightful comments. Seminar participants atDuke University, Santa Clara University, Stanford University, theUniversity of Michigan, and Washington University (St. Louis) havegreatly assisted in the refining of our ideas. Last, but not least, weare grateful to the referees and editors for thoughtful and timelyreview. Any errors remain the responsibility of the authors.



out ¯•r*(t � j) � max{(1 � A ) f (t) � l (t),0 j j j

in(1 � X )r (t � 1)} for j � 0, . . , h (30)j�1 j�1

¯ ¯ ¯• •where l (t) � I(t � 1) and l (t) � l (t) � r*(t0 j j�1 0

out� j � 1) � (1 � A )f (t). (31)j�1 j�1

The formal proof is a straightforward application of Kuhn-Tuckerconditions (cf. Rockafellar 1972). See Tsay (1995) for details. In fact,this solution is readily apparent from the problem’s economic struc-ture. (F-OLFC) without the upside constraints is essentially anMRP-style lot-sizing problem with minimum lot sizes. With no fixed costper lot and a holding cost for any material taken earlier than abso-lutely necessary, a lot-for-lot policy (modified for minimum lot sizerequirements) will be appropriate. The sequential algorithm statedin Equations (30) and (31) does precisely this, with the construct

extrapolating the beginning inventory for period (t � j).l (t)j

While above we have computed the desired future replenish-ments, denoted by ( (t � 1), . . , (t � h)), the present decision isr* r*0 0

{r(t)}, which is not uniquely determined by (F-OLFC). Because anrj(t) (in conjunction with the input flexibility parameters) simplystakes out a region within which (t � j) may lie, there will bemanyr*0{r(t)} that can enable the above ( (t � 1), . . , (t � h)). Since {r(t)}r* r*0 0

defines the lower IR bounds in subsequent periods, aminimal choiceof each rj(t) reduces the risk of unnecessary future inventory. (20)requires (t � j) � (1 � )rj(t) (one of the two constraints weinr* A0 j

relaxed earlier), so choosing an rj(t) � (t � j)/(1 � ) is neces-inr* A0 j

sary. To guarantee this without violating (19), we select:

in•r (t) � max[r*(t � j)/(1 � A ),j 0 j

in(1 � x )r (t � 1)] for j � 0, . . , h (32)j�1 j�1

The policy that results from applying this rule every period maybe stated in a more compact and analytically convenient form thatgives {r(t)} as a direct function of { f(t)}, bypassing the intermediatecalculation of in (30) and (31). Detailed(r*(t � 1), . . , r*(t � h))0 0

proof of this equivalence is omitted, however the general idea is asfollows. Direct substitution of (30) and (31) into (32) is followed bya straightforward but tedious inductive argument that (as de-l (t)j

fined in (31)) and lj(t) (as defined in (23)) are equivalent for all jwhen(32) is applied at every t.To show admissibility, we first prove Lemma 1, which states a

property of lj(t).

Lemma 1.In rolling from period (t � 1) to period t, if: (a) I(t � 1) � 0; (b) { f (t)}obeys the upside of the output IR constraints; and (c) the {r(t)} generatedby the MC policy obeys the downside of the input IR constraints, then lj(t)� lj�1(t � 1) for all j � 0.

Proof of Lemma 1. This property follows from induction on j.Details are omitted due to space limitations. Instead we offer thefollowing intuition. From the period (t � 1) perspective, lj�1(t � 1)is the most conservative (i.e., lowest) estimate for the period (t � j)

inventory. That is, it assumes maximal demand and minimal re-plenishment in all intervening periods. One period’s demand andschedule revision outcome is resolved with each horizon roll, andcannot result in inventory any lower than in the extreme scenario.Admissibility requires that if all updates to { f(t)} obey their IR

constraints, then for all t, I(t) � 0 and replenishment side IR con-straints are observed. Proof is by induction on t. At period (t � 1),

(21) implies rj�1(t � 1) � Tj�1(t � 1) [(1 � fj�1(t � 1) �out•� A )j�1

lj�1(t � 1)]/(1 � for all j � 0, which may be rewritten as (1inA )j�1

� rj�1(t � 1) � [(1 � (1 � fj�1(t � 1) � lj�1(t � 1)]/in out out� ) A ) � )j�1 j j�1

(1 � (see (9) and (11)). Since fj(t) � (1 � ) fj�1(t � 1) (IRin outA ) �j j�1

constraint) and lj(t) � lj�1(t � 1) (Lemma 1), this suggests (1 �

)rj�1(t � 1) � [(1 � fj(t) � lj(t)]/(1 � Tj(t). Thus,in out in •� A ) A ) �j�1 j j

rj(t) max[Tj(t), (1 � � 1)] � (1 � , soin in•� x )r (t � )r (t � 1)j�1 j�1 j�1 j�1

the upper bound in (19) is obeyed. Furthermore, rj(t) � Tj(t) for all j

� 0 by construction. At j � 0, this is r0(t) � T0(t) f0(t) � I(t � 1),•�

or equivalently, 0� I(t � 1)� r0(t)� f0(t) I(t). Thus, admissibility•�

conditions are satisfied at every t. �

Proof of Proposition 2. The MC policy can be stated as follows:

1 in•r (t) � max [(1 � X )T (t � (k � j))]j k�j k kin1 � Xj

for j � 0, with T () from (22) (33)k

The equivalence of this more analytically convenient form can beverified by induction on j.We next establish that inventory is non-increasing with time. Us-

ing (33) at j � 0:

1•r (t) � max0 k�0in1 � X0

outf (t � k)(1 � A ) � l (t � k)k k kin(1 � X )k� �in1 � Ak

out in1 � A 1 � Xk kout� max f (t � k) (1 � X ) � f (t)k�0 k k 0� � �� in out1 � A 1 � Xk k

The former inequality holds because lk() is non-negative and inX �0

. The latter is due to the output CF constraint and [(1 � �out0 A )/(1k

(1 � � � 1, which follows from condition (d).in in outA ) X )/(1 X )]k k k

Thus I(t) I(t � 1) � r0(t) � f0(t) � I(t � 1). Furthermore, I(t)•�

remains non-negative by the admissibility of the MC policy. So ifthe inventory is initialized at zero, it will remain there.The results for the specific case of (Ain, Xin) � (Aout, Xout) follow

from induction on j. We have shown that I(t) � I(t � 1) � 0 for all

t � 1. As I(t) I(t � 1) � r0(t) � f0(t) for all t � 1, this implies r0(t)•�

� f0(t). Also, l0(t) I(t � 1) � 0 for all t � 1.•�

Next, suppose that lj�1(t) � 0 and rj�1(t) � fj�1(t) for some j �

1. Then

in out �•l (t) � [l (t) � (1 � X )r (t) � (1 � A )f (t)]j j�1 j�1 j�1 j�1 j�1

in out �� [� (X � A )r (t)] � 0j�1 j�1 j�1



We also know that lj(t) � lj�q(t � q) � 0 for all q � 0, where the firstinequality is due to Lemma 1 and the second reflects the non-negativity of these entities. Consequently, lj�q(t � q) � 0 for all q �

0. Or, with the change of variable k � j � q, lk(t � (k � j)) � 0 forall k � j. Then, beginning with (33), we have

1r (t) � maxj k�jin1 � Xj

out(1 � A )f (t � (k � j)) � l (t � (k � j))k k kin(1 � X )k� �in1 � Ak

1 out� max [(1 � X )f (t � (k � j))]k�j k kout1 � Xj

out(1 � X )f (t)j j� � f (t)jout1 � Xj

The second equality is due to (33) and the assumption that inA �k

and for all k. By the lower output IR constraint, fk(tout in outA X � Xk k k

� (k � j))� (1� ) fk�1(t � (k � 1� j)) for all k, or equivalently,outxk�1

(1 � ) fk(t � (k � j)) � (1 � ) fk�1(t � (k � 1 � j)). Thisout outX Xk k�1

delivers the third equality as the maximization must then occur at k� j. �

Proof of Proposition 3. The proof, as detailed in Tsay (1995),entails a single, purely mechanical iteration through the MC policy,and is omitted due to space limitations. �

Proof of Proposition 4. The explicit functional forms of the dif-ferences are computed in a tedious but straightforwardmanner fromthe results of Proposition 3. �

Appendix 2. Analysis of Semi-Flex Node PolicyOur approach to obtaining a reasonable and computationally effi-cient policy for the semi-flex node will be as follows. The solutionto (S-OLFC) with (27) and (28) relaxed is relatively straightforwardto obtain. We will then consider several alternative heuristic ap-proaches for reconciling this with (27) and (28), and select one foruse in network performance analysis based on numerical simulationstudies.Noting that I(t � j) � I(t � 1) � � q) �j j� r (t � l (t �q�0 0 q�0 0

and defining Sj (I(t � 1) � r0(t � q)) and Dj(t)j j• •q) � � � �q�0 q�0

l0(t � j), the objective in (S-OLFC) can be restated asE[G(Sj � Dj(t))|{l(t)}]. If (27) and (28) are re-hmin �{r(t)},(S , . . ,S ) j�00 h

laxed, then clearly � l0(t) and �* •S*(t) S (t) � argmin E[G(Sj0 S jj

Dj(t))|{l(t)}] for j � 1 will be optimal since the summation in theobjective can be decomposed. The corresponding optimal (t � j)r*0would then be obtained as � � I(t � 1) and � j) �r*(t) S*(t) r*(t0 0 0

� for j � 1. However, in general the attainment of thisS*(t) S* (t)j j�1

solution will be obstructed by some of the constraints. We thereforeseek a feasible point that is “close” to this ideal in some sense. Ourcandidate heuristics each have two steps: (Step 1) projecting

into a feasible (r0(t � 1), . . , r0(t � h)), and (Step 2)(S*(t), . . , S*(t))0 h

constructing {r(t)} to declare to the supplier based on this (r0(t �

1), . . , r0(t � h)). Below are two proposed alternatives for each step.

Step 1: (Option a) Component-wise projection. By the above argu-ment, the ideal would be to achieve r0(t) � � I(t � 1) and r0(tS*(t)0

� j) � � for j � 1. However, (27) and (28) togetherS*(t) S* (t)j j�1

require that (1 � � 1) � r0(t � j) � (1 � �in inX )r (t A )r (tj�1 j�1 j�1 j�1

1) for all j. So one approach is to get as close as possible term-wise,subject to this constraint, i.e.,

in(S*(t) � I(t � 1)) � [(1 � X )r (t � 1),0 1 1in(1 � A )r (t � 1)] for j � 01 1r (t � j) �0 in(S*(t) � S* (t)) � [(1 � X )r (t � 1),j j�1 j�1 j�1� in(1 � A )r (t � 1)] for j � 1j�1 j�1

(Option b) Lexicographic projection. Here the projection is per-formed sequentially, with the index j target taking into account whathas been installed for all preceding terms. So, for all j,

j�1

r (t � j) � S*(t) � I(t � 1) � r (t � q)0 j � 0� � ��q�0

in in� [(1 � X )r (t � 1), (1 � A )r (t � 1)]j�1 j�1 j�1 j�1

The rationale for this approach is that the consequences of decisionvariables for near-term replenishments exceed those for periods fur-ther off. Also, the latitude for change is less broad for periods closerin. So it makes sense to first position r0(t) as close to its ideal valueas possible, then compensate for discrepancies in that match whenr0(t � 1) is selected, and so on.

Step 2: (Option a) Minimum commitment. This is the same approach

as at the flex node: max[r0(t � j)/(1 � (1 �in in•r (t) � A ), x )r (tj j j�1 j�1

� 1)] for j � 0, . . , h. The (r0(t � 1), . . , r0(t � h)) chosen at Step 1takes into account the relative impacts of overage and underage.Here we install the (component-wise) minimum allowable {r(t)} thatrenders those targets attainable.

(Option b): Centering. The selection of rj(t) induces [(1 � (1inX )r (t),j j

� as the feasible range for r0(t � j). This option positionsinA )r (t)]j j

that interval so that the target r0(t � j) sits as close to the midpoint

(rj(t)[(1 � � (1 � as is allowed by (27): rj(t) r0(t �in in •X ) A )]/2) �j j

j)/[(2 � � � [(1 � � 1), (1 � (t �in in in inA X )/2] x )r (t � )rj j j�1 j�1 j�1 j�1

1)]. Whereas minimum commitment logic was used at the flex nodebecause maximum potential customer requests are already incor-porated into the targets, at a semi-flex node the updates to {l(t)} areunconstrained. There is uncertainty as to the direction and extentthat the desired r0(t � j) will move going forward in time, so thismethod tries to keep the latest target at the middle of the windowto leave room to track it in either direction.The above alternatives suggest the following four distinct heuris-

tics, labeled SF1–SF4:

Step 2: (r0(t), . . , r0(t � h))→ {r(t)}

Min.commitment Centering

Step 1: Component-wise SF1 SF2→(S*(t), . . , S*(t))0 h

(r0(t), . . , r0(t � h))Lexicographic SF3 SF4



We compare these methods via numerical simulation, using theEWMA process defined in §3. For this process, an unbiased andminimum mean-squared-error estimate of period (t � k) demand isprovided by setting lk(t) � E[l0(t � � for k � 1 (thek)|l (t)] l (t)1 1

last equality is true since l0(t � k) � � d � , cf.k�1l (t) � n n1 m�1 t�m t�k

Box et al 1994). Cumulative demand is Dj(t) � l0(t) � �j • l (t)1

(dn � , a normal variate with moments E[Dj(t)] � l0(t)j�1� 1)nn�0 t�j�n

� and Var[Dj(t)] � jr2[d2(j � 1)(2j � 1)/6 � d(j � 1) � 1].j • l (t)1

(Calculation of the latter uses identities n2 � k(k � 1)(2k � 1)/k�n�1

6 and n � k(k � 1)/2.)k�n�1

We assume G(x) � co[x]� � cu[x]�, where co and cu are respec-tively the linear holding and backorder costs, in which case the

are easily obtained. Specifically, and, by newsven-S*(t) S*(t) � l (t)j 0 0

dor logic (cf. Heyman and Sobel 1984), �1S*(t) � F (c /(c � c ))j D (t) u o uj

where () is the distribution of Dj(t). For the EWMA process, theFD (t)j

above analysis suggests that S*(t) � l (t) � j • l (t) � (j j •j 0 1

where j � U�1(cu/(co �2r) d (j � 1)(2j � 1)/6 � d(j � 1) � 1cu)) and U() is the standard normal distribution function.We compare the heuristics over scenarios distinguished by values

used for d, r, and (Ain, Xin): d � {0.3, 0.7}, r � {10, 20}, and (Ain, Xin)� {SY, UD, DD} as described below. Profile SY has Ain � Xin �

{0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50}, symmetrical inupside and downside flexibility. UD is upside dominant, withAin �

{0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50} and Xin � {0.00,0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45}.DD is downside dom-inant, with Ain � {0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40,0.45} and Xin � {0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50}.Cost parameters (co, cu) � (30, 150) are used. The performance ofeach heuristic is evaluated by the average cost over 100 samplepaths, each path representing 500 periods. � 100 in all cases.l (0)1

The outcomes of the 12 scenarios support the following conclu-sions, with numerical details omitted due to space limitations (seeTsay 1995). SF3 and SF4 are each uniformly superior to both SF1 andSF2 by far, with results that are statistically significant with p-valuesno greater than 1 � 10�17 in all cases (and typically even lower). SoLexicographic projection dominates Component-wise projection forStep 1 regardless of the option taken at Step 2, presumably for itshandling of the interrelationships between periods. There is no dom-inant approach at Step 2, with relative performance varyingwith theflexibility structure. We thus select SF3 as the semi-flex node oper-ating policy, acknowledging the existence of alternatives that areequally easy to implement and give superior performance in somesettings.

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Accepted by Paul Zipkin; received January 26, 1998. This paper has been with the authors 45 days for 2 revisions. The average review cycle time was 32.3days.

1523-4614/00/0201/0068$05.001526-5498 electronic ISSN

Manufacturing & Service Operations Management �2000 INFORMSVol. 2, No. 1, Winter 2000, pp. 68–83 68

Optimizing Strategic Safety Stock Placementin Supply Chains

Stephen C. Graves • Sean P. WillemsLeaders for Manufacturing Program and A. P. Sloan School of Management, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139-4307, [email protected] of Business Administration, University of Cincinnati, Cincinnati, Ohio 45221-0130

Manufacturing managers face increasing pressure to reduce inventories across the supplychain. However, in complex supply chains, it is not always obviouswhere to hold safety

stock to minimize inventory costs and provide a high level of service to the final customer. Inthis paper we develop a framework for modeling strategic safety stock in a supply chain thatis subject to demand or forecast uncertainty. Key assumptions are that we can model thesupply chain as a network, that each stage in the supply chain operates with a periodic-reviewbase-stock policy, that demand is bounded, and that there is a guaranteed service time betweenevery stage and its customers. We develop an optimization algorithm for the placement ofstrategic safety stock for supply chains that can be modeled as spanning trees. Our assump-tions allow us to capture the stochastic nature of the problem and formulate it as a determin-istic optimization. As a partial validation of the model, we describe its successful applicationby product flow teams at Eastman Kodak. We discuss how these flow teams have used themodel to reduce finished goods inventory, target cycle time reduction efforts, and determinecomponent inventories. We conclude with a list of needs to enhance the utility of the model.(Base-Stock Policy; Dynamic Programming Application;Multi-echelon Inventory System;Multi-StageSupply-Chain Application; Safety Stock Optimization)

1. IntroductionManufacturing firms are subject to pressure to do ev-erything faster, cheaper, and better. Firms are expectedto continue to improve customer service by increasingon-time deliveries and reducing delivery lead-times.At the same time, they must provide this service morecheaply and utilize fewer assets. Increasingly, firmsneed to do this for a global marketplace.This pressure to improve forces companies to look

at their operations from a supply-chain perspectiveand to seek improvements from better coordinationand communication across the supply chain. A supply-chain perspective is essential to avoid some of the localsuboptimization that occurs when each step in a pro-cess operates independently with its own metrics and

rewards. Using a supply chain as a focusing mecha-nism challenges an organization to examine cross-functional solutions to address some of the barriersthat inhibit improvements.The primary intent of this research is to develop a

tactical tool to help cross-functional teams in their ef-forts to model and improve a supply chain. We pro-vide a framework for modeling a supply chain anddevelop an approach, within the framework, to opti-mize the inventory in a supply chain. More specifi-cally, we provide an optimization algorithm for find-ing the optimal placement of safety stock in a supplychain, modeled as a spanning tree and subject to un-certain demand. Key assumptions for the optimizationare that each stage of the supply chain operates with a

GRAVES AND WILLEMSOptimizing Strategic Safety Stock Placement in Supply Chains

Manufacturing & Service Operations ManagementVol. 2, No. 1, Winter 2000, pp. 68–83 69

periodic-review, base-stock policy, that each stagequotes a guaranteed service time to its customers, andthat demand is bounded.We refer to this effort as the placement of “strategic”

safety stock. As will be seen, the optimization modelleads to the determination of where to place decou-pling inventories that protect one part of the supplychain from another. In particular, a decoupling safetystock is an inventory large enough to permit the down-stream portion of the supply chain to operate indepen-dently from the upstream, provided that the upstreamportion replenishes the external demand. In this sense,the determination of where to place these decouplingpoints in a supply chain is a major design decision andis “strategic” in nature. Furthermore, this terminologyis consistent with that used in industry.In order to have an opportunity to test the research

and validate its utility for industry, we have built acommercial-quality software application to implementthe model described in this paper. The software can bedownloaded from our website, http://web.mit.edu/lfmrg3/www/.In the remainder of this section we briefly discuss

related literature. In §2, we present our framework formodeling a supply chain by stating and discussing thekey assumptions. We introduce the model for a singlestage in §3; this serves as the building block for themulti-stage model described in §4. In §5 we developthe optimization algorithm for safety stock placementin a supply chain modeled as a spanning tree. We pres-ent an overview of our application experience with themodel in §6, and conclude in §7 with thoughts on howto improve the tool.

Related Literature.There is an extensive literature on inventory modelsfor multi-stage or multi-echelon systems with uncer-tain demand; much of this literature is applicable tosupply chains as now defined. We refer the reader tothe survey articles by Axsater (1993), Federgruen(1993), Inderfurth (1994), van Houtum et al. (1996), andDiks et al. (1996). Within this vast literature, we men-tion two sets of papers that are most related to ourwork.First, we note the work by Simpson (1958) who de-

termined optimal safety stocks for a supply chainmod-eled as a serial network. Our work is based on similar

assumptions about the demand process and about theinternal control policies for the supply chain. Ourworkis also closely related to that of Inderfurth (1991, 1993),Inderfurth and Minner (1998), and Minner (1997), whoalso build off Simpson’s framework for optimizingsafety stocks in a supply chain. We extend the work ofSimpson and of Inderfurth and Minner by treating amore general network, namely spanning trees.We alsoprovide a different, and we believe richer, interpreta-tion of the framework and its applicability to practice.We provide new results in the appendix on the formof the optimal policies when we relax a constraint onthe internal control policy for the supply chain.Second, our work is closely related in intent to Lee

and Billington (1993), Glasserman and Tayur (1995),and Ettl et al. (2000). Each of these papers examines thedetermination of the optimal base-stock levels in a sup-ply chain, and tries to do so in a way that is applicableto practice. Glasserman and Tayur (1995) show how touse simulation and infinitesimal perturbation analysisto find the optimal base-stock levels for capacitatedmulti-stage systems. Both Lee and Billington (1993)and Ettl et al. (2000) develop performance evaluationmodels of a multi-stage inventory system, where thekey challenge is how to approximate the replenish-ment lead-times within the supply chain. They thenformulate and solve a nonlinear optimization problemthat minimizes the supply chain’s inventory costs sub-ject to user-specified requirements on the customer ser-vice level. Our work is similar in that we also assumebase-stock policies and focus on minimizing the inven-tory requirements in a supply chain. The resultingmodels and algorithms are much different, though,due to different assumptions about the demand pro-cess and different constraints on service levels withinthe supply chain.

2. Assumptions

Multi-Stage Network.We model a supply chain as a network where nodesare stages in the supply chain and arcs denote that anupstream stage supplies a downstream stage. A stagerepresents a major processing function in the supplychain. A stage might represent the procurement of araw material, or the production of a component, or the


Manufacturing & Service Operations Management70 Vol. 2, No. 1, Winter 2000, pp. 68–83

manufacture of a subassembly, or the assembly andtest of a finished good, or the transportation of a fin-ished product from a central distribution center to aregional warehouse. Each stage is a potential locationfor holding a safety-stock inventory of the item pro-cessed at the stage.We associate with each arc a scalar �ij to indicate

how many units of the upstream component i are re-quired per downstream unit j. If a stage is connectedto several upstream stages, then its production activityis an assembly requiring inputs from each of the up-stream stages. A stage that is connected to multipledownstream stages is either a distribution node or aproduction activity that produces a common compo-nent for multiple internal customers.

Production Lead-Times.For each stage, we assume a known deterministic pro-duction lead-time; call it Ti. When a stage reorders, theproduction lead-time, is the time from when all of theinputs are available until production is completed andavailable to serve demand. The production lead-timeincludes the waiting and processing time at the stage,plus any transportation time to put the item into in-ventory. For instance, suppose stage k requires inputsfrom stage i and j; then for a production request madeat time t, stage k completes the production at time t �

Tk, provided that there are adequate supplies of i andj at time t.We assume that the production lead-time is not im-

pacted by the size of the order; hence, in effect, weassume that there are no capacity constraints that limitproduction at a stage.

Periodic-Review Base-Stock Replenishment Policy.We assume that all stages operate with a periodic-review base-stock replenishment policy with a com-mon review period. For each period, each stage ob-serves demand either from an external customer orfrom its downstream stages, and places orders on itssuppliers to replenish the observed demand. There isno time delay in ordering; hence, in each period theordering policy passes the external customer demandback up the supply chain so that all stages see the cus-tomer demand.

