An application of designing products and processes for supply chain management

An application of designing products and processes forsupply chain management

AMIT GARG

Nonstop Solutions, Inc., 550 California Avenue, San Francisco, CA 94104, USAEmail: [email protected]

Received October 1996

In this paper we describe an application of designing products and processes for supply chain management at a large electronicsproducts manufacturer. The objective of our research project was to reduce the costs of complexity resulting from a proliferation ofparts and processes in the manufacturer's supply chain. In order to perform this analysis, we developed the Supply Chain Modelingand Analysis Tool (SCMAT). SCMAT models decentralized supply chains and is less data-intensive and yet more general thanprevious work in this area.

1. Introduction

In this paper we describe the Supply Chain Modeling andAnalysis Tool (SCMAT) that we developed and its ap-plication to designing products and processes for supplychain management at a Large Electronics Manufacturer(LEM) [1]. Although SCMAT was developed with ourapplication in mind, it can be used to perform a host ofstrategic-level analyses of fairly general supply chains. Inparticular, the types of analyses managers could useSCMAT to perform include: inventory-service leveltrade-o�s, sourcing, location, and transportation trade-o�s, e�ects of capacity limitations, impact of lot sizes,and designing products/processes for supply chain man-agement.At the very outset, we would like to de®ne the use of

the term process design. Process design can be performedat an operational and at a tactical/strategic level. At anoperational level, process design involves the design ofproduction processes required to manufacture the prod-ucts [2]. At a tactical/strategic level, process design in-volves the design of business processes such asdistribution, packaging, transportation, order entry andmanagement, etc. [3]. In this application, although westudied process design at both operational and tactical/strategic levels, we focused primarily on the tactical/strategic-level process design.Our research was largely motivated by the non-avail-

ability of several types of input data required by existingmodels of decentralized supply chains and by a few ap-plication-speci®c requirements. These constraints resultedin a model that has several implementation and practical

advantages over existing models. Since our model re-quires fewer types of input data, the time required tocollect data is less, resulting in faster implementation. Inaddition, since inaccuracies are inherent in input data formost applications, models using fewer types of input datatend to be more accurate and more robust. Theoreticalcontributions of our model are detailed in Section 2. Ourpaper contributes to the practical aspect of research byoutlining a methodology for approaching projects ondesigning products and processes for supply chain man-agement at a large corporation.This paper is organized as follows. In Section 2 we

review relevant literature. Section 3 describes our e�ortsin designing products for supply chain management at theLEM. Section 4 describes SCMAT, the model used toevaluate di�erent product and process design alternatives.Section 5 presents the results of our analyses. Finally, inSection 6 we conclude by describing areas for futureresearch.

2. Literature review

Multi-site production and inventory networks have beenstudied by many researchers. Such networks can beoperated in a centralized or decentralized mode. Somepapers that model production and inventory networksunder a centralized control policy include those of Clarkand Scarf [4], Federgruen and Zipkin [5], and Rosling [6].Total costs of a system operating under centralized con-trol are usually lower than those of a decentralized system[7]. However, centralized policies can be very complicated

0740-817X Ó 1999 ``IIE''

IIE Transactions (1999) 31, 417±429

for most general systems. And in practice, most supplychains still operate in a decentralized mode.Several researchers including (Graves [8], Cohen and

Lee [9], and Lee and Billington [10]), have modeled de-centralized networks. All these models require threebroad classes of inputs: end-product demand and servicerequirements, bill of materials and routing data, and site-speci®c part data. Di�erences in the three models citedabove stem from di�erences in the inventory policies theyhave considered, the type of distribution for the end-product demands, and the types of network studied. Thekey decision variables are the inventory policy parametersfor each part-site combination. Given inventory policyparameters for each part-site combination, one canevaluate the performance of the network.Graves [8] modeled decentralized production networks.

His model uses a periodic-review, base-stock inventorypolicy and assumes demands to be stationary and nor-mally distributed. The key to his solution approach isaggregating the results of single-site inventory models inorder to evaluate the performance of the network. In thismodel, production rate at a site is variable and moreoveris a decision variable. Although this model can be used toperform di�erent types of supply chain analyses, assum-ing total ¯exibility in varying production rates at each siteis a limitation of this model.Cohen and Lee [9] modeled a more general decentral-

ized network. In their network, the manufacturing sitehas multiple inputs, and its outputs feed a divergentdistribution network. The distribution sub-model in theirnetwork uses an (s, S) policy at each site, while themanufacturing sub-model operates under an (nQ, R)-in-ventory policy. Cohen and Lee assume end-product de-mands to follow a stationary Poisson process.Cohen and Lee solved their problem by decomposing

the network into several sub-models: material control,production, and distribution. The material control sub-model is an assembly system that supplies material to themanufacturing sites. Outputs from manufacturing sitesare fed into a divergent ¯ow distribution sub-model. Thekey linkage between manufacturing and distribution isthe manufacturing lead time that becomes the replen-ishment lead time for the distribution sub-model. How-ever, their network is limited to a single manufacturingsite.Lee and Billington [10], henceforth referred to as LB,

developed an analytical model for a decentralized supplychain. Their model assumes each site operates under aperiodic-review, base-stock inventory policy and the de-mands for end-products are normally distributed. A keyassumption in the model is that the replenishment leadtime for a product at a node comprises the material leadtime, the production lead time, and the delay time. LikeGraves, LB also evaluate performance of the network byaggregating the solutions of multiple single-product, sin-gle-site inventory problems. One