Demand Process.Without loss of generality, we assume that external de-mand occurs only at nodes that have no successors,

which we term demand nodes or stages. For each de-mand node j, we assume that the end-item demandcomes from a stationary process for which the averagedemand per period is lj.An internal stage has only internal customers or suc-

cessors; its demand at time t is the sum of the ordersplaced by its immediate successors. Since each stageorders according to a base-stock policy, the demand atinternal stage i is:

d (t) � � d (t)i � ij j(i,j)�A

where dj(t) denotes the realized demand at stage j inperiod t and A is the arc set for the network represen-tation of the supply chain. For every arc (i, j)� A, stagej orders an amount �ij dj(t) from upstream stage i intime period t. The average demand rate for stage i is:

l � � l .i � ij j(i,j)�A

We assume that demand at each stage j is boundedby the function Dj(s), for s � 1, 2, 3, . . . Mj, where Mj

is the maximum replenishment time for the stage.1

That is, for any period t and for s � 1, 2, 3, . . . Mj, wehave

D (s) � d (t � s � 1) � d (t � s � 2) � • • • � d (t).j j j j

We defineDj(0)� 0 and assume thatDj(s) is increasingand concave on s � 1, 2, 3, . . . Mj.

Discussion of Assumption of Bounded Demand.The assumption of bounded demand is contrary tomost of the literature on stochastic-demand inventorymodels, and as such, is controversial. We need to dis-cuss this assumption in the context of the intent of theresearch, namely to provide tactical guidance forwhere to position safety stock in a supply chain.We presume that it is possible to establish a mean-

ingful upper bound on demand over varying horizonsfor each end item. By meaningful, we mean in the con-text of setting safety stocks: the safety stock is set tocover all demand realizations that fall within the upperbounds. If demand exceeds the upper bounds, then thesafety stock, by design, is not adequate. In such ex-traordinary cases, a manager resorts to other tactics to

1The maximum replenishment time for node j is defined asMj � Tj

� max {Mi | i:(i,j) � A}.



handle the excess demand. A manager might use ex-pediting, subcontracting, premium freight transporta-tion, and/or overtime to accommodate the windfall ofdemand. In specifying the demand bounds, a managerindicates explicitly a preference for how demand vari-ation should be handled—what range is covered bysafety stock andwhat range is handled by other actionsor responses.As an example, consider a typical assumptionwhere

demand for end item j is normally distributed eachperiod and i.i.d., with mean l and standard deviationr. Then, for the purposes of positioning safety stock, amanager might specify the demand bounds at the de-mand node by:

D (s) � sl � kr s (1)�j

where k reflects the percentage of time that the safetystock covers the demand variation. The choice of k in-dicates how frequently the manager is willing to resortto other tactics to cover demand variability.In some contexts there may be natural bounds on

the end-item demand. For instance, suppose the enditem is a component or subassembly for a manufac-turing process whose production is limited by capacityconstraints or by a frozen master schedule. An exam-ple is a supply chain that supplies components to anautomobile assembly line or an OEM subassembly toa system integrator. In these cases, bounded demandfor the component corresponds to the maximum con-sumption by the manufacturing process over varioustime horizons.For each internal stage we assume that we can also

establish meaningful demand bounds. If stage i has asingle successor, say stage j, then Di(s) � �ij Dj(s) forall relevant s. For stages with more than one successor,we require some judgment for deciding how to com-bine the demand bounds for the downstream stages toobtain a relevant demand bound for the upstreamstage for the purposes of positioning the safety stock.One possibility is just to sum the downstream demandbounds; however, this approach assumes that there isno risk pooling from combining the demand of mul-tiple end items. An alternative approach is to assumethat there will be some relative reduction in variabilityas we combine demand streams, i.e., some risk pool-ing. For instance, we might infer the demand boundsfor internal stages by means of an expression like

pD (s) � sl � p {� (D (s)� � sl )} (2)i i � ij j j�(i,j)�A

where p � 1 is a given constant. Larger values of pcorrespond to more risk pooling. Setting p � 1 modelsthe case of no risk pooling. If we were to model theend-item demand bounds by Equation (1), then settingp � 2 equates to combining standard deviations of in-dependent demand streams.We do not attempt to model what happens when

actual demand exceeds the bounds. When this hap-pens, we assume that the supply chain responds withan equally extraordinary measure, as noted above. Weregard this as beyond the scope of the model, giventhe stated intention to provide tactical decision sup-port. See Kimball (1988), Simpson (1958), and Graves(1988) for further discussion of this assumption.Finally we note that there are no assumptions made

about the demand distribution.

Guaranteed Service Times.We assume that each demand node j promises a guar-anteed service time Sj by which the stage j will satisfycustomer demand. That is, the customer demand attime t, dj(t), must be filled by time t � Sj. Furthermore,we assume that stage j provides 100% service for thespecified service time: stage j delivers exactly dj(t) tothe customer at time t � Sj.Similarly, an internal stage i quotes and guarantees

a service time Sij for each downstream stage j, (i, j) �

A. Given a base-stock policy, stage j places an orderequal to �ij dj(t) on stage i at time t; then stage i deliversexactly this amount to stage j at time t � Sij.For the initial development, we assume that stage i

quotes the same service time to all of its downstreamcustomers; that is, Sij � Si for each downstream stagej, (i, j) � A. Graves and Willems (1998) describe howto extend the model to permit customer-specific ser-vice times. In brief, if there is more than one down-stream customer, we can insert zero-cost, zero-production lead-time dummy nodes between a stageand its customers to enable the stage to quote differentservice times to each of its customers. The stage quotesthe same service time to the dummy nodes and eachdummy node is free to quote any valid service time toits customer stage.



The service times for both the end items and the in-ternal stages are decision variables for the optimizationmodel, as will be seen in §4. However, as a model in-put, we may impose bounds on the service times foreach stage. In particular, we assume that for each enditem we are given a maximum service time, presum-ably set by the marketplace.

Discussion of Assumption of Guaranteed ServiceTimes.We assume that there are no violations of the guar-anteed service times; each stage provides perfect or100% service to its customers. As such, we do not ex-plicitly model a tradeoff between possible shortagecosts and the costs for holding inventory. Rather, wepose the problem as being how to place safety stocksacross the supply chain to provide 100% service for theassumed bounded demand with the least inventoryholding cost.In defense of this assumption, it is often very diffi-

cult in practice to assess shortage costs for an externalcustomer. Similarly, when we have asked managersfor their desired service level, more often than not theresponse is that there should be no stock-outs for ex-ternal customers. We have found that managers seemmore comfortable with the notion of 100% service forsome range of demand; they accept the fact that if de-mand exceeds this range, they will have shortages un-less they can somehow expand the response capabilityof their supply chain. The assumptions for the modelpresented herein are consistent with this perspective.For an internal customer, guaranteed service times

need not be optimal in terms of least inventory costs.Indeed we show in the Appendix how to relax thisassumption for a serial network, and report the costimpact of this assumption for a set of 36 test problems:the safety stock holding cost is 26% higher on average,while the total inventory cost is 4% higher on average.However, guaranteed service times are very practicalin contexts where there is a need to coordinate replen-ishments. For instance, any assembly or subassemblystage requires the concurrent availability of multiplecomponents, not all of which might be explicitly in-cluded in the model. When we assume guaranteed ser-vice times, we make the challenge of coordinating theavailability of these components much easier.

3. Single-Stage ModelIn this section we present the single-stage model (seeKimball 1988 or Simpson 1958) that serves as the build-ing block for modeling a multi-stage supply chain.

Inventory ModelWe assume the inventory system starts at time 0 withinitial inventory Ij(0). Given our assumptions, we canexpress the finished inventory at stage j at the end ofperiod t as

I (t) � B � d (t � SI � T , t � S ) (3)j j j j j j

where Bj � Ij(0) � 0 denotes the base stock, dj(a, b)denotes the demand at stage j over the time interval(a, b], and SIj is the inbound service time for stage j.Since we assume a discrete-time demand process, weunderstand dj(a, b) to be

d (a, b) � d (a � 1) � d (a � 2) � • • •� d (b)j j j j

for a � b, where dj(t) � 0 for t � 0. When a � b, wedefine dj(a, b) � 0.The inbound service time SIj is the time for stage j

to get supplies from its immediate suppliers and tocommence production. In period t, stage j places anorder equal to �ij dj(t) on each upstream stage i forwhich �ij � 0. Stage j cannot start production to re-plenish dj(t) until all inputs have been received; thuswe have SIj � max {Si| i:(i, j) � A}.We permit SIj � max {Si | i:(i, j) � A} to allow for

the possibility that the replenishment time for the in-ventory at stage j is less than its service time Sj; that is,the case when

max {S |i:(i, j) � A} � T � S .i j j

In this case we would delay the orders to the suppliersby Sj � max {Si| i:(i, j) � A} � Tj periods, so that thesupplies arrive exactly when needed. To account forthis case, we set the inbound service time so that theeffective replenishment time for the inventory at stagej, namely SIj � Tj, equals the service time Sj, i.e., SIj �Tj � Sj. Thus, we define the inbound service time as

SI � max{S � T , max {S |i:(i, j) � A}}.j j j i

If the inbound service time is such that SIj � Si forsome (i, j) � A, then by convention stage j delays or-ders from stage i by SIj � Si periods to avoid unnec-essary inventory.



Now, to explain Equation (3), we observe that in pe-riod t stage j completes the replenishment of the de-mand observed in period t � SIj � Tj. By the end ofperiod t, the cumulative replenishment to the inven-tory at stage j equals dj(0, t � SIj � Tj). In period t,stage j fills the demand observed in time period t � Sjfrom its inventory. By the end of period t the cumu-lative shipment from the inventory at stage j equalsdj(0, t � Sj). The difference between the cumulativereplenishment and the cumulative shipment is the in-ventory shortfall, dj(t � SIj � Tj, t � Sj). The on-handinventory at stage j is the initial inventory or base stockminus the inventory shortfall, as given by Equation (3).

Determination of Base Stock.In order for stage j to provide 100% service to its cus-tomers, we require that Ij(t) � 0; from (3) we see thatthis requirement equates to

B � d (t � SI � T , t � S ).j j j j j

Since demand is bounded, we can satisfy the aboverequirement with the least inventory by setting thebase stock as follows:

B � D (s) where s � SI � T � S . (4)j j j j j

Any smaller value does not assure that Ij(t) � 0, andthus cannot guarantee 100% service.In words, the base stock equals the maximum pos-

sible demand over the net replenishment time for thestage. The net replenishment time for stage j is the re-plenishment time (SIj � Tj) minus its service time (Sj).At any time t, stage j has filled its customers’ demandthrough time t � Sj, but has only been replenished fordemand through time t � SIj � Tj. The base stockmustcover this time interval of exposure, namely the netreplenishment time.

Safety Stock Model.We use Equations (3) and (4) to find the expected in-ventory level E[Ij]:

E[I ] � B � E[d (t � SI � T , t � S )]j j j j j j

� D (SI � T � S ) � (SI � T � S )l . (5)j j j j j j j j

The expected inventory represents the safety stockheld at stage j, and depends on the net replenishmenttime and the demand bound. As an example, suppose

the demand bound is given by Equation (1); then thesafety stock is E[Ij] � kr .SI � T � S� j j j

Pipeline Inventory.In addition to the safety stock, wemaywant to accountfor the in-process or pipeline stock at the stage. Follow-ing the development for Equation (3), we observe thatthe work-in-process inventory at time t is given by

W (t) � d (t � SI � T , t � SI ).j j j j j

That is, the work-in-process corresponds to Tj periodsof demand. The expected work-in-process dependsonly on the lead-time at stage j and is not a functionof the service times:

E[W ] � T l .j j j

Hence, in posing an optimization problem in the nextsection, we ignore the pipeline inventory and onlymodel the safety stock. This is not to say that the work-in-process is not a significant part of the inventory ina supply chain. But for the purposes of this work, weassume that the lead-time of a stage, as well as thedemand rate, are input parameters and thus the pipe-line stock is predetermined. Nevertheless, in any ap-plication, we account for both the safety stock and thepipeline stock as both will contribute to the total sup-ply chain inventory.

4. Multi-Stage ModelTo model the multi-stage system, we use Equation (5)for every stage, but where the inbound service time isa function of the outbound service times for the up-stream stages; to wit, the model for stage j is

E[I ] � D (SI � T � S ) � (SI � T � S )l , (6)j j j j j j j j j

SI � T � S � 0, (7)j j j

SI � S � 0 for all (i, j) � A. (8)j i

Equation (6) expresses the expected safety stock as afunction of the net replenishment time. Equation (7)assures that the net replenishment time is nonnegative.Equation (8) constrains the inbound service time toequal or exceed the service times for the upstreamstages.We assume that the production lead-times, the

means and bounds of the demand processes, and the



maximum service times for the demand nodes areknown input parameters. This suggests the followingoptimization problem P for finding the optimal servicetimes:

N

P min h {D (SI � T � S )�(SI � T � S )l }� j j j j j j j j jj�1

s. t. S � SI � T for j � 1, 2, . . . , N,j j j

SI � S � 0 for all (i, j) � A,j i

S � s for all demand nodes j,j j

S , SI � 0 and integer for j � 1, 2, . . . , N,j j

where hj denotes the per-unit holding cost for inven-tory at stage j, and sj is the maximum service time fordemand node j. The objective of problem P is to min-imize the holding cost for the safety stock in the supplychain. The constraints assure that the net replenish-ment times are nonnegative, the inbound service timeequals the maximum supplier service time, and theend-item stages satisfy their service guarantee. The de-cision variables are the service times.Problem P is a nonlinear optimization problem. The

objective function is a concave function, provided thatthe demand bound Dj( ) is a concave function for eachstage j. The feasible region is convex but not necessarilybounded; however, one can show that the optimal ser-vice times need not exceed the sum of the productionlead-times, provided thatDj( ) is a nondecreasing func-tion for each stage j. Thus, problem P is the minimi-zation of a concave function over a closed, boundedconvex set. An optimum for such problems is at anextreme point of the feasible region, e.g., Luenberger1973.Simpson (1958) considered a serial-line supply

chain, where he assumed that the guaranteed servicetime for the external customer is zero. Simpsonshowed that there is an optimal extreme point solutionfor P for which Si � 0 or Si � Si�1 � Ti, where stagei � 1 supplies stage i. Thus, there is an “all or nothing”optimal solution; a stage either has no safety stock (Si� Si�1 � Ti) or has sufficient safety stock (Si � 0) tode-couple it from its downstream stage. Gallego andZipkin (1999) provide supporting evidence that “all ornothing” policies can be near optimal in serial systems

under more traditional assumptions where demand isnot bounded.Graves (1988) observed that the serial-line problem

can be solved as a shortest path problem. In a series ofpapers. Inderfurth (1991, 1993), Inderfurth andMinner(1998), and Minner (1997) show how to solve problemP by dynamic programming when the supply chain isan assembly network or a distribution network.Graves and Willems (1996) developed similar resultsfor assembly and distribution networks. In the nextsection we present a dynamic programming algorithmfor the more general case of a spanning tree.

5. Algorithm for Spanning TreeWe describe in this section how to solve P by dynamicprogramming when the underlying network for thesupply chain is a spanning tree, like in the Figure 1.We solve P by decomposing the problem into N

stages whereN is the number of nodes in the spanningtree and there is one stage for each node. For a span-ning tree, there is not a readily-apparent ordering ofthe nodes by which the algorithm would proceed. In-deed, we label the nodes in a spanning tree (and thussequence the algorithm) so that there will be a singlestate variable for the dynamic programming recursion.However, the state variable for the dynamic programwill be either the inbound service time at a stage or itsoutbound service time, where the determination de-pends on the topology of the network.We first present the algorithm for labeling the nodes.

Next we present the functional equations for the dy-namic programming recursions, and then state thealgorithm.

Labeling the Nodes.The algorithm for labeling or re-numbering the nodesis as follows:1. Start with all nodes in the unlabeled set, U.2. Set k :� 1.3. Find a node i � U such that node i is adjacent to

at most one other node inU. That is, the degree of nodei is 0 or 1 in the subgraph with node set U and arc setA defined on U.4. Remove node i from set U and insert into the la-

beled set L; label node i with index k.



Figure 2 Renumbered Spanning Tree

Figure 1 Spanning Tree

5. Stop if U is empty: otherwise set k :� k � 1 andrepeat steps 3– 4.For a spanning tree, there is always an unlabeled nodein step 3 that is adjacent to at most one other unlabelednode. As a consequence, the algorithm will eventuallylabel all of the nodes inN iterations. Indeed, each nodelabeled in the first N � 1 steps is adjacent to exactlyone other node in set U. Thus, the nodes with labels 1,2, . . . , N � 1 have one adjacent node with a higherlabel, denoted by p(k) for k � 1, 2, . . . , N � 1. NodeN has no adjacent nodes with larger labels.As an illustration, we renumber the nodes in Figure

1 to produce Figure 2. Note that the labeling is notunique as there may be multiple choices for node i instep 3.For each node k we define Nk to be the subset of

nodes {1, 2, . . . , k} that are connected to k on the sub-graph with node set {1, 2, . . . , k}. We will use Nk toexplain the dynamic programming recursion. We candetermine Nk by the following equation:

N � {k} � � N � � N .k i ji�k,(i,k)�A j�k,(k,j)�A

For instance, in Figure 2, Nk is {3} for k � 3, {1, 2, 3, 9}for k � 9, {1, 2, 3, 4, 5, 9, 11} for k � 11, and {6, 7, 8,10, 12} for k � 12. We can compute Nk as part of thelabeling algorithm.

Functional Equations.The dynamic program evaluates a functional equationfor each node in the spanning tree, where we have re-numbered the nodes as described above. There are twoforms for the functional equation. First, the functionfk(S) is the minimum holding cost for safety stock in asubnetwork with node set Nk, where we assume thatthe outbound service time for stage k is S. Second, thefunction gk(SI) is the minimum holding cost for safetystock in a subnetwork with node set Nk, where we as-sume that the inbound service time for stage k is SI.At node k (or stage k) for 1 � k � N � 1, the algo-

rithm determines either fk(S) or gk(SI), depending uponthe location of the node with higher label that is adja-cent to k. If p(k) is downstream [upstream] of node k,then we evaluate fk(S) [gk(SI)]. For nodeN, we can eval-uate either functional equation.To develop the functional equations, we first define

the minimum inventory holding cost for the subnet-work with node set Nk as a function of both the in-bound service time and the outbound service time atnode k:

c (S, SI) � h {D (SI � T � S) � (SI � T � S)l }k k k k k k

� f (SI) � g (S).� i � j(i,k)�A (k,j)�Ai�k j�k

The first term is the holding cost for the safety stock atnode k as a function of S and SI.The second term corresponds to the nodes inNk that

are upstream of k. For each node i that supplies nodek, we include the minimum inventory holding costs forthe subnetwork with node set Ni, as a function of SI.The inbound service time to node k, SI, is an upperbound for the outbound service time for node i. Wecan show that fi( ), the inventory holding costs for thesubnetwork with node set Ni, is nonincreasing in theservice time at node i. Hence, we equate the outboundservice time at i to the inbound service time at k with-out loss of generality.The third term corresponds to the nodes in Nk that

are downstream of k. For each node j, j � Nk and (k, j)� A, we include the minimum inventory holding costsfor the subnetwork with node set Nj, as a function ofS. The outbound service time for node k, S, is a lowerbound for the inbound service time for node j. We can



show that gj( ), the inventory holding costs for the sub-network with node set Nj, is nondecreasing in the in-bound service time to node j; and thus we equate theinbound service time at j to the outbound service timeat k without loss of generality.We solve the following optimization by enumera-

tion to find the functional equation fk(S):

f (S) � min{c (S, SI)}k kSI

s. t. max (0, S � T ) � SI � M � T , and SI integer,k k k

where Mk is the maximum replenishment time fornode k. The lower bound on SI comes from P, whilethe definition of Mk gives the upper bound.The functional equation for gk(SI) is very similar in

structure:

g (SI) � min{c (S, SI)}k kS

s. t. 0 � S � SI � T , and S integer.k

If node k is a demand node, then we also constrain Sby its maximum service time, i.e., S � sk. The mini-mization can be done by enumeration.

Dynamic Program.The dynamic programming algorithm is now asfollows:1. For k :� 1 to N � 12. If p(k) is downstream of k, evaluate fk(S) for S

� 0, 1, . . ., Mk.3. If p(k) is upstream of k, evaluate gk(SI) for SI �

0, 1, . . ., Mk � Tk.4. For k :� N evaluate gk(SI) for SI � 0, 1, . . . , Mk

� Tk.5. Minimize gN(SI) for SI � 0, 1, . . . , MN � TN to

obtain the optimal objective function value.This procedure finds the optimal objective functionvalue; we can find an optimal set of service times bythe standard backtracking procedure for a dynamicprogram.To summarize, at each stage of the dynamic pro-

gram, we find the minimum inventory holding costsfor the subnetwork with node set Nk, as a function ofa state variable. The state variable depends on the di-rection of the arc that connects the subnetwork Nk to

the rest of the network. When the connecting arc orig-inates in Nk, then the state variable is the outboundservice time (step 2); otherwise, the state variable is theinbound service time (step 3). We number the nodesso that we have previously determined the functionsrequired to evaluate either fk(S) or gk(SI). At stage N(step 4), we determine the inventory costs for the entirenetwork as a function of the inbound service time tonode N. At step 5, we optimize over the inbound ser-vice time to find the optimal inventory cost.The computational complexity of the algorithm is of

order NM2 where M is the maximum service time,which is bounded by the sum of the production lead-times . We have implemented the algorithm forN T� jj�1

a PC in the C�� programming language. The runtimes for real problems with 25 to 30 nodes are effec-tively instantaneous on a Pentium PC with a 100 meg-ahertz Intel processor.

6. ApplicationThis section presents an application of the model at theEastman Kodak Company. Starting in 1995, Kodak hasapplied the model to more than eleven finished prod-ucts across two of its assembly sites within its equip-ment division. We first present the model’s applicationto the internal supply chain for a high-end digital cam-era,2 and then summarize Kodak’s financial results, asof 1997 year-end.

Product Background.The key subassemblies for the digital camera are a tra-ditional 35 mm camera, an imager, and a circuit-boardassembly. The 35mm camera is procured from an out-side vendor. The imager (a charge-coupled device) andthe circuit-board assembly are produced internally.The 35mm camera supplies the lens, shutter, and focusfunctions for the digital camera. The imager capturesand digitizes the picture, and the circuit-board assem-bly processes and stores the image. To produce the dig-ital camera, the back of the 35mm camera is removedand replaced with a housing containing the imager

2The data presented in this section has been altered to protect pro-prietary information. However, the resulting qualitative relation-ships and insights drawn from this example are the same as theywould be from using the actual data.



Figure 3 Implemented Safety Stock Policy for Digital Camera. AllStages Have a Circle That Denotes the Processing Activityat the Stage. A Triangle Denotes That the Stage Holds aSafety Stock of Its Finished Goods

and circuit board. The camera is then tested to makesure that there are no defects in the imager. Once thecamera passes the quality tests, the product is shippedto the distribution center. From the distribution center,the camera is shipped to the final customers, which forour purposes are high-end photography shops andcomputer superstores.In Figure 3, we provide a high-level depiction of this

supply chain. In addition to the three key subassem-blies, we include the remaining parts in order to ac-curately represent the product’s cost structure; sincethere are nearly 100 additional parts in a camera, mod-eling them in any level of detail would greatly expandthe size of the model. Hence, we group these parts intotwo aggregate stages of the supply chain, where onestage represents all of the parts with long procurementlead-times (greater than 60 days) and the other stagerepresents the short lead-time parts (less than 60 days).We also aggregate certain operations. As seen in Fig-

ure 3, we combine the build operation for a camerawith the test operation and the packing operation. Theimager stage and circuit board stage are also aggre-gates as each represents the flow through a separate

department. The circuit-board stage entails circuitboard assembly and test. The imager stage consists ofa semiconductor operation to produce wafers, fol-lowed by packaging and testing of the semiconductors,followed by an assembly operation.

Implementation Approach.The product’s supply chain crosses several functionalboundaries within Kodak. Functional areas like circuit-board assembly and imager assembly are separate de-partments and act as suppliers to an assembly groupthat performs final assembly and test. Distribution is aseparate organization and owns the product once itleaves the final assembly area. To improve coordina-tion across the departments, the equipment division atKodak has set up product flow teams with the generalcharge to optimize their supply chains.The product flow team for the digital camera relied

on the model to identify opportunities for better co-ordination and improved asset utilization. The teamimplemented the model in phases. The implementa-tion strategy was to start simple and get experiencewith the model; once there was some evidence of theutility of the model, the team extended the applicationin increments to capture more and more of the supplychain.

Phase One.The initial goal was to optimize the safety stock levelsfor the stages that were under the direct control of thefinal assembly area. The decision to start with the finalassembly area was based on the product’s high mate-rial cost and its relatively simple supply chain struc-ture, as described above. The (disguised) costs andproduction lead-times are:

Table 1 Phase One Digital Camera Information

Stage Name Production Lead-Time Cost Added

Camera 60 750Imager 60 950Circuit Board 40 650Other Parts LT � 60 days 60 150Other Parts LT � 60 days 150 200Build/Test/Pack 6 250Transfer to DC 2 50Ship to Customer 3 0



Figure 4 Digital Camera Supply Chain

The demand bound was estimated by Equation (1)where l � 11, r � 7 and k � 1.645. From looking athistorical demand and future demand estimates, Ko-dak felt that this function realistically captured therange of demand for which they wanted to use safetystock.This demand characterization excluded large one-

time orders from the government and some large cor-porations. These orders are typically for 200 – 300 unitswith delivery scheduled less than a month from whenthe order is placed. However, since there is advancewarning about these orders and they are independentof the other demand for the product, we developed aseparate anticipatory stock policy to deal with large,infrequent orders.Marketing determined that the maximum service

time to the final customer is five days.Finally, the assembly group imposed the constraint

that a safety stock of imagers must be held on-site atfinal assembly. Therefore, we set the service time forthe imager stage to be zero; the effect of this constraintincreased the total safety stock cost by 8.7%.In the optimal solution, the subassembly stages, the

aggregate parts stages, and the build/test/packagestages hold safety stocks and quote zero service times.The ship-to-distribution and ship-to-customer stageseach quote their maximum feasible service times, twoand five days, respectively. The annual holding costfor the safety stock is $78,000. Thus, the optimal solu-tion holds an inventory of components, subassemblies,and completed cameras at the manufacturing site, butholds no inventory in the distribution center. In effect,the distribution center would act only as an order pro-cessing and transshipment center. This is feasible sinceit is possible to get the product from the assembly areathrough the distribution center and to the final cus-tomer within the maximum service time of five days.The product flow team decided to explore some

near-optimal solutions because they felt that therewere some additional organizational constraints notcaptured in the model; in particular, distributionwould want to hold safety stock on-site. To amelioratethe situation, the team suggested that both manufac-turing and distribution hold safety stock and quotezero service times. However, the model showed thatthe cost for the safety stock would increase to $89,000.

The team also investigated a policy in which the dis-tribution center would hold safety stock but the manu-facturing site would not. The safety stock cost for thispolicy was $81,000, which was deemed to be accept-able as it was quite close to the unconstrained opti-mum and satisfied distribution’s desire to hold inven-tory. This policy, as shown in Figure 3, wasimplemented at the end of phase one of theapplication.

Phase Two.After the initial phase, the product flow team ex-panded the model to incorporate the internal supplychain for the imager. The resulting supply chain isshown in Figure 4.Prior to this study, safety stocks of (in-process) im-

agers had been held at each stage of the supply chain.By application of the model, the team decided to re-move safety stocks from two stages in the supply chainfor the imagers, as shown in the figure. This requiredsome increase in the downstream safety stocks of fin-ished imagers, but overall the supply chain’s safetystock of imagers (measured in terms of finished ima-gers) was more than halved.Now that the model has been successfully piloted

with an internal supplier, the product flow team is inthe process of extending the model to incorporateother key internal and external suppliers.

Financial Results.Table 2 contains the financial summary for two assem-bly sites that use the model. Site A has applied the



Table 2 Financial Summary for Assembly Sites A and B

Assembly Site A Y/E 95 Y/E 96 Y/E 97

Worldwide FGI $6.7m $3.3m $3.6mRaw Material & WIP $5.7m $5.6m $2.9mDelivery Performance 80% 94% 97%Manufacturing Operation MTS RTO RTO

Assembly Site BWorldwide FGI $4.0m $4.0m $3.2mRaw Material & WIP $4.5m $1.6m $2.5mDelivery Performance Unavailable 78% 94%Manufacturing Operation MTS RTO RTO

model to each of its eight products and Site B has ap-plied the model to each of its three product families.The sales volume has remained relatively constantover the three years.At the start of 1996, the sites moved from a make-

to-schedule (MTS) to a replenish-to-order (RTO) sys-tem. The modeling effort began at the end of 1995 andwas used to help guide the transition to replenish-to-order. The increase in worldwide finished goods in-ventory for 1997 is due to a marketing promotion thatwas underway in Europe. By our estimate, this pro-motion increased the finished goods inventory by asmuch as $0.5 million. In the first year of the project,the emphasis was on reducing the areas directly underthe control of final assembly. Over the following year,the effort was on reducing the raw material costs andWIP in the manufacturing supply chain. The totalvalue of the inventory for these products has been re-duced by over one third over the two years.Kodak’s product flow teams have also used the

model for a variety of purposes other than settingsafety stocks. Some products have tens of componentswith long procurement lead-times. The model hashelped to prioritize the suppliers with whom to workto reduce these lead-times. The teams have used themodel to determine the cost effectiveness of lead-timereduction efforts in manufacturing. One can comparethe investment required to reduce a lead-time versusthe cost savings from the reductions in pipeline andsafety stock cost. Finally, manufacturing and market-ing personnel have used the model to help quantifythe cost of quoting a specific maximum service time to

the final customer. With the model, the supply-chainteam can accurately estimate the costs of a one-day,one-week, or two-week guaranteed service time to thecustomer, and weigh the costs of the policy against themarketing benefits of the policy.Another benefit of the model is that it provides a

common, objective framework with which a cross-functional supply-chain team can work. In particular,we note that it provides a standard terminology andset of assumptions for these teams to use as they worktogether to improve or optimize a supply chain. Assuch the model has been a very effective communica-tion vehicle or platform.

7. ConclusionIn this paper we introduce and develop a model forpositioning safety stock in a supply chain. We modelthe supply chain as a network, where the nodes of thenetwork are the stages of a supply chain. We assumethat each stage uses a base-stock policy to control itsinventory. We also assume that each stage quotes aservice time to its customers, both internal and exter-nal, and that each stage provides 100% service for thesequoted service times. Finally we assume that externalcustomer demand is bounded.We show how to evaluate the inventory require-

ments at each stage as a function of the service times.For supply chains that can be modeled as spanningtrees, we develop an optimization algorithm for find-ing the service times that minimize the holding cost forthe safety stock in the supply chain.As a form of validation, we describe an application

of the model at Kodak to an internal supply chain fora digital camera. This application helped Kodak to re-position its inventories in this supply chain to reduceits inventory and increase its service performance. Inparticular, Kodak realized the benefit from creating afew strategic locations to hold safety stocks rather thanspreading the safety stock across the entire supplychain. We have also applied the model to a number ofother related products at Kodak and at two other com-panies (Black 1998, Coughlin 1998, Felch 1997, Wala1999).As with any research, we end with a number of un-

resolved issues and new questions. We discuss these



in the relative order of importance, based on our ex-perience in applying the research to date.

Stochastic Lead-Times. We assume that associatedwith each stage is a known, deterministic lead-time. Inpractice, this is often not true; for instance, componentprocurement times are often highly uncertain. It willbe of value to capture this in the model. We know howto extend the model in an approximate way for stagesthat procure rawmaterials or components from an out-side vendor. In effect, for such a stage we just need toapproximate its inventory requirements as a functionof the outbound service time quoted by the stage andthe stochastic procurement time. But it is less clear howto extend the model, either exactly or approximately,to permit stochastic lead-times at stages whose func-tion is not procurement.

Non-Stationary Demand. We assume that theend-item demand processes are stationary. Yet virtu-ally all of the products with which we have workedhave short lifetimes over which demand is never reallystationary. In practice, one runs the model under vari-ous (stationary) scenarios to see how sensitive thesafety stock is to the demand characterization(Coughlin 1998). Fortunately, we have found empiri-cally that where the model locates safety stock in thesupply chain is fairly insensitive to the demand. Thesize of the safety stock, though, does depend directlyon the demand characterization. We currently are con-ducting research to understand these observations bet-ter, to extend the model to treat non-stationarydemand.

Different Review Periods. We assume that eachstage operates with a base-stock policy with a commonreview period. In many supply chains different stageswill operate with different reorder frequencies. That is,whereas one stage may place replenishment orders ona daily basis, another stage may do this weekly. Inother cases, a stage may operate with a continuous-review policy so that the time between orders varies.We can extend the model to evaluate nested periodic-review base-stock polices in whichwhenever one stagereorders, all stages downstream also reorder. That is,the review period for an upstream stage is an integermultiple of the review period of its immediate custom-ers. However, we have not yet built the software to

implement this extension, as it is a major programmingtask and it may only be a partial fix to the issue.

Capacity Constraints. In the model we ignore ca-pacity constraints. For certain stages in a supply chain,the consideration of a capacity limit may be necessaryin order to get a credible model for determining safetystock requirements. At this time, we do not have agood idea of how to add this to the model.

General Networks. In this paper, we have devel-oped and implemented an optimization algorithm forsupply chains that can be modeled as spanning trees.We describe in Graves and Willems (1998) how to ex-tend this algorithm to general networks. However, wehave not done a systematic study of this extension be-yond some exploratory work. More research is neededto test and refine these ideas as well as uncover betterapproaches.3

AppendixIn this appendix we examine the assumption that each internal stagequotes a guaranteed service time to its customers. To get some in-sight, we consider a serial system for which we can determine theoptimal policy when we relax the assumption of guaranteed servicetimes for internal customers. We then compare the inventory hold-ing costs for the optimal policies with and without this assumptionfor a small set of test problems.

Consider a serial supply chain with N stages where stage 1 is thedemand node and stage i supplies stage i � 1 for i � 2, . .. , N. Thesame assumptions hold as in the original model, except that we donot require guaranteed service times to internal customers. Thereare no restrictions on the service level that stage i provides to itscustomer, stage i � 1 for i � 2, . . . , N; rather, these internal servicelevels depend on the base stocks, which are chosen to minimize theinventory holding costs for the entire supply chain. We do assumethat stage 1 provides a 100% service level to the external customer,and, without loss of generality, we assume that the service timequoted to the external customer is zero.

For ease of presentation, we assume that �i,i�1 � 1 for i � 2, . . . ,N. We let d(t) denote the end-item demand in period t; d(a, b]; denotethe end-item demand over the time interval (a, b]; and D(s) denotethe maximum possible end-item demand over a time interval of speriods.

3This research has been supported in part by the Eastman KodakCompany; by the MIT Leaders for Manufacturing Program, a part-nership between MIT and major U.S. manufacturing firms; and bythe MIT Integrated Supply Chain Management consortium. The au-thors acknowledge and thank Dr. John Ruark who contributed sig-nificantly to this research effort; John played a lead role in devel-oping the software application for implementing the results of thisresearch. We also wish to thank the editors and referees for theirvery helpful and constructive feedback on earlier versions of thepaper.



For each stage i, we define Qi(t) to be the shortfall or backlog attime t, namely the amount that has been ordered by the stage’s cus-tomer but not yet delivered. We assume at t � 0, Ii(t) � Bi � 0 andQi(t) � 0 for all stages.

We can show for i � 1, 2, . . . , N that the on-hand inventory andbacklog at time t are:

�I (t) � [B � d(t � T , t) � Q (t � T )] ,i i i i�1 i

�Q (t) � [d(t � T , t) � Q (t � T ) � B ] , (A1)i i i�1 i i

where [x]� � max(0, x), and QN�1(t) � 0 by definition. Equation(A1) requires that each stage has a deterministic lead-time and thateach stage follows a base-stock policy in which, for each period, eachstage observes end-item demand and places a replenishment orderfor this amount. The essence of the argument is to observe that thenet inventory on hand at a stage equals the stage’s base stock minusthe inventory on order. For stage i, the inventory on order at time tis the backlog as of time t � Ti, plus all of the demand over theinterval (t � Ti, t].

From Equation (A1) we can show by induction that for i �

1,2, . . . , N,

Q (t) � max[0, d(t � T , t) � B , d(t � T � T , t)i i i i i�1

� B � B , . . . , d(t � T � T � • • •�T , t)i i�1 i i�1 N

� B � B � • • •�B ]. (A2)i i�1 N

In order for the supply chain to provide 100% service to the externalcustomer, we must never have a backlog at stage 1; thus, we mustselect base stocks so that Q1(t) � 0 for all t. From Equation (A2) wesee that Q1(t) � 0 is assured if the base stocks satisfy the followingconstraints:

B � B � • • •�B � D(T � T � • • •�T )1 2 i 1 2 i

for i � 1, 2, . . . , N. (A3)

Thus, if the base stocks satisfy Equation (A3), there will never be ashortfall at stage 1 and end-item demand will be satisfied with 100%service. As we assume that the demand bounds can be realized, thenthe constraint set (A3) provides not just sufficient but also necessaryconditions for assuring 100% service for end-item demand.

In order to select the base stocks to minimize the inventory hold-ing costs for the supply chain, we must develop an expression forthe inventory holding costs; we note from Equation (A1) that the netinventory on hand at stage i is given by:

I (t) � Q (t) � B � d(t � T , t) � Q (t � T ). (A4)i i i i i�1 i

From Equation (A4) we can write the inventory holding costs for thesupply chain as:

N N

h E[I (t)] � h {B � lT � E[Q (t)] � E[Q (t � T )]} (A5)� i i � i i i i i�1 ii�1 i�1

where l is the expected demand rate, and E[ ] denotes expectation.We now pose an optimization problem to select the base stocks;

namely, we minimize Equation (A5) subject to Equation (A3) and

nonnegativity constraints on the base stocks. After dropping con-stant terms in Equation (A5) and noting that Q1(t) � 0 for any fea-sible solution, we write the optimization as

N N

min h B � e E[Q ]� i i � i�1 ii�1 i�2

P* s.t.

B � B � • • •� B � D(T � T � • • •� T )1 2 i 1 2 i

for i � 1, 2, . . . , N,

B � 0 for i � 1, 2, . . . , N,i

where ei � hi � hi�1 is the echelon holding cost. We note fromEquation (A2) that E[Qi] is a nonlinear function of Bi, . . . , BN for i �

1, 2, . . . , N.Our main result is that there is an optimal solution to P* in which

all the constraints in Equation (A3) are binding. More formally westate the following:

Result. If the echelon holding costs are nonnegative and if D( )is a nondecreasing function, then an optimal solution to P* is givenby

B � D(T ),1 1

B � D(T � • • •� T ) � D(T � • • •� T )i 1 i 1 i�1

for i � 2, . . . , N. (A6)

Proof. We note that the solution given by Equation (A6) is non-negative and satisfies the constraints in Equation (A3) as equalities;thus it is a feasible solution to P*. To prove that this is also an optimalsolution, we will argue that there must be an optimal solution inwhich the constraints in Equation (A3) are binding.

Suppose we have a solution , . . . , such that Equation (A3)B* B*1 N

holds as a strict inequality for one or more constraints. Suppose thekth constraint is the first constraint that is not binding and that k �

N; we will treat the case when k � N later. Thus, we assume

B* � B* � • • •� B* � D(T � T � • • •� T )1 2 i 1 2 i

for i � 1, 2, . . . , k � 1 and

B* � B* � • • •� B* � D(T � T � • • •� T ).1 2 k 1 2 k

We now define a new solution , . . . , in which the first k con-B** B**1 N

straints are satisfied as equalities, and show that its objective valueis no worse than that for , . . . , :B* B*1 N

B** � B* for i � 1, . . . , N and i � k, k � 1,i i

B** � B* � Dk k

B** � B* � D,k�1 k�1

where

k k�1

D � B* � D T � D T .k � i � i� � � �i�1 i�1



We first observe that D � 0 due to the supposition that the solution, . . . , satisfies the kth constraint in Equation (A3) as a strictB* B*1 N

inequality. Thus, we have . We also see that sinceB** � 0 B** � 0k�1 k

D( ) is nondecreasing. Hence the new solution , . . . , is non-B** B**1 N

negative. By construction, the new solution satisfies the kth con-straint as an equality, and there are no changes in the remainingconstraints. Thus, the new solution , . . . , is a feasible solution.B** B**1 N

To express the objective function for the new solution, we decom-pose it into two parts. The first part of the objective function is

N N N

h B** � (�h � h )D � h B* � �e D � h B*. (A7)� i i k k�1 � i i k � i ii�1 i�1 i�1

For the second part of the objective function, let E[Qi]* and E[Qi]**denote the expected backlog at stage i for the first and second so-lution. Then we find from Equation (A2) that

E[Q ]** � E[Q ]* for i � k � 1,i i

E[Q ]* � E[Q ]** � E[Q ]* � D for i � k � 1, andi i i

E[Q ]* � E[Q ]** � E[Q ]* � D.k�1 k�1 k�1

Thus, for nonnegative echelon holding costs, we can bound the sec-ond part of the objective function as follows:

N N

� e E[Q ]** � � e E[Q ]* � e D. (A8)� i�1 i � i�1 i ki�2 i�2

By combining Equations (A7) and (A8), we see that the objectivefunction for the second solution is no greater than the objective forthe first. Thus, we have found a new solution in which the first kconstraints in Equation (A3) are binding and whose objective valueis no worse than that for the first solution. This argument can becontinued in this fashion to construct a solution in which the first N� 1 constraints in Equation (A3) are binding and whose objectivevalue is no worse than that for the solution , . . . , . The argumentB* B*1 N

for the case when k � N is similar in structure but easier; we justhave to reduce the base stock for stage N until the Nth constraint isbinding, which can be donewith no penalty to the objective function.

Hence, there is a feasible solution that satisfies all the constraintsin Equation (A3) as equalities and that has an objective value nohigher than that for the solution , . . . , . Furthermore, this newB* B*1 N

solution must be given by Equation (A6), as it is easy to see that itis the unique binding solution to Equation (A3). Finally we concludethat Equation (A6) must be an optimal solution, as its objective valueequals or is less than that for any interior solution , . . . , . ThisB* B*1 N

completes the proof.We note that the optimal base-stock policy does not depend at all

on the holding costs. All we need to know is that the holding costsdo not decrease as we move down the supply chain, closer to thecustomer. We also note that this result generalizes to assembly sys-tems by means of the transformation given by Rosling (1989);namely, we can transform an assembly system into an equivalentserial system, and the result applies.

We use this result to compare the performance of the base stockpolicies with and without the assumption of guaranteed service

times for internal customers. The test problems were all for a 3-stageserial system; the problems differed according to their demand pro-cess, their production lead-times, and their holding costs.

For the demand process, we start with a Poisson demand distri-bution with mean k and with a specified percentile � to truncate thedemand. For each time window of length s, we set the demandbound D(s) as the smallest integer such that the cumulative proba-bility for the Poisson random variable with mean ks exceeds �. Wethen normalize the demand distribution over the truncated range.We consider four possible demand processes: k � 10, � � 0.90; k �

10, � � 0.98; k � 50, � � 0.90; k � 50, � � 0.98.We permit three settings for the production lead-times and three

settings for the holding costs, as follows:

(T , T , T ) � (4, 4, 4); (1, 3, 8); (8, 3, 1).1 2 3

(h , h , h ) � (1, 0.5, 0.2); (1, 0.66, 0.33); (1, 0.8, 0.5).1 2 3

By evaluating all combinations we have a total of 36 test prob-lems. For each test problem we determine the optimal policy for themodel with guaranteed internal service times and the optimal policy(given by the result above) for the model without this requirement.For each instance, we evaluate the base stocks, the safety-stock hold-ing cost and the total inventory holding cost. The safety stock hold-ing cost is given by the objective function of P for the model withguaranteed internal service times and by Equation (A5) for themodel without this requirement. The total inventory holding cost isthe sum of the safety-stock holding cost plus the pipeline-stock hold-ing cost. The expected pipeline stock at stage i is lTi; we assume thatthe holding cost for the pipeline stock at stage i is (hi � hi�1)/2.

For the 36 test problems we find that the safety-stock holding costfor the model with guaranteed internal service times is on average26% higher than that for the model without this requirement; therange is between 7% and 43%. The size of the gap is insensitive tothe choice of demand process. However, the gap becomes larger asthe production lead-time at stage 1 increases and as the echelon hold-ing cost at stage 1 increases.

The impact on the total inventory holding cost is less dramatic.The difference in holding costs is 4% on average, with a range fromless than 1% to 14%. The gap increases as the holding cost of thepipeline stock decreases, namely as the production lead-time at stage1 decreases and as the demand rate decreases.

From the limited computational study we see that there can be asignificant increase in safety stock due to the assumption of guar-anteed internal service times. Relative to the total inventory, thisincrease does not look as large. Nevertheless, there is a cost in termsof higher inventories from the requirement of guaranteed internalservice times. This cost needs to be considered in light of the practicalbenefits, as discussed in the body of the paper, from imposing thisrequirement. Based on our observations from industrial projects, thisrequirement, and the resulting increase in safety stock, has not beenan issue as the assumption of guaranteed internal service times isalready ingrained in practice.

ReferencesAxsater S. 1993. Continuous review policies formulti-level inventory

systems with stochastic demand. S. C. Graves. A. H. Rinnooy



Kan, P. H. Zipkin, eds. Handbooks in Oper. Res. and ManagementSci. Vol 4., Logistics of Production and Inventory. North-HollandPublishing Company, Amsterdam, The Netherlands. Chapter4.

Black, B. E. June 1998. Utilizing the principles and implications ofthe base stock model to improve supply chain performance.S.M. Thesis, Leaders for Manufacturing Program, MIT, Cam-bridge, MA.

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Ettl, M., G. E. Feigin, G. Y. Lin, D. D. Yao. 2000. A supply networkmodel with base-stock control and service requirements. Oper.Res. 48, March-April.

Federgruen, A. 1993. Centralized planning models for multi-echeloninventory systems under uncertainty. S. C. Graves, A. H.Rinnooy Kan, P. H. Zipkin, eds. Handbooks in Operations Re-search and Management Science, Vol 4., Logistics of Production andInventory, North-Holland Publishing Company, Amsterdam,The Netherlands, Chapter 3.

Felch, J. A. June 1997. Supply chain modeling for inventory analysis.S. M. Thesis, Leaders for Manufacturing Program, MIT, Cam-bridge, MA.

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Glasserman, P., S. Tayur. 1995. Sensitivity analysis for base-stocklevels in multiechelon production-inventory systems Manage-ment Sci. 41 263–281.

Graves, S. C. 1988. Safety stocks in manufacturing systems. J. ofManufacturing and Oper. Management 1 67–101.

——, D. B. Kletter, W. B. Hetzel. 1998. A dynamic model for require-ments planning with application to supply chain optimization.Oper. Res. 46 S35–S49.

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——,——. August 1998. Optimizing strategic safety stock placementin supply chains. Working Paper available from http://web.mit.edu/sgraves/www/papers/.

Inderfurth, K. 1991. Safety stock optimization in multi-stage inven-tory systems. Interna. J. of Production Econom. 24 103–113.

—— 1993. Valuation of leadtime reduction inmulti-stageproductionsystems. G. Fandel, T. Gulledge, A. Jones, eds. Oper. Res. inProduction Planning and Inventory Control, Springer, Berlin, Ger-many, 1993, 413–427.

—— 1994. Safety stocks in multistage, divergent inventory systems:A survey. International J. of Production Economics 35 321–329.

——, S. Minner. 1998. Safety stocks in multi-stage inventory systemsunder different service measures. European J. Oper. Res. 106 57–73.

Kimball, G. E. 1988. General principles of inventory control. J. ofManufacturing and Oper. Management. 1 119–130.

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ordination in stochastic multi-echelon systems. European J. ofOper. Res., 95 1–23.

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The consulting Senior Editor for this manuscript was Paul Zipkin. This manuscript was received on June 1, 1998, and was with the authors 127 days for3 revisions. The average review cycle time was 53 days.

A DYNAMIC MODEL FOR REQUIREMENTS PLANNING WITHAPPLICATION TO SUPPLY CHAIN OPTIMIZATION

STEPHEN C. GRAVESMassachusetts Institute of Technology, Cambridge, Massachusetts

DAVID B. KLETTERBooz, Allen & Hamilton Inc., New York, New York

WILLIAM B. HETZELMerck & Co., Inc., Whitehouse Station, New Jersey

(Received March 1994; revisions received July 1995, January 1996; accepted June 1996.)

This paper develops a new model for studying requirements planning in multistage production-inventory systems. We first charac-terize how most industrial planning systems work, and we then develop a mathematical model to capture some of the key dynamicsin the planning process. Our approach is to use a model for a single production stage as a building block for modeling a network ofstages. We show how to analyze the single-stage model to determine the production smoothness and stability for a production stageand the inventory requirements. We also show how to optimize the tradeoff between production capacity and inventory for a singlestage. We then can model the multistage supply chain using the single stage as a building block. We illustrate the multistage modelwith an industrial application, and we conclude with some thoughts on a research agenda.

Most discrete parts manufacturing firms plan theirproduction with MRP (materials requirements

planning) systems, or at least, with logic based on the un-derlying assumptions of MRP. A typical planning systemstarts with a multiperiod forecast of demand for each fin-ished good or end item. The planning system then devel-ops a production plan (or master schedule) for each enditem to meet the demand forecast. These production plansfor the end items, after offsetting for lead times, then actas the requirement forecasts for the components needed toproduce the end items. The requirements forecast for eachcomponent gets translated into production plans for thecomponent, similar to how the production plan for the enditems was created. The planning system continues in thisway, developing requirement forecasts and productionplans for each level of the bill of materials.

Implicit in this planning process are assumptions aboutthe production and demand process. The production planis developed assuming that the forecast is accurate and willnot change. Within the production process, requirementsare generated assuming that there are deterministic pro-duction lead times and deterministic yields. Needless tosay, these assumptions of a benevolent world do not matchreality. Inevitably, the forecast changes, and uncertaintiesin the production process arise that result in deviationsfrom the plan. To respond to these changes, most planningsystems will completely revise their plan after some timeperiod, say a week or a month. Again, the planning processstarts with the (new) forecast and repeats the steps neces-

sary to regenerate a plan for each level in the products’bills of materials.

The intent of this paper is to present a model that cap-tures the basic flavor of this planning process, and does soin such a way that it can be used to look at varioustradeoffs within the production and planning systems. Inparticular, we model the forecasts for the planning systemas a stochastic process. In this way, we try to represent adynamic input to the planning system, namely, how fore-casts change and evolve over time. The forecast process isa key input for the model. Another key is how the fore-casts get converted into production plans or master sched-ules. We model this process as a linear system, with whichwe can represent the logic for MRP systems and fromwhich we get significant analytical tractability. Finally, themodel is structured so that it can describe multistageproduction-inventory systems.

We are not aware of very much work that is directlyrelated to the dynamic modeling of requirements planning.Baker (1993) provides a nice survey and critique of theliterature relevant to the general topic of requirementsplanning. However, most of the work deals with specificissues like lot sizing or determination of buffer levels. Kar-markar (1993) discusses tactical issues of lot sizing, orderrelease, and lead times in the context of dynamic planningsystems. But neither of these papers reports on work thatattempts to model a dynamic forecast process. One excep-tion is Graves et al. (1986), in which we modeled a two-stage production-inventory system with a dynamic forecast

Subject classifications: Inventory/production: multi-item, multi-stage supply chain with uncertain demand and dynamic forecast revisions; application to film production;dynamic requirements planning and supply chain optimization.

Area of review: MANUFACTURING OPERATIONS.

S35Operations Research 0030-364X/98/4603-S035 $05.00Vol. 46, Supp. No. 3, May–June 1998 q 1998 INFORMS

process. In contrast with the present paper, Graves et al.(1986) focused on issues of how to disaggregate an aggre-gate plan in the two-stage context. Although this paperdoes not consider the disaggregation issue, it does providea more powerful model that is applicable to general multi-stage systems.

Another exception is Heath and Jackson (1994), whoconsidered the same dynamic forecast process as this pa-per as part of a simulation model that was used to analyzesafety stock levels in a multiproduct production/distribu-tion system.

The model for converting the forecast into a productionplan is related to earlier work by the first coauthor, in thatit uses linear systems for a production-inventory context.(See Graves 1986, 1988a, 1988b, 1988c, and Fine andGraves 1989.)

Lastly, we note Lee and Billington (1993), who developa model for supply chain optimization and describe itsapplication at Hewlett Packard. Our work complementstheir work but differs as we try to model the process ofrequirements planning.

In the next section we develop the model for a singleproduction-inventory stage. As part of the development,we present our model for the forecast process, and wedevelop the analyses to generate three performance mea-sures for the stage: production smoothness, productionstability, and inventory requirements. In the second sectionwe examine an optimization for the tradeoff between pro-duction capacity and inventory for a single stage. Althoughthe development is somewhat involved, the final results aresurprisingly simple and, we believe, of interest. This sec-tion can be omitted by the reader without loss of continu-ity. In the third section, we show how the model for thesingle stage can serve as a building block in modeling ageneral acyclic network of multiple stages. We report onan application of the model to a supply-chain study in thefourth section. The application demonstrates the value of asystem-wide perspective for optimizing the supply chain. Inthe final section we briefly summarize the paper, and thenlay out a research agenda for further work.

1. SINGLE-STAGE MODEL

In this section we present the model for a single produc-tion stage that produces one (aggregate) product andserves demand from a finished good inventory. The single-stage model serves as a building block for creating modelsof multistage, multiitem systems. We first describe theforecast process and state our assumptions about how theforecast evolves over time. We then give a model for de-termining the schedule for production outputs from theproduction stage, and we show how to manipulate thismodel to obtain three measures of interest: (1) the produc-tion variance as a measure of production smoothing, (2)the inventory variance as a measure of safety stock, and (3)the stability of the production schedule as a measure forthe forecast process passed on to any upstream stages.

1.1. Forecast Process

We assume that there is a forecast horizon H such that ineach time period t we have forecasts for the requirementsfor the next H periods. Let ft(t 1 i) be the forecast madeat time t for the requirements in period t 1 i, i 5 1, 2, . . .H. We denote the demand observed in period t by ft(t),the forecast made in period t for requirements in period t.Beyond the forecast horizon, there is no specific informa-tion about requirements. In effect, for i . H we assumethat ft(t 1 i) 5 m, where m equals the long-run averagedemand rate.

We propose a stochastic model of this forecast processand show that the forecasts are unbiased, the forecastsimprove as they are revised, and the forecast error overthe forecast horizon matches the inherent variability in thedemand process.

We assume that, each period, we generate a new set offorecasts ft(t 1 i) that incorporates new information aboutfuture demand. We define the updates of the forecastsfrom period to period by the forecast revision, Dft(t 1 i):

Df t ~t 1 i! 5 f t ~t 1 i! 2 f t21 ~t 1 i! for i 5 0, 1, . . . H,

(1)

where ft21(t 1 H) 5 m by assumption.Let Dft be the vector for the revisions to the forecast

process, where Dft(t 1 i) is the i 1 1st element, i 5 0, 1,2, . . . H. We assume that Dft is an i.i.d. random vector withE[Dft] 5 0 and Var[Dft] 5 ¥, the covariance matrix. Thus,for a fixed index i, Dft(t 1 i) is an i.i.d. random variableover time t with zero mean, and the forecast process is amartingale. We note that if we can observe the forecastprocess, then we can assess whether or not the forecastsare unbiased (i.e., E[Dft] 5 0) with independent revisionsand we can estimate the covariance matrix ¥.

This model of the forecast process is the same as that ofGraves et al. (1986) and Heath and Jackson (1994). Wehave validated this model as part of field studies at AT&Tand at Kodak. And this forecast model is descriptive of theforecast process at nearly all of the discrete-part manufac-turing contexts we have encountered.

The i-period forecast error is the difference between theactual demand in period t and the forecast of this demandmade i periods earlier:

f t ~t! 2 f t2i ~t! 5 Df t ~t! 1 Df t21 ~t! 1 · · · 1 Df t2i11 ~t!.

We can now demonstrate the following properties forthis model of the forecast process:

1. The i-period forecast, ft2i(t), is an unbiased estimate ofdemand in period t.

2. The variance of the i-period forecast error is no greaterthan the variance of the (i 1 1)-period forecast error,for i 5 1, 2, . . . H.

3. The trace of the covariance matrix ¥ equals the vari-ance of the demand process.

S36 / GRAVES, KLETTER, AND HETZEL

We see that the first property must be true by observingthat the expectation of the i-period forecast error is zero,since E[Dft2s(t)] 5 0, for s 5 0, . . . i 2 1.

We now prove the second property. Since Dft2s(t) fors 5 0, 1, . . . i 2 1 are independent random variables, thevariance of the i-period forecast error is given by:

Var @ f t ~t! 2 f t2i ~t!# 5 Var ~Df t ~t!! 1 Var ~Df t21 ~t!!

1 · · · 1 Var ~Df t2i11 ~t!!

5 s 02 1 s 1

2 1 · · · 1 s i2,

where sj2 5 Var(Dft2j(t)) is the j 1 1st element on the

diagonal of the covariance matrix ¥, for j 5 0, 1, . . . H.Thus, since sj

2 Ä 0 for j 5 0, 1, . . . H, each forecastrevision improves the forecast, in that it reduces the vari-ance of the forecast error.

For the third property, we observe from the above ex-pression that the variance of the (H 1 1)-period forecasterror equals s0

2 1 s12 1 . . . 1 sH

2 , i.e., the trace of ¥. Sinceby assumption ft2H21(t) 5 m, we have,

Var @ f t ~t! 2 f t2H21 ~t!# 5 Var @ f t ~t!# ,

which proves the third property.Since the demand variance is an exogenous parameter,

this imposes a constraint on the forecast process: namely,the variance of the forecast error over the forecast horizonmust equal the demand variance.

1.2. Schedule for Production Outputs

Given the forecast vector for period t, we need to convertit into a schedule or plan for production. This is oftentermed the master schedule. We focus on production out-puts from the production stage. Later we will discuss howto translate a plan for production outputs into productionstarts. Production starts will be of interest, since they serveas the requirements forecast for the next upstream produc-tion stage.

Let Ft(t 1 i) equal the planned production outputs forperiod t 1 i as of period t, where Ft(t) is the actual pro-duction completed in period t. We assume that the pro-duction plan extends out only for the next H periods, andthat beyond this horizon the plan is just to produce theaverage demand, that is, Ft(t 1 i) 5 m for i . H.

Each period, after we obtain the new forecast, we up-date or revise the plan for production outputs. We defineDFt(t 1 i) as the plan revision:

DF t ~t 1 i! 5 F t ~t 1 i! 2 F t21 ~t 1 i! .

From this definition and the fact that Ft(t 1 i) 5 m for i .H, we see that:

F t ~t 1 i! 5 m 1 DF t1i2H ~t 1 i! 1 · · · 1 DF t ~t 1 i!

for i 5 0, 1, . . . H . (2)

Thus to model the production plan, we need to modelthe plan revision DFt(t 1 i). To do this, we first define theinventory process. For It being the inventory at time t, theinventory balance equation is:

I t 5 I t21 1 F t ~t! 2 f t ~t!. (3)

The planned inventory at time t 1 i is the expected level ofinventory in a future period given the current forecast andthe current production plan as of time t:

I t ~t 1 i! 5 I t 1 F t ~t 1 1! 1 · · · 1 F t ~t 1 i!

2 f t ~t 1 1! 2 · · · 2 f t ~t 1 i!. (4)

We assume that for each time t, we set the productionplan Ft(t 1 i), i 5 0, 1, . . . H, so that the planned inven-tory at the end of the planning horizon, It(t 1 H), is agiven constant. That is, we will set the production plan andmaintain it from period to period so that the end-of-horizon inventory neither grows nor decreases, but re-mains constant. We term the level to which the inventoryis targeted as the safety stock. In a later section we willdiscuss how to set this level. For now, all we need to knowis that this level remains constant.

From (3) and (4), we obtain by equating It21(t 2 1 1 H)and It(t 1 H) that:

DF t ~t! 1 DF t ~t 1 1! 1 · · · 1 DF t ~t 1 H!

5 Df t ~t! 1 Df t ~t 1 1! 1 · · · 1 Df t ~t 1 H!. (5)

That is, to assure that the end-of-horizon inventory re-mains constant, we require that the cumulative revision tothe production plan should equal the cumulative forecastrevision in each period.

Each period we revise the production schedule to ensure(5). To do this, we model the schedule update as a linearsystem:

DF t ~t 1 i! 5 Oj50

H

w ij Df t ~t 1 j! for i 5 0, 1, . . . H, (6)

where wij denotes how the forecast revision affects theschedule. In particular, wij is the proportion of the forecastrevision for period t 1 j that is added to the schedule ofproduction outputs for period t 1 i.

We expect that 0 ¶ wij ¶ 1. To ensure that (5) is true,we require that for each j:

Oi50

H

w ij 5 1.

We refer to wij as a weight or proportion. We can inter-pret these weights either as decision variables in a pre-scriptive model or as parameters in a descriptive model.On the one hand, we can view these weights as control orsmoothing parameters and use the model for prescription.To smooth production we set the weights wij for a fixed jto be as nearly constant as possible (e.g., wij 5 1/(H 1 1)for i 5 0, 1, . . . H). To minimize inventory, we set theweights so that the production plan tracks the forecast asclosely as possible (e.g., for fixed j, wij 5 1 for i 5 j and wij

5 0 otherwise). In this way, specification of the weightspermits one to balance the tradeoff between productionsmoothing and inventory requirements, as will be seen.

On the other hand, we can view the weights as parame-ters for a descriptive model of an existing planning system.

S37GRAVES, KLETTER, AND HETZEL /

In particular, we can use (6) to model how most imple-mentations of MRP systems translate forecast revisionsinto schedule revisions. For instance, in the simplest caseat time t the schedule is frozen for periods t 1 j, j 5 0, 1,2 . . . k for some value of k , H, and is totally free tochange for periods t 1 j, j 5 k 1 1, . . . H. Then, anyrevision to the forecast within the frozen zone results in aschedule revision for the first period beyond the frozenzone; i.e., for 0 ¶ j ¶ k, wij 5 0 for i Þ k 1 1 and wij 5 1for i 5 k 1 1. Any revision to the forecast beyond thefrozen zone results in a one-for-one schedule revision inthe same period: for k 1 1 ¶ j ¶ H, wij 5 1 for i 5 j andwij 5 0 otherwise. Occasionally there is an intermediatezone (between the frozen and free zones) in which changesto the schedule are permitted but are restricted in size,e.g., no more than 10% increase or decrease in the sched-uled amount. The model given by (6) cannot exactly cap-ture this policy, but it can approximate its behavior byusing fractional weights.

In matrix notation, we can rewrite (6) as:

DF t 5 W Df t , (7)

where W 5 {wij} is an (H 1 1) 3 (H 1 1) matrix, andDFt and Dft are column vectors with elements DFt(t 1 i)and Dft(t 1 i), for i 5 0, 1, . . . H. From this, we observethat DFt is an independent random vector, has zero mean,and has a covariance matrix W¥W*. (We will see later thatthis is an important observation for the extension to mul-tiple stages: we will derive the forecast revision for up-stream stages from DFt.)

We can express the production plan in matrix notationby:

F t 5 B F t21 1 mU H11 1 DF t , (8)

where Ft(t 1 i) is the i 1 1st element of the vector Ft fori 5 0, 1, . . . H; UH11 is a unit vector with ui 5 0 for i 51, . . . H and uH11 5 1; and B is a matrix with elementsbij 5 1 for j 5 i 1 1, and bij 5 0 else. Premultiplying acolumn vector by B replaces the ith element in the vectorwith the i 1 1st element and replaces the last element witha zero.

From (7) and (8) and repeated substitution, we obtain:

F t 5 BF t21 1 mU H11 1 WDf t (9)

5 B H11F t2H21 1 m 1 Oi50

HB iWDf t2i ,

where m is the vector with each element equal to m, andthe superscript i in Bi denotes the ith power of B. We cansimplify (9) by noting that premultiplying an (H 1 1) 3 1vector by BH11 gives the null vector:

F t 5 m 1 Oi50

HB iWDf t2i . (10)

1.3. Measures of Interest

There are three categories of measures for the single-stagemodel: the smoothness of the production outputs, the

safety stock for the end-item inventory, and the stability ofthe production plan.

The smoothness of the production outputs is of interestbecause more variable (less smooth) production is ex-pected to require more production resources or capacity.Furthermore, we can influence the smoothness of produc-tion via our inventory and control policies.

An output of the model is the variability of the inventoryprocess, which will dictate how much safety stock is neededto ensure an acceptable service level. If the inventory pro-cess is more variable, more safety stock will be needed.

The stability of the production output plan is of interest,since the output plan determines the plan for productionstarts, which determines the requirements forecast for up-stream stages. We will, in effect, equate the stability of theproduction plan to the accuracy of the forecast process forthe upstream stages. More stability means a more accurateforecast process upstream. This measure is critical as wetry to understand the workings of a multistage system,since the inventory requirements and the variability of theproduction outputs for a stage will depend heavily on theaccuracy of the forecast process.

We first develop the measures for production smoothingand for the stability of the production plan. We will need amore extensive development to obtain the variability of theinventory process in order to set the safety stock.

Production Smoothing. A common measure of productionsmoothing is the variance of the production output,Var[Ft(t)]. From (10) we see immediately that the randomvector Ft has mean m and has a covariance matrix given by:

Var ~F t ! 5 Oi50

H

B iW¥W*B* i. (11)

We can use the covariance matrix to obtain the first mea-sure of the production smoothing, Var[Ft(t)]. Indeed, onecan show that:

Var @F t ~t!# 5 tr~W¥W*!, (12)

where tr(A) is the trace of matrix A.A second measure of production smoothing is given by

Ft(t) 2 Ft21(t 2 1), the change in production outputs fromone period to the next. In matrix notation we see from (9)that:

F t 2 F t21 5 mU H11 1 WDf t 2 @I 2 B#F t21 ,

where I is the identity matrix. Since Dft and Ft21 are inde-pendent of each other, we find that the covariance matrixfor Ft 2 Ft21 is given by:

Var ~F t 2 F t21 ! 5 W¥W*

1Oi50

H

@I 2B#B iW¥W*B* i@I 2B#9 (13)

5 ~I 2 B! Var ~F t ! 1 Var ~F t !~I 2 B!9 .


From this covariance matrix, we can determine the secondmeasure of production smoothing, namely Var [Ft(t) 2Ft21(t 2 1)].

Production Stability. For the stability of the productionplan, we use DFt: the random vector for the one-periodrevision to the production plan, which is the basis for therevision to the forecast of requirements for upstreamstages. (The production starts, as described earlier, wouldgenerate the actual forecast seen by the upstream stages;but since the starts are usually just the production planoffset by the lead time, we can use the revision to theproduction plan for defining stability.) From (7) we obtainits expectation and covariance matrix:

E~DF t ! 5 0(14)

Var ~DF t ! 5 W¥W*.

We propose the covariance matrix W¥W* as a measure ofthe stability of the production plan. A more stable produc-tion plan will have a smaller covariance matrix, and willyield more accurate forecasts for the upstream stages.When analyzing the upstream stages, the dynamics of therequirements depend upon this covariance matrix. In thissense, for the upstream stages, the covariance matrix in(14) is analogous to ¥ for the downstream stage, namely itis the covariance matrix for the relevant requirementsforecast process.

One measure of the size of a covariance matrix is itstrace. We note that with this interpretation the tr(W¥W*)signifies not only the stability of the production plan, butalso the variance of the requirements forecast for the up-stream stages over the planning horizon. Furthermore, wesee that, according to the proposed measures (12) and(14), smoothing production is essentially equivalent to sta-bilizing the production plan and requirements forecast forthe upstream stages.

Inventory. We focus on the end-item inventory for thesingle stage, namely the random variable It given in (3).We assume that the requirements for the single stage areto be met from the end-item inventory and that typicalservice expectations apply, e.g., the inventory should stockout in no more than 2% of the periods, or that the inven-tory should provide a 97% fill rate. We will find the expec-tation E(It) and variance Var(It), from which we candetermine the safety stock required to achieve a desiredservice level, under suitable distributional assumptions.For instance, if the forecast errors are normally distrib-uted, then we will see that It has a normal distribution. Fora desired service level expressed as the stockout probabil-ity, we need to set the safety stock level so that:

E~I t ! . ks~I t !, (15)

where k is such to ensure the service level, and s( ) de-notes the standard deviation.

Recall that in (4) we defined It(t 1 i) to be the plannedinventory level in period t 1 i as of time t; that is, It(t 1 i)

is the expected inventory in period t 1 i, where the expec-tation is as of period t. For notational convenience, It(t)denotes the actual inventory in period t, i.e., It(t) is thesame as It. As stated in the earlier development of (5), weassume that the end-of-horizon inventory It(t 1 H) is tar-geted to equal some constant, which we call the safetystock and denote by ss.

The inventory flow equation for the planned inventoryis:

I t ~t 1 i! 5 I t ~t! 1 F t ~t 1 1! 1 · · · 1 F t ~t 1 i!

2 f t ~t 1 1! 2 · · · 2 f t ~t 1 i!. (16)

Define DIt(t 1 i) 5 It(t 1 i) 2 It21(t 1 i). From (3) and(16), we find that

DI t ~t 1 i! 5 DF t ~t! 1 · · · 1 DF t ~t 1 i!

2 Df t ~t! 2 · · · 2 Df t ~t 1 i!,

for i 5 0, 1, . . . H 2 1. By assumption, since we keep theinventory constant at ss beyond the horizon, we have:

DI t ~t 1 H! 5 I t ~t 1 H! 2 I t21 ~t 1 H! 5 ss 2 ss 5 0.

In matrix notation, let It be an (H 1 1) 3 1 columnvector with It(t 1 i) as its i 1 1st element. Then

DI t 5 T@DF t 2 Df t #,

where T is a matrix with element tij 5 1 for i $ j and tij 5 0else. We can now write the inventory random vector as

I t 5 T@DF t 2 Df t # 1 BI t21 1 ssU H11 , (17)

where we use the fact that It21(t 1 H) 5 ss. We cansimplify (17) by repeated substitution, by substitution of(7), and by noting that premultiplication of an (H 1 1) 31 vector by BH11 gives the null vector:

I t 5 Oi50

H

B iT@W 2 I#Df t2i 1 ss , (18)

where ss denotes the column vector with each elementequal to ss. From (18) we see that the random vector It hasmean equal to ss, and has a covariance matrix given by:

Var ~I t ! 5 Oi50

H

B iT@W 2 I#¥@W 2 I#9T*B* i. (19)

We can use (19) to find Var[It(t)], which is necessary todetermine how to set the safety stock level ss. From (19),we can show with some effort that

Var @I t ~t!# 5 tr~T@W 2 I#¥@W 2 I#9T*! 5 Ok50

H Oi50

k Oj50

k

q ij ,

(20)

where

Q 5 $q ij % 5 @W 2 I#¥@W 2 I#9 .

Now from (15) we set the safety stock by ss 5 k s[It(t)],where s[It(t)] is obtained from (20) and k is such to pro-vide the desired service level from the inventory.


2. OPTIMAL WEIGHTS FOR SINGLE-STAGEMODEL

For a single stage it is natural to wonder how to choose theweights in (6) that determine how a forecast revision isconverted into a revision of the production plan. To gainsome insight into this question, we pose and solve an opti-mization problem for choosing the weights for the simplecase of uncorrelated demand. The tradeoff between pro-duction smoothing in the stage and the end-item inventoryrequirements should govern the choice of weights. Thistradeoff is the basis for stating the optimization problem:

Min s@F t ~t!# ,

subject to:s@I t ~t!# < K,

Oi50

H

w ij 5 1 ; j. (21)

The optimization problem minimizes productionsmoothing, as given by the standard deviation of the pro-duction output, subject to a constraint on the standarddeviation of the inventory and the requirement that theweights sum to one. We interpret the objective as minimiz-ing required production capacity. We view the nominalcapacity required at the stage as being the expected pro-duction requirements, plus some number of standard devi-ations. (See Graves 1988a for further discussion.) Theconstraint on the standard deviation of the inventory iseffectively a constraint on the amount of safety stock re-quired, where we assume that the safety stock is a multipleof s[It(t)]. An alternative formulation would be to mini-mize the standard deviation of the inventory, equivalentlyminimize the safety stock, subject to a constraint on thestandard deviation of the production output.

There are no restrictions in the optimization on theweights, other than the convexity constraint. We have notimposed any nonnegativity constraints, nor any restrictionson the weights due to a fixed production lead time. Rather,we allow the weights to be totally free. In this sense, theoptimization will produce a lower bound for the case withfixed lead times.

To develop some insights on the optimal weights, wetransform the original optimization problem (21) into anequivalent form by restating it in terms of the variances ofthe production and inventory variables:

Min Var @F t ~t!# , (21a)

subject to:

Var @I t ~t!# < K 2,

Oi50

H

w ij 5 1 ; j.

To analyze this equivalent problem, we consider the La-grangian relaxation:

L~l! 5 Min Var @F t ~t!# 1 l Var @I t ~t!# 2 lK 2, (21b)

subject to:

Oi50

H

w ij 5 1 ; j.

By solving this problem over a range of positive values forthe Lagrange multiplier l, we can find the tradeoff surfacebetween production smoothing and inventory require-ments for a single stage. We will also obtain some intuitionfor the form of the optimal weighting function.

In the remainder of this section we will focus on solving(21b). To solve (21a), and equivalently (21), we wouldneed to search over l until the solution to (21b) satisfiesthe relaxed constraint.

We only consider the case when the covariance matrixfor the forecast revision process is diagonal. That is, theforecast revisions are uncorrelated, and Var[Dft] 5 ¥ 5{si

2}, where si2 5 Var[Dft(t 1 i)] is the i 1 1st element on

the diagonal, i 5 0, . . . H.For this case, we can simplify (12) and (20) to be:

Var @F t ~t!# 5 Oi50

H Oj50

H

~w ij s j !2, (12*)

and

Var @I t ~t!# 5 Oi50

H Oj50

H

~b ij s j !2, (20*)

where

b ij 5 w 1j 1 · · · 1 w ij for i , j, (22)5 w 1j 1 · · · 1 w ij 2 1 for i > j.

By substituting (12*) and (20*) into (21b), we observethat the minimization problem separates into H 1 1 sub-problems, one for each period j:

L~l! 5 Oj50

H

L j ~l! 2 lK 2, (21c)

where

L j ~l! 5 Min Oi50

H

~w ij s j !2 1 l O

i50

H

~b ij s j !2, (23)

subject to:

Oi50

H

w ij 5 1.

We now characterize the solution to Lj(l) with a series ofpropositions.

Proposition 1. The optimal weights in (23) are indepen-dent of sj

2.

Proof. Each term in the objective function of Lj(l) in (23)is proportional to sj

2, which can then be factored out. □

Thus, we can determine the optimal weights in the La-grangian (21b) without knowing the covariance matrix forthe forecast revision. We only need to know that the co-variance matrix is diagonal. However, to solve the original


problem, (21) or (21a), does require knowledge of thecovariances to ensure satisfaction of the inventoryconstraint.

The Kuhn-Tucker conditions for (23) consist of the con-vexity constraint over the weights, plus the following set ofequations:

w ij 1 l Ok5i

H

~w 0j 1 . . . 1 w kj 2 u kj ! 5 g for i 5 0, . . . H ,

(24)

where ukj 5 1 if k $ j, ukj 5 0 if k , j, and g is the(scaled) dual variable for the single convexity constraint in(23). Since (23) is a convex program, the Kuhn-Tuckerconditions are both sufficient and necessary, and they iden-tify a unique solution.

To find the solution, we equate (24) for i 2 1 and i toobtain:

w ij 5 w i21, j 1 l~w 0j 1 · · · 1 w i21, j 2 u i21, j !

for i 5 1, . . . H. (25)

We can construct a solution to (24) by selecting a value forw0j and repeatedly applying (25). To satisfy the convexityconstraint, we could search over values for w0j. Alterna-tively, we describe in the next two propositions how to findw0j analytically.

Proposition 2. For a given value of l, the solution to (25)for wij is a linear function of w0j given by:

w ij 5 P i ~l!w 0j for i 5 0, 2 . . . j , (26a)

w ij 5 P i ~l!w 0j 2 R i2j ~l! for i 5 j 1 1, . . . H , (26b)

where Pi(l) is a polynomial in l of degree i, and Ri2j(l) isa polynomial in l of degree i 2 j. In particular, we canshow by induction that for n 5 0, 1, . . . H,

P n ~l! 5 Oi50

n ~n 1 i!!

~2i!!~n 2 i!!l i,

and that for n 5 1, 2, . . . H 2 j,

R n ~l! 5 Oi51

n ~n 1 i 2 1!!

~2i 2 1!!~n 2 i!!l i.

Proposition 3. The optimal choice for w0j that solves (23)is given by:

w 0j 51 1 ¥ i5j11

H R i2j ~l!

¥ i50H P i ~l!

5P H2j ~l!

¥ i50H P i ~l!

, (27)

which simplifies to:

w 0j 5

¥ i50H2j ~H 2 j 1 i!!

~2i!!~H 2 j 2 i!!l i

¥ i50H ~H 1 i 2 1!!

~2i 1 1!!~H 2 i!!l i

.

Proof. From Proposition 2 we can rewrite the convexityconstraint as follows:

1 5 Oi50

H

w ij 5 Oi50

H

P i ~l!w 0j 2 Oi5j11

H

R i2j ~l!.

We can now use this to express w0j in terms of Pi(l) andRi(l), as given in the proposition. We simplify the expres-sion for w0j by substituting the following for Ri(l):

Oi51

n

R i ~l! 5 Oi51

n ~n 1 i!!

~2i!!~n 2 i!!l i 5 P n ~l! 2 1,

which is found by an induction argument. Similarly, we cansimplify (27) by noting that

Oi50

n

P i ~l! 5 Oi50

n ~n 1 i 1 1!!

~2i 1 1!!~n 2 i!!l i. □

Having found the optimal choice of w0j, we obtain theremaining weights by iteratively solving (25). We see im-mediately from Proposition 3 that for positive l, w0j ispositive; we can similarly show that wHj is positive. Fromthese facts, we can obtain the following proposition byexamining the first differences for the optimal weights.

Proposition 4. The optimal weights wij are positive, in-creasing, and strictly convex over the range i 5 0, 1, . . . j.The optimal weights wij are positive, decreasing, and strictlyconvex over the range i 5 j, j 1 1, . . . H.

Proposition 5. The matrix of optimal weights is symmetricabout the off-diagonal, i.e., wij 5 wH2j,H2i.

Proof. This can be shown by substitution of (27) into(26). □

Proposition 6. The optimal weights are such that wij 5wH2i,H2j.

Proof. Since the optimal weights satisfy the convexity con-straint, we can substitute the convexity constraint into (25)and rewrite, after some rearrangement, as:

w i21, j 5 w ij 1 l~w ij 1 · · · 1 w Hj 2 ~1 2 u i21, j !!

for i 5 1, . . . H. (28)

From (28), by a similar development as used to find (26),we can express the weights as linear functions of wHj:

w ij 5 P H2i ~l!w Hj for i 5 j, . . . H , (29a)

w ij 5 P H2i ~l!w Hj 2 R j2i ~l! for i 5 0, 1, . . . j 2 1.

(29b)

In order for the weights to sum to one, we then find that:

w Hj 5P j ~l!

¥ i50H P i ~l!

. (30)

From (29) and (30), we establish the result. □

Proposition 7. The matrix of optimal weights is symmetricabout the diagonal; i.e., wij 5 wji.

Proof. This follows immediately from Propositions 5 and6. □


Figure 1 shows the form of the optimal weights for var-ious values of j for l 5 1 and H 5 12. Table I lists theactual values for the optimal weights. From the table weobserve that the matrix of optimal weights is symmetricabout both diagonals, as stated in the propositions above.Furthermore, for a fixed index j, the weights increase geo-metrically to a maximum at wjj, and then decay geometri-cally over the rest of the column.

Figures 2 and 3 show the form of the optimal weights forl 5 4 and l 5 0.25 at H 5 12. Intuitively, we would expectthat as l increases to `, wjj goes to 1 and wij goes to 0 fori Þ j (no production smoothing), and as l decreases to 0,wij goes to 1/(H 1 1) (maximum production smoothing).At l 5 4 and l 5 0.25 we already begin to observe thisbehavior.

Proposition 8. The optimal objective value for the La-grangian function in (23) is given by Lj(l) 5 wjjsj

2 for j 50, 1, . . . H.

Our proof of Proposition 8 involves quite a bit of unat-tractive and nonintuitive algebra. (See Kletter 1994 for the

details.) The basic structure of the proof is as follows: werewrite the right-hand side of (23) strictly in terms of w0j andl for a given j by repeatedly applying (26) and factoring outsj

2. We then show that this expression equals wjj, where wjj isalso expressed in terms of w0j and l. This is achieved byreplacing w0j with the expression given in (27), expressing allterms as polynomials in l, and then manipulating the bino-mial coefficients until they are shown to be equal.

The value of Proposition 8 is that it provides a relativelyquick way to evaluate the objective function of the Lagrang-ians, namely (21b) and (23). Also, we show next how to get agood approximation of wjj, which will then yield an analyticexpression for the objective function of the Lagrangian.

Suppose we define the first difference Dwij 5 wij 2wi21, j; we can use (25) to express Dwij by:

Dw ij 5 Dw i21, j 1 lw i21, j

for i 5 1, 2, . . . H and i Þ j 1 1,

Dw j11, j 5 Dw jj 1 lw jj 2 l .

Figure 1. Optimal weights for l 5 1 and various j.

Table IOptimal Weights for l 5 1, H 5 12

j0 1 2 3 4 5 6 7 8 9 10 11 12

0 0.6180 0.2361 0.0902 0.0344 0.0132 0.0050 0.0019 0.0007 0.0003 1.1E-04 4.1E-05 1.6E-05 8.2E-061 0.2361 0.4721 0.1803 0.0689 0.0263 0.0101 0.0038 0.0015 0.0006 0.0002 8.2E-05 3.3E-05 1.6E-052 0.0902 0.1803 0.4508 0.1722 0.0658 0.0251 0.0096 0.0037 0.0014 0.0005 0.0002 8.2E-05 4.1E-053 0.0344 0.0689 0.1722 0.4477 0.1710 0.0653 0.0250 0.0095 0.0036 0.0014 0.0005 0.0002 1.1E-044 0.0132 0.0263 0.0658 0.1710 0.4473 0.1709 0.0653 0.0249 0.0095 0.0036 0.0014 0.0006 0.00035 0.0050 0.0101 0.0251 0.0653 0.1709 0.4472 0.1708 0.0653 0.0249 0.0095 0.0037 0.0015 0.0007

i 6 0.0019 0.0038 0.0096 0.0250 0.0653 0.1708 0.4472 0.1708 0.0653 0.0250 0.0096 0.0038 0.00197 0.0007 0.0015 0.0037 0.0095 0.0249 0.0653 0.1708 0.4472 0.1709 0.0653 0.0251 0.0101 0.00508 0.0003 0.0006 0.0014 0.0036 0.0095 0.0249 0.0653 0.1709 0.4473 0.1710 0.0658 0.0263 0.01329 1.1E-04 0.0002 0.0005 0.0014 0.0036 0.0095 0.0250 0.0653 0.1710 0.4477 0.1722 0.0689 0.0344

10 4.1E-05 8.2E-05 0.0002 0.0005 0.0014 0.0037 0.0096 0.0251 0.0658 0.1722 0.4508 0.1803 0.090211 1.6E-05 3.3E-05 8.2E-05 0.0002 0.0006 0.0015 0.0038 0.0101 0.0263 0.0689 0.1803 0.4721 0.236112 8.2E-06 1.6E-05 4.1E-05 1.1E-04 0.0003 0.0007 0.0019 0.0050 0.0132 0.0344 0.0902 0.2361 0.6180


To get an approximate solution to these first differenceequations, suppose we look at a limiting case where weallow both H and j to grow. In effect, we let the range bei 5 . . . 22, 21, 0, 1, 2, . . . , except for i 5 j 1 1. Then inthe limit, the solution to these difference equations is:

w j1k, j 5 w j2k, j 5 a@~1 2 a!/~1 1 a!# k

for k 5 0, 1, 2,. . ., (31)

where a 5 =l/(l 1 4).Furthermore, this solution satisfies the convexity con-

straint over the weights. From (31) we see that in the limit:

Y the optimal weights are symmetric about wjj;Y the optimal weights decline geometrically on either side

of wjj;Y the value of the maximum weight wjj is independent of

j; andY the maximum weight wjj is a simple monotonic function

of l, that approaches 1 as l increases.

We can see from Figure 1 and Table I that for l 5 1,the optimal weights already begin to approach the limit atH 5 12. In particular, we observe that, except at the endpoints j 5 0 and j 5 H, wjj ' a 5 =l/(l 1 4) 5 =1/5 '0.4472 and wj11, j 5 wj21, j 5 a[(1 2 a)/(1 1 a)] ' 0.1708.

The limit provides a simple approximation to the objec-tive function of the Lagrangian relaxation. Using Proposi-tion 8 and (31), we find that for large values of H we canapproximate (21b) by:

L~l! 5 Min Var @F t ~t!# 1 l Var @I t ~t!# 2 lK 2

< tr~¥! Îl/~l 1 4! 2 lK 2.

This simplification is helpful for finding the value of l thatmaximizes the Lagrangian, and thus solves the originaloptimization problem (21).

We end this section with an interesting and perhapsuseful result.

Figure 2. Optimal weights for l 5 4 and various j.

Figure 3. Optimal weights for l 5 0.25 and various j.


Proposition 9. The optimal weight matrix is the inverseof a tridiagonal matrix C, with c00 5 cHH 5 (l 1 1)/l, c01

5 c10 5 cH,H21 5 cH21,H 5 21/l, and with (ci,i21, cii,ci,i11) given by (21/l, (l 1 2)/l, 21/l) for i 5 1, 2, . . .H 2 1.

Proposition 9 can be proved by construction through aseries of careful matrix operations. (See Kletter 1994 fordetails.) Our proof simply shows that inverting the matrixC gives W, as specified in (29). This is accomplished byfirst factoring C into LDL*, where L is bidiagonal, since Cis symmetric and tridiagonal, and then inverting to obtain(LDL*)21 5 (L*)21 D21 L21. Since the diagonal matrix Dand the bidiagonal matrix L are both easily inverted, wethen compute the product and simplify to show that cij

21 5wij for all i and j.

One significance of Proposition 9 is that it makes thecomputation of the optimal weight matrix even easier.

3. EXTENSION TO MULTISTAGE SYSTEMS

In the previous sections we developed a single-stage modelof requirements planning. We now discuss how this single-stage model can serve as a building block in modeling ageneral acyclic network of multiple stages.

To begin, we state the assumptions and introduce someadditional notation that will be necessary for our discus-sion.

Assumption 1. The production system is an acyclic net-work with n distinct stages, m of which produce end-items,where m , n. We index the stages so that if stage i isdownstream from stage j, then i , j. In addition, the enditem stages are numbered 1, 2, . . . m.

Assumption 2. The forecast processes at the end-itemstages are mutually independent.

Assumption 3. Each downstream stage is effectively de-coupled from the upstream stages, i.e., there is always ade-quate (raw material ) inventory for a stage to make itsproduction starts. This is an approximation that is likely tobe reasonable if each stage operates with an inventory pol-icy in which stockouts are rare.

Assumption 4. Each stage operates according to the as-sumptions for the single-stage model.

Namely, let fi, i 5 1, . . . n, be the forecast vector foreach stage i; for simplicity, we will omit the subscript t inthis section. Note that for i 5 1, . . . m, fi is an exogenousrandom vector, whereas for i 5 m 1 1, . . . n, fi will be aderived forecast. Let Fi, i 5 1, . . . n, be the output planfor each stage i. Thus, by Assumption 4, there is a weightmatrix Wi, and DFi 5 Wi Dfi for each stage i.

To link the requirements of a downstream stage to anupstream stage, we need to model the production starts orreleases into each stage. We assume that in each periodeach stage i, i 5 1, 2, . . . n, must translate its planned

production outputs Fi into a plan of production starts, callit Gi, over some planning horizon.

Assumption 5. At each stage, we model production startsas a linear system of production outputs: Gi 5 AiFi forsome matrix Ai.

We can set Ai to model a variety of real-world consider-ations as well as production policies. For instance, wemight use the matrix Ai to model production leadtimes,where production starts are just the production outputsoffset by the leadtime, to model yield factors within theproduction stage (e.g., need to start 1.2 units to get output1.0), or to model the fact that production starts occur on adifferent time scale (biweekly rather than weekly) from theproduction outputs. We can also model a constant work-in-process policy where production starts for the periodexactly equal production outputs. Indeed, in this way, forgeneral multistage systems we can use this general ap-proach to compare push policies—where starts equalplanned output L periods from now—with pull policies,where starts “replace” the outputs produced in the currentperiod.

Assumption 6. At each stage we know how many units ofinput are required for one unit of output. Without loss ofgenerality, we assume that one unit of input is required forone unit of output at each stage.

The single-stage model that we wish to use as a buildingblock takes as input a dynamic forecast process of therequirements for the stage. We now show that, given theassumptions above, the forecast process at each stage inthe multistage network satisfies the assumptions of thesingle-stage model. In particular, we show the followingproposition:

Proposition 10. At each stage i, the forecast revision Dfi

can be expressed as a linear combination of Df1, . . . Dfm ;

Dfi 5 ¥j51m Mij Df j for some matrices Mij. By Assumption 2,

this implies that Dfi is an i.i.d. random vector.

We will demonstrate this proposition by an inductionargument. The proposition is true by assumption for theend-item stages 1, . . . m. Suppose this proposition is truefor stages i 5 1, . . . j 2 1; we will now show that it is truefor Df j. Let Sj be the index set of immediate successors tostage j. The forecast process for outputs of an upstreamstage j . m is

f j 5 Ok[Sj

G k .

Accordingly, we can write

Df j 5 Ok[Sj

DG k .

We note by Assumption 6 that DGk 5 Ak DFk 5 Ak Wk Dfk,and by the induction hypothesis that each Dfk is a linearcombination of Df1, . . . Dfm. Thus, we can see that eachDGk for k [ Sj is a linear combination of Df1, . . . Dfm, and


hence so is Df j. This completes the induction argument,showing that each Df j is an i.i.d. random vector. □

We have thus shown that at each stage we have pre-served the essential requirement that the forecast revisionsare i.i.d. random vectors, and thus, that the assumptionsfor the forecast process of the single-stage model are sat-isfied at each stage in the multistage network. This is animportant result because it means that we can now model anacyclic multistage system by just replicating the single-stagemodel. In this sense, the single-stage model serves as abuilding block.

4. CASE STUDY

In this section we describe an industrial application of theDynamic Requirements Planning (DRP) model from athesis internship performed by one coauthor (Hetzel) atthe Eastman Kodak Company. The internship was con-ducted as part of MIT’s Leaders for Manufacturing Pro-gram and ran from June 1992 to December 1992. (SeeHetzel 1993 for more details on the application.)

The general charge for the thesis was to investigate cycletime reduction within the context of the film manufactur-ing processes at Kodak. As part of the internship, Hetzeljoined an internal supply chain optimization team that wasinvestigating opportunities for better coordination over aspecific supply chain, including issues of cycle time andinventory reduction. One open issue facing the team wasthat of strategic inventory placement: how much inventorywas needed, and where should it be placed across a multi-stage supply chain. Hetzel identified this as an opportunityto apply the DRP model, and the team agreed that it wasan appropriate tool for their task of strategic inventoryplacement. The only alternative considered was to developa simulation: since the DRP model was already availablefrom the authors in a software package, developing a sim-ulation would have required extensive additional work.

The goal of the supply chain analysis was to determinethe optimal safety stock levels between each stage in thefilm making supply chain. The underlying concept is thatlooking at one stage of the supply chain in isolation isinherently suboptimal. All the stages in the supply chainare interconnected by information flows. In short, the in-ventory and production policies that are best for one stagemay not be optimal for the supply chain as a whole.

In the case study, the team was able to address thissituation by using the DRP model to consider all stages inthe supply chain. Their recommendations challenged theconventional targets and performance measures for indi-vidual divisions (stages). For example, an upstream stage,roll coating, faced a corporate-wide mandate to lower in-ventories. However, by using the DRP model, the teamdiscovered that roll coating needed to increase inventoriesto provide the desired service to the next stage. When rollcoating holds sufficient inventory to provide a high level ofservice, downstream stages can hold less, resulting in a netsavings for the corporation. Overall, the analysis deter-

mined that inventories for the products of the case studycould be reduced by 20%. This example highlights theimportance of considering the entire supply chain whensetting inventory and production policies.

The rest of this section will describe the supply chain forthe case study, provide the results from the DRP model,and comment on implementation issues.

4.1. Supply Chain for Case Study

In Figure 4 we give a simplified version of the process forfilm making. Roll coating transforms raw chemicals into aroll of film base. Sensitizing coats the film base with asilver halide emulsion. Then finishing cuts and packagesthe sensitized rolls into finished products. The structure ofthis supply chain has three interesting characteristics. First,the number of items grows dramatically from stage tostage; one film base might result in 5 to 10 different sensi-tized rolls, which might lead to a hundred or more finishedgoods. Second, there is a rapid growth in the value of theproduct due to added material (e.g., silver) and nature ofthe processes. Third, there is a gradual decrease in theleadtimes across the supply chain.

For the case study, the supply chain optimization teamfocused on a single film base (called a support). That sin-gle base becomes three different sensitized film codes be-cause it can be coated with three different emulsions. Thethree film codes can be finished (slit, chopped, and pack-aged) into 24 different finished good items. Figure 5 illus-trates the supply chain for the case study. This particularproduct “tree” was chosen because it is high volume, it hasrelatively few end items (24 total), and it represents a“typical product” that the team felt would make a usefulpilot program.

It is important to note that the case study does establisharbitrary bounds on the supply chain. The case study startswith the creation of a film base in roll coating and excludesthe upstream raw material stages such as chemical, gelatin,and polymer production. The case study ends with thefinishing process and arrival at the Central DistributionCenter, and ignores the rest of the distribution system.Besides being bounded at both ends, the case study’s sup-ply chain is also simplified. In reality, the sensitizing andfinishing stages have materials flowing into them such asemulsion and packaging components. Even though thesematerials require inventory management, they are as-sumed to be available with 100% service, and were notexplicitly incorporated into the model.

Figure 4. Simplified version of film manufacturing supplychain.


4.2. Data Collection

Parameterizing the DRP model required an extensive datacollection effort. For each item in the chain, the teamgathered data on the item’s leadtime, unit cost, inventoryholding cost, manufacturing frequency, and desired servicelevel. For each end item, they needed the planning hori-zon, the average demand level, and a time history of theforecast process.

The leadtime and manufacturing frequency were mod-eled through the weight matrix (W). Since the team didnot consider production smoothing, in the absence of lead-time and production frequency considerations, the weightmatrix is simply the identity matrix. A leadtime of L peri-ods is then captured by forcing the first L rows of W to bezero. To represent a production frequency of once everytwo weeks, W would then be modified so that every otherrow was zero. It should be noted that this method of cap-turing production frequency is only an approximation.

From the forecast histories, the team estimated the di-agonal elements of the covariance matrix (¥) for the fore-cast revision process Dft; the off-diagonal elements wereassumed to be zero. Associated with each “branch” linking

different stages they calculated a historical “goes into fac-tor” to capture any yield loss or conversion factors. Thisinformation was used to construct the matrix A, as de-scribed in Section 3.

A side benefit of applying the DRP model was that thedata collection effort identified some potential issues alongthe supply chain. For example, in the course of reviewingthe forecast data, the team discovered that the forecastsvaried in a systematic way that led to a reevaluation of theforecasting process. In addition, collecting data enhancedsupply chain communication and allowed the team to re-solve a discrepancy in the annual planned volumes be-tween two of the stages.

4.3. Results

The team used the DRP model to develop a base caserecommendation on inventory placements. The 24 finisheditems were grouped into nine product aggregates, wherethe product aggregates shared common production pro-cesses and had similar demand histories. Service levelswere set at 95% for each stage. The weight matrices werenot optimized and were set to reflect each stage’s leadtimeand manufacturing frequency. In order to reflect Kodak’scurrent scheduling systems, there was no productionsmoothing across weeks.

The DRP model showed the potential to lower inven-tory across the case study product “tree” by 20%, as shownin Figure 6. Note that, in general, inventories can bepushed upstream where they are in a strategic positionbecause: (1) the inventory is common to the greatest num-ber of finished end items desired by the customer, and (2)the inventory is at its lowest value added and thus at itslowest carrying cost. In fact, the inventory levels of rollcoating’s “Support 1” actually need to increase to providesavings for the supply chain as a whole.

The definition of “inventory” as it is used in this resultssection is important. The inventory changes and the com-parisons in Figure 6 represent average inventories. Aver-age inventory for each item includes the safety stockcalculated by the DRP model, plus the cycle stock due toproduction batching, plus the pipeline stock from transportneeds.

Besides the required safety stocks, the DRP model alsoprovided information on the variance of the productionrequirements at each stage of the supply chain. The supplychain optimization team used this variance to determinethe “surge” production capability needed for any stage.For instance, they might set the surge capability to be theproduction level that would cover the production require-ments 95% (1.645 standard deviations above the mean,assuming normal forecast errors) of the time.

4.4. Validation

Before the DRP model recommendations could be imple-mented, the team needed to develop confidence in theresults. Therefore, multiple scenarios were run to test the

Figure 5. Supply chain analysis case study.


sensitivity to various parameters, including the service lev-els, the leadtimes, and the size (variance) of the forecasterrors. (See Hetzel 1993 for details.)

The main barrier that the team had to overcome wasunderstanding how the DRP model works. This was ac-complished by exercising the model for different scenarios,especially conservative ones; by displaying all the inputdata and its sources for validation; by keeping the model(relatively) simple, e.g., assuming a diagonal covariancematrix and limiting the size of the explosion; by comparingthe model results versus current inventory levels; and byacknowledging the model’s shortcomings. Finally, a keysuccess factor was that the model was implemented on apersonal computer, provided a graphical interface for rep-resenting and visualizing the supply chain, and provided analmost instantaneous response. In addition to the analyticmodel, a Monte Carlo simulation was used that simplyworked through the mechanics of the analytic model for arandomly generated demand stream, reporting on perfor-mance measures of interest. The simulation allowed as-sessment of model assumptions, thereby validating theanalysis. For example, constraints were added to the simu-lation that enforced production capacities and prohibitedproduction from beginning if no raw material was onhand.

No model is perfect, and no description of a model iscomplete without a list of shortcomings. The supply chainoptimization team identified three weaknesses: (1) theDRP model does not account for lead time variability, (2)

it assumes stationary average demand over time, and (3) itcannot accommodate a large product explosion. Whereasthe first two are inherent assumptions for the model, thelatter concern was due to a limitation in the software thatcould be easily overcome. However, we expect that in mostpractical situations a team should probably not be workingat any greater level of detail than the case study, say, lessthan 25 items. Keeping the model at an aggregate levelboth reinforces the fundamental guiding principles andalso makes implementation simpler.

4.5. Implementation

Once all of the supply chain analysis requirements werecomplete, the supply chain optimization team added localintelligence about specific customers and manufacturingissues for each item that could not be captured by themodel. After reaching an understanding about how all ofthe model’s proposed changes would impact the supplychain, the team decided to implement a pilot program,with the intention of moving to the more aggressive “basecase” if there were no service problems.

The pilot program only involved the inventories of threeitems. The plan raised the one roll coating item’s inventoryby 20%, and it lowered two sensitizing items’ inventories,each by 60%. The plan was implemented in early 1993.The savings were captured in the 1993 Annual OperatingPlan for the case study’s line of business. As of the end ofApril 1993, not a single end-customer order had been

Figure 6. Results from the supply chain analysis—strategic inventory placement. Note: Excludes emulsion and chemicalsinventories. Excludes regional distribution center (RDC) inventories. Includes all WIP, cycle, and in-transitstocks. Finished goods inventories are gross CDC averages. Service levels are at 95% for all stages. There is noproduction smoothing.


missed on the pilot product due to stockouts or inventoryshortages. The team then implemented the remaining rec-ommendations over the course of 1993.

5. CONCLUSIONS

This paper presents a new model of the requirementsplanning process. We first describe in detail how to modela single production-inventory stage as a linear system, andprovide the analysis for determining performance mea-sures on production smoothness, production stability, andinventory requirements. We also show how to optimize thetradeoff between production smoothness and inventory fora single stage.

To model a multistage system, we can use the single-stage model as a building block. The structure of thesingle-stage model makes it very easy to link single-stagemodels together to represent the multistage system. Inparticular, each single-stage model takes as input a fore-cast of demand requirements and converts this forecastinto a production plan. In the context of a network ofproduction stages, the production plan from a downstreamstage acts as the demand forecast for an upstream stage. Inthis way, we can cascade the single-stage models to modela multistage system.

We also report on an application of the model withinthe context of a supply chain study. The DRP model wasused as a tool to help determine inventory placementacross a multistage supply chain. This illustration providessome evidence of the value of taking a corporate-wide viewby optimizing the supply chain rather than suboptimizingeach of the pieces.

One outgrowth from the case study is a better under-standing of industry needs, and where the DRP model isweak. Based on this experience, as well as observationsfrom industry, we identify the following research topics.

● Nonstationary demand. A stationary demand process isnot an accurate model for the demand experienced bymany products. Common nonstationary effects include sea-sonal effects, end-of-quarter or end-of-year effects (the“hockey stick”), and short-product life cycles. Some ofthese nonstationarities get masked when products are ag-gregated into families or product groups. Nevertheless, animportant enhancement to the model would be to capture,in some way, nonstationary demand processes.

● Service-Level Assumptions. In extending the single-stage model to a multistage setting, we assume that therewill be sufficient inventory to decouple the stages. In effect,we assume that the service levels will be set to assure ahigh level of service, and in the model analysis, we ignorethe downstream consequences of an upstream stockout;i.e., starvation of inputs. These assumptions raise twoquestions. One is, what are the consequences of ignoringthe internal stockouts, and the second is, what should theinternal service levels be. Graves (1988a) provides some

justification for these assumptions in a related setting. Andsimulation tests that we have done confirm that ignoringthe internal stockouts in the analysis, when service levelsare high, does not distort the results of the model. But theissue remains as to how to set the service levels. The liter-ature on multiechelon distribution systems (e.g., Jackson1988, Schwarz 1989, Graves 1995) suggests that, from asystem perspective, it often may be better to have lowlevels of internal service.

● Guidelines for Consolidating Stages. On a related note,we conjecture that, in some instances, the best policy maybe to remove the inventory between an upstream anddownstream stage, and thus consolidate these stages forplanning purposes (Simpson 1958). Rather than have twostages separated by an inventory buffer, we would have one(combined) stage, albeit with a longer leadtime. Within amultistage system, depending on the leadtimes and holdingcosts, it may be optimal to consolidate some of the stages.We expect it would be helpful to have guidelines for deter-mining what stages are good candidates for consolidation.

● Multistage Optimization. The paper describes the opti-mization of the tradeoff between capacity and inventory ina single stage for a diagonal covariance matrix: It would beinteresting to explore how this development extends tonondiagonal covariance matrices, as well as to a multistagesystem. In particular, we would like to develop guidelinesfor setting the weight matrix W for each stage. Further-more, one could explore how to choose among alternativeproduction release policies, such as pull versus push, in amultistage setting.

● Production Assumptions. The model has a highly-simplified model of the production process. The modelsets the production outputs, and these outputs are trans-lated into production starts (e.g., by a leadtime offset).With this model, we can represent fixed lead times, yieldloss factors, batch setup frequencies, as well as uncertaintythat can be modeled as an additive factor. Nevertheless,there are issues as to the validity or appropriateness of thisrepresentation and the sensitivity of the model results tothese assumptions. It would certainly be useful to have aricher model of the production process. For instance, itwould be useful to capture the nonlinear congestion effectsdue to multiple items competing for a shared resource.

ACKNOWLEDGMENT

We wish to thank Chris Athaide for his contributions inthe initial stages of this research, the IBM Thomas J.Watson Research Center and the NSF Strategic Manufac-turing Initiative for financial support for this research,MIT’s Leaders for Manufacturing Program for its supportand resources to complete and apply this research, and thereferees for their helpful and constructive comments on anearlier draft.


REFERENCES

BAKER, K. R. 1993. Requirements Planning. In Handbooks inOperations Research and Management Science, Vol. 4, Lo-gistics of Production and Inventory. S. C. Graves, A. H.Rinnooy Kan and P. H. Zipkin (eds.), North-Holland,Amsterdam.

FINE, C. H. AND S. C. GRAVES. 1989. A Tactical PlanningModel for Manufacturing Subcomponents in MainframeComputers. J. Manuf. and Opns. Mgmt. 2, 1, 4–34.

GRAVES, S. C., H. C. MEAL, S. DASU, AND Y. QIU. 1986.Two-Stage Production Planning in a Dynamic Environ-ment. In Multi-Stage Production Planning and InventoryControl. S. Axsater, C. Schneeweiss and E. Silver, (eds.),Lecture Notes in Economics and Mathematical Systems,Springer-Verlag, Berlin, 266, 9–43.

GRAVES, S. C. 1986. A Tactical Planning Model for a JobShop. Opns. Res. 34, 4, 522–533.

GRAVES, S. C. 1988a. Safety Stocks in Manufacturing Systems.J. Manuf. and Opns. Mgmt. 1, 1, 67–101.

GRAVES, S. C. 1988b. Determining the Spares and StaffingLevels for a Repair Depot. J. Manuf. and Opns. Mgmt. 1,2, 227–241.

GRAVES, S. C. 1988c. Extensions to a Tactical Planning Modelfor a Job Shop. Proceedings of the 27th IEEE Conferenceon Decision and Control, Austin, Texas, December.

GRAVES, S. C. 1996. A Multiechelon Inventory Model withFixed Replenishment Intervals. Mgmt. Sci. 42, 1–18.

HEATH, D. C. AND P. L. JACKSON. 1994. Modeling the Evolu-tion of Demand Forecasts with Application to SafetyStock Analysis in Production/Distribution Systems. IIETrans. 26, 3, 17–30.

HETZEL, W. B. 1993. Cycle Time Reduction and StrategicInventory Placement Across a Multistage Process. MITMaster’s Thesis.

JACKSON, P. L. 1988. Stock Allocation in a Two-Echelon Dis-tribution System or ‘What to Do Until Your Ship ComesIn’. Mgmt. Sci. 34, 7, 880–895.

KARMARKAR, U. S. 1993. Manufacturing Lead Times, OrderRelease and Capacity Loading. In Handbooks in Opera-tions Research and Management Science, Vol. 4, Logisticsof Production and Inventory. S. C. Graves, A. H. RinnooyKan and P. H. Zipkin (eds.), North-Holland, Amsterdam.

KLETTER, D. B. 1994. Proofs of P8 and P9. Technical Appendix.LEE, H. L. AND C. BILLINGTON. 1993. Material Management in

Decentralized Supply Chains. Opns. Res. 41, 5, 835–847.SCHWARZ, L. B. 1989. A Model for Assessing the Value of

Warehouse Risk-Pooling: Risk-Pooling over Outside-Supplier Leadtimes. Mgmt. Sci. 35, 828–842.

SIMPSON, K. F. 1958. In-Process Inventories. Opns. Res. 6,863–873.


O R P R A C T I C E

DEVELOPMENT OF A RAPID-RESPONSE SUPPLY CHAIN ATCATERPILLAR

UDAY RAO, ALAN SCHELLER-WOLF, and SRIDHAR TAYURGraduate School of Industrial Administration, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213-3890

[email protected] ● [email protected] ● [email protected]

(Received August 1998; revision received April 1999; accepted August 1999)

As part of its growth strategy, Caterpillar Inc. is launching a new P2000 product line of “compact” construction equipment andworktools. In anticipation of this, they asked the authors to construct and analyze potential P2000 supply chain configurations. Usingdecomposition and results from network flow theory, inventory theory, and simulation theory, we were able to provide solutions tothis problem for different supply chain scenarios provided by Caterpillar. Novel features of our model include expedited deliveries,partial backlogging of orders, and realized sales that are responsive to service. Caterpillar made their decision regarding the P2000supply chain based on our recommendations.

1. INTRODUCTION

We describe an operations research application supportingthe design and deployment of a distribution logistics sys-tem for a new product line at Caterpillar Inc. (Cat). Afterdecomposing the problem, we apply network flow tech-niques, recent results from inventory theory, and simula-tion based optimization (Infinitesimal PerturbationAnalysis or IPA; see Glasserman and Tayur 1995) to arriveat a solution. In addition to such standard features asmultiple-echelons, capacity constraints, uncertain demand,lead times, and multiple products, our problem also hasthe following novel features:

1. Dealers in Caterpillar’s distribution network can or-der from dual suppliers. There is a low cost regular alter-native and a high-speed, expedited supplier. We determinethe optimal replenishment paths for each (dealer, product)pair using a deterministic minimization of cost or timeover the supply network.

2. The magnitude of captured demand is sensitive toservice response time. In each time period (day, week), thenumber of lost sales depends on the customer service pro-vided. A certain percentage of new customers renege if notimmediately served, while a different percentage of waitingcustomers are lost if forced to continue to wait.

To our knowledge, this is the first time a problem of thisscope and complexity has been solved in this manner. Inparticular, the use of IPA to establish inventory levels inan industrial problem of this magnitude appears to beunprecedented.

In this paper, we describe the development of an opti-mization engine for use in designing Caterpillar’s supplychain. We also detail the collection of data for the engine,provide the results of our optimization, and report on asensitivity analysis of our output. Specifically, after describ-ing the problem environment in §2, we provide details ofthe modeling and analysis in §§3 and 4, respectively. Wereport our results in §5, after which we present some con-cluding remarks in §6.

Throughout the paper, numerical data is disguised tomaintain confidentiality.

2. THE P2000 SUPPLY CHAIN

To exploit anticipated growth in the small constructionindustry, Caterpillar Inc., the world’s leading producer ofconstruction and mining equipment, decided to introducea new “compact” product line, the P2000, starting in 1999.This decision has been widely reported in the business andindustry news media, with articles appearing in publica-tions such as The Financial Times (Feb. 12, 1998) andBusiness Week (March 9, 1998).

One reason for the media’s interest is that the P2000 notonly represents a new product line, but also a new strategyfor Caterpillar’s construction equipment division. Caterpil-lar’s traditional product line consists of large, low-volume,high-margin, customized machines costing $500,000 ormore. Cat is a well-known leader in this market, with fewlarge competitors (such as Komatsu Ltd.). The P2000 fam-ily encompasses smaller, medium- to high-volume, stan-dardized products selling for as little as $20,000 per

Subject classifications: Professional: OR/MS implementation. Inventory/production: applications. Industries: machinery.Area of review: OR PRACTICE.

189Operations Research, q 2000 INFORMS 0030-364X/00/4802-0189 $05.00Vol. 48, No. 2, March–April 2000, pp. 189–204 1526-5463 electronic ISSN

machine. Specifically, the P2000 product line consists ofseveral models of three different machines including aSkid-Steer Loader (“SSL”), a Compact Wheel Loader(“CWL”), and a Mini-Hydraulic Excavator (“MHE”), aswell as some 40 worktools (such as buckets, fork sets, andgrapples). Designed for use with one or more particularmachines such as a Skid-Steer Loader, worktools can besold as attachments to both competitor’s machines andCaterpillar’s. Worktools thus provide a means to enter themarket independent of P2000 machine sales. This compactproduct segment currently has many entrenched marketleaders including BobCat (trademark of the Melroe Com-pany, a unit of Ingersoll-Rand Company), Deere & Com-pany, and Case Corporation.

Both strategic and operational considerations motivateda careful analysis of the P2000 supply chain before itsdeployment. Cat feared that the P2000 family might not fitwell in their current large equipment supply chain. Theywanted to develop a network for the P2000 that couldmaximize profits, capture market share, and provide flexi-bility. They contacted the authors for assistance in deter-mining a configuration which could fulfill these goals. Weexamined two study years: an initial year, 2000, and, fouryears later, in 2004. Although the product launch is in1999, this first year is considered a “ramp-up,” making theyear 2000 demand more suitable for supply chain designand analysis. The two study years differ in the volume offorecast demand, price and cost parameters, and in routingrestrictions. Compared to the year 2000, the routing re-strictions generally were relaxed in year 2004, because Cat-erpillar expected to develop new processing capabilities.

The P2000 products are sourced, manufactured, and as-sembled in approximately 20 locations throughout NorthAmerica and Europe. Sanford (North Carolina) and Leic-ester (England) are key production centers for machines.Worktools come into Caterpillar’s supply chain fromsource-locations in the United Kingdom, the United States,Mexico, Sweden, Germany, and Finland. Each worktoolhas a single source of supply. In North America, P2000products will be sold by a network of 190 Caterpillar deal-ers serving 58 districts in the United States and Canada.

2.1. P2000 Strategy

Specific concerns motivated Caterpillar’s focus on theirP2000 supply chain. The international nature of the chain,coupled with the weight of the equipment, created thepotential for large lead times and shipping costs. Caterpil-lar previously had made the decision not to compete onprice. Rather, in keeping with their core philosophy, qual-ity and service were areas in which Cat would differentiatethe P2000. Thus, long lead times were particularly worri-some. The dealer surveys reinforced this concern, implyingthat Caterpillar’s future products would be highly substi-tutable with those of the competition—primarily Bobcat.

Therefore, Caterpillar believed it crucial that they cap-ture customer demand for their P2000 products as soon asthe demand materialized. By not forcing potential custom-

ers to wait for delivery, Cat would establish a reputationfor product availability. This would not only generate de-mand for the P2000, but also allow Cat to steal demand fortheir competitors’ (substitutable) products. Cat wanted toidentify both a minimum cost channel for a product in thesupply chain and an additional channel for expedited de-livery. The expedited channel, likely one with a highercost, would be used if dealer inventory levels dropped pre-cipitously. This was the genesis of the dual supply modeswithin the supply chain.

To address the objective of maximizing profit subject tocapturing a satisfactory portion of market demand, weconstructed our model so that poor service (product avail-ability) led to lower sales. We felt that Cat would haveconsidered accepting a lower profit, higher inventory solu-tion (for year 2000) to gain “market presence” for theirproducts. To this end, we were prepared to develop a trade-off curve between year 2000 profit and market penetration.

2.2. The Worktool Problem

The nature of the P2000 line dictates that final manufac-turing and testing of some worktools take place at specificnodes of the supply chain. This constrains certain work-tools to pass through selected processing facilities. Similarly,the presence of international import/export facilities such asbonded warehouses (which permit Caterpillar to forgo payingduties while storing their products in transit) requires certainitems from overseas to pass through selected customs loca-tions. With these factors in mind, Caterpillar determined thatthey could use up to seven additional transshipment loca-tions—intermediary nodes between the source and thedealer—in North America, in addition to direct shipment(DS) of worktools from sources to dealers.

The seven possible transshipment locations for work-tools in the United States are grouped into two disjointsets: three Tool Facilities (TFs, which had not yet beenconstructed) and four Parts Distribution Centers (PDCs,which already were handling Caterpillar products). TheTool Facilities potentially would be located in Sanford(North Carolina), Laredo (Texas), and Indianapolis (Indi-ana). The Parts Distribution Centers are located in Mor-ton (Illinois), Miami (Florida), Denver (Colorado), andYork (Pennsylvania). In addition, other transshipment lo-cations are included in the supply network. For instance,there is a UK tool facility in Leicester, a UK PDC inDesford, and a European tool facility in Belgium (with aPDC in Grimbergen) that feed worktools made in Europeto North America. However, we were not given the optionof excluding these nodes from the supply chain.

There were four primary options for the North Ameri-can worktool supply chain:

1. Use of all TFs and PDCs;2. Use of PDCs only;3. Use of TFs only; and4. Use of neither TFs or PDCs, thus allowing only direct

shipment (DS).

190 / RAO, SCHELLER-WOLF, AND TAYUR

Secondary options included using the PDCs with one ortwo supporting TFs, or vice-versa. For example, one mightadd the Sanford Tool Facility to the PDCs to performcertain manufacturing or quality check operations withinthe supply chain.

For each of these four primary scenarios, we were askedto determine:

1. Supply path(s) from each worktool’s source to everydealer region in Caterpillar’s network;

2. Inventory levels and ordering policies at all pointsalong these paths;

3. Revenues, costs, and profits, and their breakdown byproduct, geographical regions, and nodes (these costs ex-cluded the fixed cost of constructing the tool facilities); and

4. The expected percentage of demand captured.

Our analysis showed that the optimal supply chain con-figuration comprised the PDCs and the Sanford TF. Thisconfiguration yields an estimated profit several million dol-lars higher than the TF-Only or DS-Only options, whilecapturing virtually all of the potential P2000 demand.These comparisons are particularly salient because politi-cal considerations prompted Caterpillar to initially favorthe TF-Only and DS-Only scenarios over inclusion of thePDCs.

2.3. The Machine Problem

The nodes in the distribution network for machines werepreviously determined by Caterpillar. Thus, for this fixednetwork, the machine problem requires finding inventorylevels that maximize profits while capturing no less than aspecified percentage of the customer demand. This isequivalent to evaluating a single worktool scenario. There-fore, we will concentrate on the worktool problem in thispaper, although we provide illustrations of some of theresults for machines in §5.

The extant network for machines included two manufac-turing plants and five North American storage facilities.The Sanford plant manufactured only SSL machines, andthe Leicester, UK plant was responsible for CWL andMHE machines. The five storage facilities, used exclusivelyfor machines, were: (1) Houston (Texas), Savannah (Geor-gia), and Harrisburg (Pennsylvania), which served the U.S.market; and (2) the bonded warehouses at Portland (Ore-gon) and Harrisburg, which served the rest of NorthAmerica (primarily Canada).

3. MODELING THE P2000 PROBLEM

Many papers have addressed aspects of material flow man-agement, but few have considered modeling entire supplychains. We approach Caterpillar’s problem in a spirit sim-ilar to that of Lee and Billington (1993) and Feigin (1998).Lee and Billington describe their experience with a decen-tralized DeskJet printer supply chain at Hewlett Packard.They point out some of the challenges in modeling supplynetworks, and take advantage of various approximations to

model a single site in the network. Feigin (1998) also usesapproximations in his analysis of the trade-off between ser-vice levels and inventory investment in large supply chains.See Tayur et al. (1998) for a compilation of recent ad-vances in supply chain management.

Finding optimal solutions to the individual componentsof Caterpillar’s problem, such as the material routing orinventory replenishment subsystem, is in itself extremelydifficult. The deterministic version of the routing problem,a min-cost, multicommodity, network flow problem withnonconvex costs, is the subject of a parallel work by Keski-nocak et al. (1998). This problem is comparable, from acomplexity viewpoint, to the transportation routing prob-lem for next day, second day, and deferred delivery ofpackages considered by Barnhart and Schneur (1996). Sep-arately, Scheller-Wolf and Tayur (1998) consider the de-termination of optimal policies for the inventory problemwith expedited orders. They use IPA to find such optimallevels within the class of order-up-to (or base-stock) poli-cies. Because the P2000 supply chain problem includessubproblems of these types, we believe its exact solution isunobtainable using currently available methodologies.

To make Caterpillar’s problem tractable, while main-taining the model’s validity, we reduced its scope in a vari-ety of ways. This permitted us to arrive at a good solutionin a reasonable amount of time. We could then conduct asensitivity analysis on the robustness of the solution tochanges in model parameters.

3.1. Model Assumptions and Justification

Our primary assumptions are listed below.Problem Decomposition: Figure 1 summarizes our prob-

lem modeling and solution procedure. We decompose theproblem into a network routing problem (§3.2) and a sto-chastic inventory problem (§3.3). The routing problem ig-nores safety stock levels throughout the network.Consequently, it may overlook some inventory risk poolingopportunities offered by overlapping routes for the sameworktool and different destinations. However, for Caterpil-lar’s problem, the relative cost of inventory is small com-pared to the transportation costs (see Figure 5).Significantly, without this decomposition, Caterpillar’sproblem falls in the realm of stochastic nonlinear integerprogramming (SNLIP; Horst and Tuy 1993, Birge andLouveaux 1997). Efficient approaches for SNLIP problemsof this scale currently are not available.

Network Decomposition: Decomposing the network intodealer nodes and transshipment nodes enables us to solvethe resulting subproblems as single-stage inventory sys-tems. The accuracy of known approximations (Glasserman1997, Tayur 1992), and the need to quickly evaluate multi-ple products over different scenarios, motivated this deci-sion. If required, we could have modeled the entirenetwork as one multiechelon inventory system using tech-niques similar to those described in §3.3. Glasserman andTayur (1995) provide the theoretical base for such a mul-tiechelon model.

191RAO, SCHELLER-WOLF, AND TAYUR /

Dealer Aggregation: We solve the problem for one “typi-cal” aggregate dealer within 21 regions in North America,rather than by individually considering all 190 dealers. Cat-erpillar accepted this because:

1. The dealers were roughly uniformly distributedwithin each region.

2. The sourcing and transportation costs up to each re-gion would likely dominate the variation in costs betweendealers within a region.

3. Any increase in cost resolution resulting from a moredetailed model probably would be voided by the compara-tively less exact estimates of individual dealer demand andlogistics cost.

The use of the more detailed network structure wouldpose no theoretical problems for our methodology, but itwould increase computational times.

Disaggregation of Sourcing: Caterpillar uses a uniquesource for each worktool.

Decomposition by Products: As a “base model,” we de-couple the distribution of different products or productfamilies throughout the network—worktools from eachother and from machines. We then solve the routing andinventory problem for each machine and worktool inde-pendently, aggregating the results. Caterpillar agreed tothis based on the fact that worktools and machines were toinitially use disparate distribution networks. This decou-pling disregards the possible dependencies between de-mands for particular products.

In our “refined model,” we consider situations in whichworktools and machines could share transportation. Work-tools may use the flatbed transportation normally used for

the machines at rates, times, and capacities different fromthe normal closed-van mode for worktools. Section 5.1.1describes the results of this generalization in more detail.

Demand Modeling: We model demand in one of twoways. If the estimated mean daily demand for a product isless than one unit, we approximate it with a Bernoullirandom variable. Otherwise, daily demand is modeled us-ing a truncated Normal. The data Caterpillar provided didnot suggest any specific demand distribution. We con-ducted preliminary tests with alternative demand distribu-tions, including the uniform and exponential. These didnot change the overall optimal network configuration. Wemodel the distribution of transship node demand with anormal random variable, because this demand is an aggre-gate over the dealers supplied by this transshipment nodeaccording to lot-for-lot ordering policies. Caterpillaragreed that our demand models were acceptable.

We treat demand for different products at differentdealer districts as uncorrelated and time stationary. Thiswas acceptable to Caterpillar. If they had desired (andbeen able to provide data on correlation/seasonality), wecould have incorporated this in our simulation based opti-mization model. See Kapuscinski and Tayur (1998).

Lost Sales: Based on information provided by Caterpil-lar dealers, we use a two-parameter model for customerimpatience (see §3.3.1). We adopt this model because thedata indicates that customers fall into two categories: thosewho leave immediately and those who are willing to wait afixed amount of time. Had the data indicated more seg-mented behavior, additional parameters could have beenestimated and used.

Preliminary experimentation (borne out by our final re-sults) indicated that the optimal percentage of lost sales issmall. This is a consequence of the fact that the inventorycost rate for a worktool is significantly smaller than theunit profit. This leads us to believe that alternative meth-ods of modeling lost sales (e.g., stochastic parameters, dis-crete parameters) would lead to qualitatively similarresults.

Continuity: For simulation purposes, inventory was ap-proximated by a continuous variable. Due to the observedunimodal nature of the model’s profits, this continuity as-sumption was relaxed by searching the adjacent integralvalues after arriving at an optimal inventory level.

In summary, this paper presents a general methodologyapplied to a specific problem at Caterpillar. At the core ofthis methodology are a network routing problem and sim-ulation based recursions (Appendix A). This methodologyremains valid for many of the model enhancements men-tioned above, such as a more detailed dealer network withcorrelated demand among products and regions. We de-veloped our specific model in an iterative fashion based onperiodic interaction with Caterpillar. As new data arrivedand new features were needed, we updated the model.Our final result is a recommendation to Caterpillar regard-ing the configuration of their supply chain. To the extentpossible, we investigated alternatives to our assumptions,

Figure 1. Flow chart outlining problem decompositionand solution steps.


consistently finding that they did not affect the final recom-mendation. (For example, see the paragraph preceding§5.1.1.)

Our technique’s greatest value lies in its ability to pro-vide a good solution and perform what-if analyses whileincorporating uncertainty over a large and complex prob-lem. The question of whether our specific model, or evenCaterpillar’s data, is an accurate representation of theproblem is a valid one. In the absence of comparable mod-els and solution techniques we are unable to answer thisquestion conclusively. Caterpillar is satisfied with our ef-forts; they are currently implementing a supply chain con-figuration based upon our work.

3.2. The Product Routing Model

For each product and dealer combination, we model thesupply chain as a collection of nodes (sources, dealers, andtransshipment points) and edges (connecting the nodes).Each edge has an appropriate lead time and cost compo-nent: overseas shipment on freighters at container rates,shipment within North America by closed vans or flatbedtrucks at either truckload (TL), or less-than-truckload(LTL) rates and times. Trucking rates depend on thesource and destination, and the product’s weight and vol-ume. Likewise, each node has times and costs associatedwith it. Inventory costs accrue at varying rates for differentlocations and products throughout the network, as do han-dling times and costs. Certain nodes are precluded fromholding any inventory—we treat them as instantaneoustransshipment points.

To accomplish the dual objectives of the supply chain—maximum profit and a high service level—up to two pathsare found for each product and dealer node within thenetwork. The first is the minimum cost or regular path. Thesecond has the smallest lead time from the dealer to thenext level up in the supply chain. This next level up islinked to the source node by a min-cost path. Together,these two links form the expedited path. When the last linkin the minimum-cost path also has the shortest deliverylead time for direct shipment from any transshipmentpoint to the dealer, only this single dominating path isused. The majority of shipments of a product are assumedto flow along the regular path. Worktools would utilize theexpedited link in situations where unexpectedly large de-mand had caused dealer inventory to drop below a speci-fied level. The inventory optimization portion of thealgorithm determines this level.

3.3. The Inventory Model

After decomposing the model by products, we decomposethe inventory system for each product into two subsystems:the dealers and the transshipment nodes. This decomposi-tion implicitly assumes that service levels at transshippoints will be high enough to avoid stock-out occurrences.We describe experiments below, based on Glasserman andTayur (1995), used to test this assumption. They validatedthis decomposition.

Our experiments compared the performance of a two-stage system under our decomposition with a two-stagesystem globally optimized using IPA. For a variety of cost,service, and demand parameters, the approximationtended to decrease inventory levels at the lower echelon.This decrease in lower echelon inventory does not ad-versely affect customer service in Caterpillar’s problem. Infact, the selected inventory levels under the decompositionattain a near 100-percent customer service level. Upperechelon inventory levels could be higher or lower thanoptimal under the decomposition, but these stock levelsalways were very close under both systems, implying simi-lar service levels to the lower echelon.

By virtue of this decomposition, we are able to modelthe dealer subsystem and the transshipment subsystem asdecoupled single-stage inventory systems. Both systemshave cost functions, lead times, demand functions, andservice measures. We measure the dealers’ customer ser-vice using captured demand (fraction of satisfied customerorders). The metric for the transshipment nodes is theprobability of not stocking out when downstream nodesplace orders. For the Markovian demand model with onereplenishment path, or dual replenishment paths havinglead times differing by no more than one period (a day inour case), an order-up-to policy is optimal. This latter re-sult was first proved by Fukuda (1964). For supply chainswhere this is not the case, order-up-to policies, though notnecessarily optimal, have the important advantage of beingsimple to implement. Based on this fact, Caterpillar de-cided that order-up-to policies would be appropriate forthe P2000.

3.3.1. Dealer Nodes. The dealer nodes face a complexstochastic problem with dual replenishment paths. We useIPA to find the optimal inventory parameters within theclass of order-up-to policies, as in Scheller-Wolf and Tayur(1998). This IPA procedure yields either one or two pa-rameters for each item and dealer location—two in thecase where the regular and expedited paths do not coin-cide, and one otherwise. These parameters specify theprofit-maximizing levels at which Cat should maintain theinventory position (IP). By definition, IP equals the inven-tory on hand plus what is on order from the supplier, lesswhat is on backorder to customers. Hence, Cat can changethe IP value by changing the order quantity. When theinventory position, IP, drops below the upper parameter, aregular order is placed to increase IP to that level. If IPdrops below the lower parameter, an expedited order isplaced to bring the inventory position up to this lower levelquickly, and then a regular order is placed to bring it up tothe upper level. The rationale behind this procedure issimple; only when unusually low inventory levels endangerthe satisfaction of customer demand is it worthwhile to paythe extra cost to use the expedited channel.

We determine inventory levels that will maximize ex-pected profit. We do this rather than minimizing total ex-pected cost because our captured demand, and thus sales


revenue, depends on the level of service provided. How-ever, even with the objective of maximizing profit, it isconceivable that the optimal inventory parameters couldpermit an unacceptably large proportion of customers tobe lost. Therefore, consistent with Caterpillar’s strategy(§2.1), we track the customer service level resulting fromour optimal parameters. If the fraction of customers lost isunacceptable to Caterpillar’s management, we could incor-porate additional penalty costs (beyond the lost revenue)for failing to satisfy customer demands. Making these pen-alty costs large forces the algorithm to find inventory levelsthat guarantee arbitrarily high service levels, assuming that

the system has sufficient capacity at the source nodes/pro-cessing facilities. (In our problem environment, there is alimit on the maximum number of units of each productthat a transship node can supply to dealers in any period.)

To efficiently model the likelihood of customers reneg-ing, we used two parameters to incorporate data providedby Caterpillar’s dealer surveys. Our parameters were basedon an aggregation of dealer surveys such as the one shownin Figure 2. The first parameter captured the probabilitythat a new customer would immediately leave should theynot find the product they want in stock. The secondparameter models the proportion of waiting customers,

Figure 2. Sample dealer survey and response.


backordered in a previous period, who depart if their de-mand is not satisfied in the current period. Caterpillar’sdealer surveys implied that a large number of customerswould immediately leave if unsatisfied, while those whochoose to remain undergo a more gradual rate of attrition.

Because both new and old customers renege, the se-quence in which waiting customers should be satisfied be-comes important. Two sequencing approaches arecommonly used: First-Come-First-Served (FCFS) or Last-Come-First-Served (LCFS). Our analysis may be used witheither service discipline. We decided to satisfy the moreimpatient customers first. For Caterpillar’s problem, weused Last-Come-First-Served (LCFS) because new cus-tomers were more likely to renege than old customers. Ifthe model predicted that a significant number of new cus-tomers would be served at the expense of those alreadywaiting, then this LCFS assumption would have to be dis-cussed with Caterpillar management.

3.3.2. Transshipment Nodes. For the transshipmentnodes, we use aggregated demand data from the dealersand the approximation methods developed in Glasserman(1997) to compute base-stock levels, and to estimate re-sulting costs. These methods take into account local de-mand characteristics and production capacities to specifyan ordering parameter that ensures a prescribed servicelevel at each node. Once specified, this single base-stockparameter determines the ordering behavior of the node.

3.4. Problem Data

At each dealer node (and for each product),

Dt 5 stochastic product demand in period t,with mean mt, variance st

2, and cvt 5 st/mt.Lm 5 delivery lead time via mode m for m 5 r,

e (regular, expedited).

cm 5 total unit purchase cost via mode m,including transportation costs.

p 5 unit selling price.h ; I 3 c 5 unit holding cost, where I 5 interest rate

and c 5 relevant purchase cost.

(Along each arc, a transportation mode m will beselected (§4.1). The value of c used for computingholding costs is the cm value corresponding to theselected mode m.)

b0 5 fraction of unsatisfied new customer demand in aperiod that is immediately lost.

b1 5 fraction of unsatisfied old customer demand in aperiod that is lost.

At each transshipment node (PDC or TF), a target ser-vice level, d, and a capacity limit, C, are specified, alongwith holding cost rate, h, unit cost c, and lead time L. Thecapacity limit, provided by Caterpillar, specifies the maxi-mum number of units of each product that a node canobtain from the supplier in any period.

In addition, as illustrated in Figure 3, for a typical work-tool, we have the following:

Product Information: Product name, whether it is aworktool or machine, unit source cost, dealer net sellingprice, weight and volume, source node ID, and any restric-tions on paths from source to dealer. For instance, a typi-cal bucket might have a source cost of $400/unit, a sellingprice of $500/unit, weight of 450 lbs/unit, with volume 32 366 3 25 cubic inches, be sourced from “CMSA” in Mexico,and have the following restrictions (based on qualitychecks, additional work requirements, and current capabil-ities at different nodes): No direct shipment from CMSAto dealers permitted in year 2000. In year 2000, CMSA willship all buckets either to a U.S. Tool Facility or to theMorton PDC for processing before shipment to any other

Figure 3. Illustrative transportation data for a typical bucket worktool.


location. In 2004, CMSA will be able to ship directly to anylocation.

In general, tool facilities can ship worktools to eachother, to PDCs, and to dealers; but PDCs can only ship toother PDCs, or directly to dealers. As compared with Figure3, routing networks for worktools and machines sourced fromEurope tend to have more nodes and higher lead times.

Transportation Data: Feasible routes, TL, and LTL ratesand times (via containers, closed vans, and flatbeds, asapplicable), along with whether specified rates are chargedper unit, by weight or volume. A TL transportation costtable for a typical bucket worktool is shown in Table 1.The entries specify the dollar cost per truckload at closedvan rates over the subnetwork consisting of the sourcenode 1 tool facilities 1 PDCs 1 a typical dealer. LTLtransportation cost tables are similar, except that the en-tries are specified in the units of $/CWT, that is, dollars per“Cent WeighT” (100 lbs). When applicable, similar tables areavailable for container rates and flatbed TL & LTL rates.

We compute transportation costs per unit of productusing TL & LTL rates and either the product’s cent weightor volume, whichever is more restrictive. For example, ifthe LTL rate for Laredo to a dealer is $13.5/CWT, thenthe LTL transportation cost for a 400-pound worktool is$13.5 3 400/100 5 $54/unit. Now suppose a maximum ofone-hundred worktools can be shipped in one truckload,based on permissible weight or volume per TL. Then, ifthe TL rate from Laredo to this dealer is $4,000/TL, theTL transportation cost is $40/unit. If both LTL and TLmodes are available, we select the lower cost option, in thiscase, TL. Note that availability of the TL option does notnecessitate shipment of each product in full truckloads.Essentially, Cat permits the TL option between locationswhere the total demand volume, summed over all prod-ucts, is expected to be large enough that the sum total oforders for different products almost always will use up theentire capacity of a truck. (This is an assumption implicitin the transportation data we were given.) Cat also pro-vided tables with LTL and TL transportation time data foreach pair of nodes in the network. Note that, even if therewas only a single mode of transport for each link, therewould be multiple paths from source to dealer with differ-ent (cost, time) attributes.

Node Information: Minimum order quantities, capacitylimits, processing costs and delays, inventory carryingcharges, desired service levels, and a list of which productsand destinations the node could serve. For example, for a

bucket worktool at the Sanford TF node, the minimumorder quantity is one; the production capacity is 800 units/week; storage capacity is `; the processing cost is $50/unitin 2000 (reduced, presumably due to learning and product/process redesign, to $20/unit in 2004); the processing delayis one day; inventory carrying charges are based on aneffective interest rate of I 5 10%; and the desired servicelevel (probability of not stocking out) is 95%. Sanford canship buckets to all other locations in the network.

Demand Data: Mean and variance of demand in 2000and 2004, for each product at each dealer. We use thisdata to specify a distribution for simulation of daily de-mand. For example, estimated year 2000 demand for abucket worktool at one dealer is 550 units. This translatesto a daily demand of 1.757 (based on 313 working days peryear), which we model as a normal random variable withmean m 5 1.757 and standard deviation s 5 0.5m 5 0.8786(corresponding to coefficient of variation, cv, of 0.5). Oursolution approach is not restricted to a particular value ofcv; we use 0.5 as a representative value that was consid-ered reasonable by Caterpillar. Note that, for normal de-mand, a cv . 0.35 generates a significant amount ofnegative demand, necessitating truncation. Hence, in thiscase, we used equations from Johnson et al. (1994) tosuitably update the demand parameters fed into the trun-cated normal generator. (In addition, we always confirmed,empirically, that the mean of the generated truncated nor-mal demand was equal to the input mean, m.) At a differ-ent dealer, the forecast average annual demand is 78 units,yielding a mean daily demand of m 5 0.249 , 1. We modelthis demand using the Bernoulli distribution with probabil-ity of nonzero demand in any day set at pd 5 0.249. Thus,daily demand is either zero or one, with mean 0.249 andstandard deviation =pd(1 2 pd) 5 0.4326, resulting in asuitably higher coefficient of variation of 1.736.

Customer Patience Parameters: b0 and b1, respectively,the proportion of unsatisfied customers who renege imme-diately and in each period thereafter. Dealer surveys (seeFigure 2) established these parameters. Typically, we useb0 5 0.4 to 0.75 and b1 5 0.15; more service sensitiveregions such as the Northeastern U.S. have higher valuesof b0. The dealer surveys implied worktools should have agreater b0 than machines because Caterpillar’s worktoolswere considered substitutable with those of their competi-tors. Preliminary sensitivity analysis also was conducted ondifferent b values to confirm that small deviations from thechosen b did not significantly affect system performance.

Table 1. Sample TL transportation cost data ($/TL).

Sanford Laredo Indianapolis Morton York Miami Denver Dealer

CMSA 1,500 500 2,500 2,000 ` ` ` `Sanford 0 ` ` 1,000 600 1,400 2,200 1,000Laredo ` 0 ` 2,100 3,000 2,500 1,800 4,000Indianapolis ` ` 0 600 500 2,000 1,700 1,700Morton ` ` ` 0 750 1,750 1,200 2,200Other PDCs ` ` ` ` ` ` ` †

Note: †400 $/TL for York, PA and ` for other PDCs (excluding Morton).


Furthermore, the IPA derivative estimates of performancemeasures with respect to b also were computed during thesimulation to assess if these derivative values were unac-ceptably large.

4. ANALYSIS AND IMPLEMENTATION

4.1. The Product Routing Model

For each product, we determine the lowest cost path fromthe source to each dealer by solving a deterministic net-work problem. Let Em be the set of arcs that permit use oftransport mode m, 6 be the unique source node for theproduct, D denote the set of dealer nodes with mean dailydemand dj for j [ D, and T be the set of transship nodes.Let xij

m be the flow from node i to node j using mode m,and let cij

m denote the corresponding cost of a unit flow. Inour model, cij

m 5 aijm 1 nj 1 IiLij

myi, where aijm is the unit

transport cost from i to j by mode m with correspondinglead time Lij

m, nj is the unit node processing cost at j, Ii isthe inventory carrying cost rate at node i, and yi is theminimum total unit product cost from 6 to i. Then theproduct routing problem can be formulated as:

minxij

mÄ0O

i, j,mc ij

mx ijm

s.t. Om

Oj:~6, j![E m

x 6jm 5 O

j[Dd j ,

O~i, j![E m

x ijm 2 O

~ j,k![E m

x jkm 5 0 for all j [ T,

Om

Oi:~i, j![E m

x ijm 5 d j for all j [ D.

In the absence of arc capacity constraints, the optimalextreme point solution to the above linear programmingproblem has the following characteristics: (1) On each arc(i, j), we use, at most, one of the modes, m* 5 argminm

cijm, corresponding to the lowest cost mode with lead time

of Lijm*. (2) There is, at most, one positive incoming flow

into each node, corresponding to the min cost path from 6to the node. (3) The arcs with positive flows define a span-ning tree rooted at 6, with leaves at each dealer in D; thisdefines a unique path from 6 to each dealer. Based onthese observations, the formulation can be simplified byreplacing the xij

m flow variables on arc (i, j) with one xij

corresponding to the lowest cost mode. Further, withoutloss of generality, each positive demand dj may be replacedby 1, because the lowest cost path for flow into dealer jwill remain unchanged. Thus, once the arc costs cij arespecified (based on the lowest cost mode), the problem isreduced to finding the lowest cost path in the networkfrom 6 to each dealer j [ D. This is facilitated by LPduality.

Let E denote the union, over all modes m, of arcs in Em.Then the dual of our product routing problem is:

maxyj Ä0

H Oj[D

y j Uy j < y i 1 c ij for all arcs ~i , j! [ E;

y 6 5 0J ,

where, as defined earlier, yj is the minimum total sourcingand transportation plus pipeline inventory cost of movingone worktool from the supplier 6 to node j. This dual maybe solved efficiently using Dijsktra’s shortest path algo-rithm (refer to Lawler (1976) for relevant theory on net-work optimization). The only difference between theproduct routing dual and a standard shortest path problemis that the arc lengths are not constant in our dual; thepipeline inventory portion of cij depends on the value ofthe decision variable yi. This does not pose any computa-tional difficulties because, in our implementation of Dijsk-tra’s algorithm, we process the nodes in a specific orderobtained using a topological sort (Aho et al. 1983) of theunderlying directed acyclic graph. That is, we renumberthe nodes of the supply chain so that, for every arc from ito j, the index of i is smaller than the index of j. If nodesare processed in increasing order of their index, the ab-sence of directed cycles in our supply network allows us tocompletely determine yi before cij is calculated, eliminatingany potential difficulties.

The model above sets each dealer’s regular supply nodeequal to the immediate predecessor of the dealer in themin-cost path from the source to the dealer. The lead timefor regular deliveries is the time for shipment from thisimmediate predecessor node to the dealer, under the as-sumption that the predecessor carries sufficient inventoryto provide a high level of service. For cases in which thepredecessor node is allowed to carry no inventory, thedelivery lead time observed by the dealer is increasedappropriately.

We determine inventory levels at transship points afterthe ordering policies at all dealers are specified. Conse-quently, the use of supply nodes by different dealers isaccounted for when transshipment inventory levels are set.

4.2. The Inventory Model

We use separate models, which are both stochastic, fordealer nodes and transshipment nodes. This is motivatedby the difference in service level definitions at dealer nodes(which face response-sensitive customer demand) andtransshipment nodes (where demand comes from captivedealers). Thus, the dealer model must incorporate the pos-sibility of lost demand, whereas the transshipment nodemodel just backlogs excess demand.

4.2.1. The Dealer Model. Data from the routing modelserves as input to the dealer inventory model, which max-imizes the total expected profit, where profit equals reve-nue minus regular and expedited sourcing costs minus on-hand inventory carrying costs. The unit sourcing cost fromnode i is yi plus additional transportation, pipeline inven-tory, and node processing costs incurred between node iand the dealer. The dealer’s regular supply node is deter-mined by the product routing model. The transshipmentnode, i, with the smallest lead time for direct material flowto the dealer is the supply node for expedited deliveries.


The variables and features of the inventory model in-clude:

Y It21 5 inventory level at end of period t 2 1 5 on-hand inventory 2 backlog.Y Xt

m 5 order placed in period t , t, for delivery viamode m, for m 5 r (regular), or m 5 e (expedited). Westore past orders for t 5 t 2 1, . . . , t 2 Lm. All ordersthat have been placed but not yet delivered contribute tothe in-transit or pipeline inventory.Y Pt21 5 ¥m5r,e ¥t5t2Lm

t21 Xtm 5 total pipeline inventory

at end of period t 2 1.Y IPt21 5 It21 1 Pt21 5 inventory position at end of

period t 2 1.Y Rt 5 X t2Lr

r 1 X t2Le

e 5 receipts in period t.

The sequence of actions in period t is:

Step 1. Determine beginning inventory level, It21, andpipeline inventories, X r, X e.

Step 2. Receive delivery of relevant pipeline inventory,Rt.

Step 3. Observe demand Dt.

Step 4. Satisfy as much demand as possible from on-hand inventory; a portion of unfilled demand is lost.

Step 5. Place new replenishment orders, Xte and Xt

r.

Step 6. Update profit.

We elaborate on Steps 4 and 5 below:

Step 4. Inventory allocation: We use inventory to satisfythe most impatient demand first. For our data set, b0 . b1,so we satisfy new demand before demand from previousperiods is satisfied (LCFS).Lost demand in period t: Let x1 5 max(0, x) and x2 5(2x)1. Then

+ t0 5 b 0 ~D t 2 I t21

1 2 R t !1 and

+ t1 5 b 1 ~I t21

2 2 ~R t 2 D t !1! 1, (1)

where +t0 and +t

1 denote, respectively, the portion of un-filled new demand and waiting customer orders in period tthat are lost. (If FCFS were used, +t

0 5 b0(Dt 2 (It21 1Rt)

1)1 and +t1 5 b1(It21

2 2 Rt)1.) With +t 5 +t

0 1 +t1,

the ending inventory level is

I t 5 I t21 1 R t 2 D t 1 + t . (2)

For example, if b0 5 0.6, b1 5 0.15, It21 5 220, Rt 5 40,and Dt 5 50, then +t

0 5 0.6(50 1 0 2 40)1 5 6 and +t1 5

0.15(20 2 0)1 5 3. Thus, as new demand exceeds receiptsby ten, six of these ten units will be lost in addition to threeof the backlogged 20 units of past demand. Total lost de-mand is +t 5 9. On the other hand, if Rt 5 60 in the aboveexample, then +t

0 5 0 (no new demand is lost) and +t1 5

1.5.

Step 5. Order-up-to policy for replenishment: Because cr ¶ce and Lr Ä Le, it follows that zr Ä ze. Thus, the inventoryposition after order placement will always be IPt 5 zr. If

the inventory position prior to ordering is IPt2 5 IPt21 2Dt 1 +t, the expedited order quantity which orders-up-toze is

X te 5 ~ z e 2 IP t2 ! 1 5 ~ z e 2 z r 1 D t 2 + t !

1. (3)

Assuming zr Ä ze, the regular order quantity is

X tr 5 min~ z r 2 z e , ~ z r 2 IP t2 !1! 5 ~D t 2 + t 2 X t

e!1.(4)

Sales in period t is

S t 5 min~I t211 1 R t , D t 2 + t 1 I t21

2 !

5x 15x1x 2

min~I t21 1 R t , D t 2 + t ! 1 I t212

5~2!

min~I t , 0! 1 D t 2 + t 1 I t212

5 I t212 2 I t

2 1 D t 2 + t .

Period t profit is

p t 5 pS t 2 hI t1 2 c e X t

e 2 c r X tr. (5)

We also measure the long-run fraction of demand that islost and the fraction of demand satisfied using the regularand expedited modes, respectively. We solve the dealerinventory model by selecting a starting value of (zr, ze) andgenerating a set of k demand scenarios. From these sce-narios, we compute the estimated expected profit pt(zr, ze)and the IPA derivative estimates dpt/dze and dpt/dzr ac-cording to their recursions (shown in Appendix A). Wethen use a subgradient-based search to find the optimalvalue of (zr, ze). Because we are using simulation, a proofof joint concavity of the profit function with respect to theparameters is desirable. This is difficult to prove for ourproblem, and, therefore, is the subject of parallel work. Ifthere were no lost sales, concavity would be relativelystraightforward to show using induction on Equations (1)–(5). Our computer experiments, illustrated in Figure 4,indicate that the profit function is concave when the base-stock levels are high (which eliminates lost sales). Whilealways unimodal over zr Ä ze, the profit does fail to bejointly concave at low (ze, zr) values, where significantnumbers of customers are lost.

Based on Figure 4 (which we generated for several prob-lem parameters) and the concavity of profit in the absenceof a loss function, we believe that, for the high servicelevels we use, the profit function is likely unimodal in theregion of interest. Assuming the dealer profit function is,in fact, unimodal, our IPA procedure converges to theoptimal values of zr and ze. Refer to Scheller-Wolf andTayur (1998) for further details.

Prior to embarking on our IPA optimization, we con-ducted the standard practice of removing initialization biasand checking for “steady-state,” based on pilot runs. Asexpected, simulation estimates of the optimal expectedprofit become more accurate as the number of simulationiterations (demand scenarios) increases. However, theseprofit estimates differed by no more than 0.2% over arange of simulation iterations between k 5 1,000 and k 510,000, while simulation run times increased by an order


of magnitude. Therefore, to strike a balance between com-putational time and accuracy, we generated 10,000 de-mand scenarios for the last two simulation runs, justbefore termination of the search for optimal (zr, ze), andused k 5 3,000 demand scenarios during the IPA search.

4.2.2. The Transshipment Node Model. Using outputfrom the product routing and dealer inventory models, wedetermine the fraction of customer demand satisfied ateach dealer using the regular and expedited modes. Byaggregating these product flows over all dealers we deter-mine the mean, m, and variance, s2, of daily demand foreach product at each transshipment node. Given these pa-rameters, the capacity C, desired service level d, and thelead time L that the transshipment node faces for deliveryfrom its source, the base-stock level at a transshipmentnode is set to

z 5 F ~L 1 1!m 1s 2

2~C 2 m!G

1 F 21~d!F ~L 1 1!s 2 1 S s 2

2~C 2 m!D 2G 1/ 2

. (6)

The first bracketed term accounts for the mean demandover the lead time and the mean shortfall, while the sec-ond term corresponds to safety stock (incorporating de-mand and shortfall variability). See Glasserman (1997) andTayur (1992) for details. Because demand at each trans-shipment node is the sum of many demands originating atdifferent dealers, we assumed that the cumulative distribu-tion function (CDF) of demand at the transshipment nodeduring its delivery lead time could be approximated by anormal distribution F[. We denote the inverse of the

CDF by F21[. This demand model ignores correlationbetween demands at different transshipment points. If cor-relation effects or deviations from the normal distributionare significant, we could use a more accurate, but compu-tationally intensive simulation model similar to the dealermodel in §4.2.1.

Inventory holding costs at each transshipment node areestimated as h[ z 2 m(L 1 0.5)]. This approximation isfairly standard (see, for instance, formula (5-1) in Hadleyand Whitin 1963). Costs over different transship nodes areaggregated to obtain estimated total costs for this subsystem.This cost then is subtracted from the simulation-based esti-mate of dealer profit for each product. This yields theproduct’s contribution to expected profit. Total systemprofit is the sum of the profit for all of the products.

5. RESULTS

For each product, solution of the routing problem tookjust a few seconds on a Sparc20 workstation. This yieldsthe min-cost (regular) path from the source to each dealerin the network. For expedited deliveries, we identified thetransshipment location that had the shortest delivery timeto the dealer. This was instantaneous. The average run-time to compute the optimal inventory levels (zr, ze) foreach product was just under 40 seconds per dealer. Calcu-lation of inventory levels at the transshipment nodes usingEquation (6) was instantaneous. We considered 21 dealerdistricts, so complete analysis of one product over all deal-ers took approximately 21 3 40 seconds or 14 minutes. Catprovided comprehensive data for 21 products over twoyears (2000 and 2004). Consequently, one run over all

Figure 4. SSL GP-bucket year 2000 profit function for a typical dealer.


products, dealers, and study years took approximately 9.8hours. For each scenario tested, we typically ran our exper-iments in under five hours on two computers working inparallel on independent sets of products or study years.

5.1. General Results

We considered several scenarios consisting of sets of per-missible transshipment nodes. The four primary scenarioswere TF & PDC, PDC-Only, TF-Only, and DS-Only, forwhich we generated solutions that were optimal within theclass of order-up-to policies for each of these networks. Asolution maximizes expected profit by setting order-up-tolevels for each of the products, locations, and both theregular and expedited modes. These inventory levels en-sure that a sufficiently high proportion of customers areserved. Our reported results include:

1. Optimal total profit for each scenario.2. The relative profit from machines and worktools.3. The geographical distribution of profit percentages

and the contribution of each product.4. The market capture percentage (100% 2 lost

sales%).5. The breakdown of cost components (source cost,

node costs, transportation costs, pipeline, and safety stockcosts).

6. Optimal transportation modes for each link in thesupply chain.

7. Product delivery lead times.8. The effect of demand volume changes from 2000 to

2004.

As noted earlier, scenario costs exclude the fixed costsassociated with construction of the tool facilities. There-fore, it comes as no surprise that the greatest expectedprofit is attained by using the entire network, as shown inTable 2. From the table, we observe that TFs will benefitless from the increased demand volume in 2004. This maybe explained by noting that the linear transportation costcomponent dominates the sublinear inventory cost compo-nent for the TF-Only scenario.

Table 2 shows that inclusion of Tool Facilities (TFs)adds about 2–3% to profit from worktools, as compared tousing the PDC-Only scenario, which is on the order ofseveral million dollars. Nevertheless, this was outweighedby the estimated costs of building and operating the toolfacilities. Therefore, Caterpillar decided to implement theuse of the PDC-Only alternative (plus the Sanford TF dueto routing constraints) along with direct shipments. Ourmodel indicates that this supply chain configuration willcapture almost 100% of the demand. This negligible lost

sales is a consequence of the fact that the incrementalinventory holding costs are significantly smaller than thelost profit due to shortages. This leads to large dealerinventories primarily supported by the regular mode, withthe expedited mode used only occasionally during highdemand periods. This behavior can be seen in Figure 4,which illustrates how profit changes with ze and zr for arepresentative worktool. For this example, the optimal lev-els are zr 5 6, ze 5 0, but many higher values of zr, ze (e.g.,zr 5 6, ze 5 1) result in near-optimal profit with little or noexpediting. However, using zr 5 3 instead of the optimalzr 5 6 reduces profit by more than 18%, primarily due to adecrease in customer service from 100 to 81 percent.

Since demand forecasts form an integral part of ouroptimization, we assessed the sensitivity of our recommen-dations to variations in input demand data. We studiedperformance measures (profit, inventory, and service lev-els) for each scenario at four distinct mean demand levels:(1) Caterpillar’s forecast m, (2) 0.8 3 m, (3) 1.2 3 m, and(4) U(0.8, 1.2) 3 m, where U(a, b) denotes a uniformrandom variate between a and b and each product’s meandemand is multiplied by a different realization of U(0.8,1.2). In all cases, the relative profitability of the differentscenarios remained remarkably insensitive to changes inmean demand. We chose 0.8 and 1.2 after discussions withCaterpillar. Caterpillar felt that a demand greater than 1.2times the forecasted mean was unrealistic, given the ag-gressive nature of their target. On the downside, if thedemand was less than 0.8 of the forecast, then a strategicdecision would be made on price and advertising thatwould lead to a new analysis of the situation. Furthermore,if deemed necessary by Caterpillar, we were prepared torerun the model for several values of mean demand tobetter estimate the robustness of our recommendations.

We also note that each mean demand is an aggregationof mean demands for different products/dealers (e.g., theproduct type we call fork sets is actually comprised ofseveral distinct, but similar, fork sets; each dealer district isan aggregation of several dealers). It is widely acceptedthat such aggregate forecasts are likely to be more reliable,and so a 620 percent variation on the mean, in general,may be an acceptable range of analysis.

5.1.1. The Refined Model. Because the PDCs performedwell for worktools, Caterpillar decided to consider an al-ternative scenario in which PDCs (and TFs) also werepermitted to act as transshipment points for machines.Their hypothesis was that, if machines were allowed toflow through the PDCs and TFs, the worktools would ben-efit from lower transportation costs resulting from cheapermodes of transport normally available only to machines.This necessitated the generation of new data that specifiedTL transportation costs and times via closed vans and flat-beds, charged at new rates for worktools. Preliminary anal-ysis showed that this would not increase total profits. Theadded node costs incurred for processing machines over-whelm any savings from combined worktool and machine

Table 2. Percentage of optimal profit acrossdifferent scenarios.

Year TF & PDC PDC-Only TF-Only DS-Only

2000 100.00 96.76 88.99 77.582004 100.00 97.98 89.38 81.23


transportation. In addition, while PDCs were already wellequipped to handle worktools, this was not the case formachines. Hence, Caterpillar decided to use the results ofthe base model, which decoupled the distribution of ma-chines and worktools.

5.2. Revenues, Costs, and Profits of First Model

The remainder of this section illustrates output analysesconducted to provide a better understanding of the resultsof our study. Caterpillar was interested in the breakdownof costs by components such as source costs, transporta-tion costs (separated by PDC and non-PDC costs), pipe-line inventory costs, and on-hand inventory costs. Thisinformation served a variety of purposes. For example, Catwas considering different contracts with dealers, some ofwhich included Caterpillar’s ownership of “consignment”inventory and/or a manufacturer buy-back option. ThusCat might be responsible for a portion of the dealer inven-tory costs. In addition, the PDC costs and profits wererequired to estimate the appropriate transfer payment tothe PDC logistics group, should they be included in thesupply chain. Consequently, these expected costs were cat-egorized as:

1. Source (cost charged by external supplier);2. Non-PDC Ship (non-PDC pipeline inventory and

transportation costs);3. PDC Outbound (costs incurred after some PDC

took receipt and control of material);4. TF Inv/Proc (on-hand inventory and node process-

ing costs at TFs); and5. Dealer Inv (costs of on-hand inventory at dealers).

We illustrate this cost breakdown in Figure 5 for thePDCs plus Sanford TF scenario. The bulk (90%) of costsare source costs; non-PDC pipeline inventory and trans-portation costs form a substantial portion (approximately7%) of the remaining 10%; PDC-Outbound is between 1%and 1.5%; and dealer inventory costs are under 1% of totalcosts. This comparatively low dealer inventory cost comesdespite the high service levels and the use of longer leadtimes (more regular shipments than expedited). The TFcost portion is negligible (,0.02%) because the SanfordTF is not used for most worktools, but is required forprocessing a select few. Together with Figure 7, Figure 5shows that inventory and costs shift from the dealer to thepipeline, with an increase in demand volume and addi-tional new routes in 2004. The increased demand volumedoes not require much increased floor space or inventoryinvestment at the dealer; however, it does require the ca-pability to handle larger volumes throughout the distribu-tion network.

Our model’s output also includes regional profits. Figure6 shows that the top six districts account for more than 60percent of total profit from projected year 2004 worktoolsales in North America. Only 18 dealer districts are showninstead of 21 because “Other Canada” actually consists offour districts including Northeast (Newfoundland and

Nova Scotia) and Southeast (Montreal and Toronto) Can-ada. Similar graphs illustrate the breakdown by volume ofmaterial flow through distribution centers, as well as thetotal profit generated by each worktool. These can be usedto prioritize operations by product and/or region, via anABC type analysis.

5.3. Inventory Levels

We now focus on the location and magnitude of inventorywithin the supply chain, and illustrate our observationsusing the TF & PDC scenario. Figure 7 demonstrates thatmost of the inventory is in transit, the transshipment nodescarry very little inventory, and the dealers carry most ofthe on-hand inventory (cycle stock and safety stock). Inyear 2000, expected total demand volume for SSL tools isabout four times the demand volume for CWL/MHE tools.However, SSL tools have less relative demand variability,so their safety stock is not proportionately larger than thatfor CWL/MHE tools. Similar numbers hold for 2004.

We also studied the breakdown of each of these inven-tories by product. For year 2000, the detailed breakdownof average dealer on-hand inventory of 811 SSL worktoolsand 320 CWL/MHE worktools is shown in Table 3. Be-cause Caterpillar was considering different contracts with

Figure 5. Worktool cost breakdown by percentage.

Note: Ninety percent of the total cost is due to manufactur-ing (termed “source” cost), approximately 7% of cost is dueto transportation and pipeline inventory prior to arrival atPDCs, 1.5% of cost is due to transportation and inventory atthe PDC and in transit from PDCs to dealers, and the remain-der is made up in inventory holding costs at the dealers. Theinventory carrying costs at the TF are negligible.


dealers, as mentioned above, these inventory breakdownswere of considerable importance. They also provide a firstestimate of which products could most benefit from im-proved logistics. For example, there are more SSL GPbuckets in inventory on average than the combined total ofall of the CWL and MHE worktools that were studied.

A similar analysis of pipeline inventory (say, in terms ofuse of TL and LTL rates and expedited vs. regular ship-ments) over the network is possible from our results, but isnot included. We also graphed the worktool and machineinventory by nondealer locations. The latter is shown in

Figure 8, which illustrates output from our analysis of thesupply chain for machines.

To summarize, our model evaluates different supply chainconfigurations along the dimensions of profit, captured de-mand, inventory parameters, and transportation mode usage.From these evaluations we obtain optimal inventory routesand levels corresponding to order-up-to policies with dualsupply modes. These evaluations proved to be robust withrespect to changes in system parameters such as demandintensity, transportation options, and customer impatience.

6. CONCLUDING REMARKS

In this paper, we develop an integrated model to analyzedifferent supply chain configurations for Caterpillar’s newline of compact construction equipment, the P2000 series.We use decomposition techniques, network optimizationtheory, inventory modeling, and simulation theory. Thenovel features of our model include dual modes of supplyfor dealer replenishments and net customer demand thatis responsive to speed of service. We were able to makerecommendations to Cat on the effects of different factorson profits. In the last quarter of 1998, Caterpillar launched

Figure 6. Year 2004 worktool profits by region.

Figure 7. Worktool inventory location within supplychain.

Table 3. Dealer inventory by worktool type 2000.

SSL Tool UnitsCWL/MHE

Tool Units

Brooms 64 Brooms 20GP Bucket 338 GP Bucket 67MP Bucket 58 MP Bucket 60Grapples 63 Grapples 64Hammers 63 Hammers 59Augers 67 Light Material 17Derivatives 84 Special Dump 33Fork Sets 74Total 811 Total 320


its P2000 line, supporting it with the supply chain—PartDistribution Centers (PDCs) plus Sanford Tool Facility—recommended by our analysis.

Without our analysis, Caterpillar likely would have im-plemented either the Direct Shipment-Only (DS-Only) orthe Tool Facilities-Only (TF-Only) option. Internal con-siderations caused Cat to question the value of inclusionof PDCs into the network. Disregarding fixed costs ofTF construction, the annual benefit of our solution overTF-Only is roughly eight percent of the maximal ex-pected profit, which is several million dollars. This com-parison does not capture the full benefit of our project.It assumes that Caterpillar would have used the optimalinventory in its implementation of the TF-Only option,which we, in fact, specify. This is significant because ourIPA optimization indicates that choosing the correct in-ventory ordering parameters can be vital. In our prob-lem, setting the levels too low increases lost sales andrequires a greater use of expediting, resulting in signifi-cantly lower profits, as shown in Figure 4. The actualbenefit of our project thus is likely to be significantlygreater than eight percent.

The work reported in this paper has concentrated on theNorth American market. A similar analysis applies to Cat-erpillar’s European market.

APPENDIX A. IPA DERIVATIVE RECURSIONS

The derivative recursions for dp/d(ze) and dp/d(zr) (usedby the gradient-based search for optimal ze and zr) are:

dp t

dz e5 p1$S t . 0%

dS t

dz e2 h1$I t . 0%

dI t

dz e

2 c e 1$X te . 0%

dX te

dz e2 c r 1$X t

r . 0%dX t

r

dz e. (A1)

dS t

dz e5 21$I t21 , 0%

dI t21

dz e

1 1$I t , 0%dI t

dz e2 1$+ t . 0%

d+ t

dz e. (A2)

dI t

dz e5

dI t21

dz e1

dR t

dz e1

d+ t

dz e,

withdR t

dz e5

dX t2Le

e

dz e1

dX t2Lr

r

dz e

~known from previous iterations!. (A3)

dX tr

dz e5 2

d+ t

dz e2

dX te

dz e

~information from (A4) and (A5) will be usedin future iterations.)

(A4)

dX te

dz e5 1$ z e . IP t2 %S 1 2

dIP t2

dz eD

5 1$ z e . z r 2 D t 1 + t %S 1 2d+ t

dz eD . (A5)

d+ t

dz e5

d+ t0

dz e1

d+ t1

dz e, where

d+ t0

dz e

5 b 0S21$I t21 . 0%dI t21

dz e2

dR t

dz eD

z 1$D t 2 I t21 2 R t . 0% andd+ t

1

dz e

5 b 1S21$I t21 , 0%dI t21

dz e2 1$R t . D t %

dR t

dz eD

z 1$+ t1 . 0%. (A6)

The order of derivative computation is in the reverseorder of the listing above. We only track derivatives of It

and Xtk (which yield all other required derivatives of Rt, +t,

etc.). Derivative recursions w.r.t. zr are identical in form,except for item (A5) above, which becomes

dX te

dz r5 1$ z e . z r 2 D t 1 + t %S21 2

d+ t

dz rD .

The validity of these simulation-based derivative estimatesfollows from arguments similar to Glasserman and Tayur(1995).

ACKNOWLEDGMENTS

We thank Dr. George Cusack at Caterpillar, Inc., Peoria,for providing us with the opportunity to work on this prob-lem, and for his insightful comments. All numbers con-tained in this article have been disguised in accordancewith a contractual agreement. We also thank the associateeditor and two anonymous referees for many suggestionsthat greatly improved the content and presentation of thispaper.

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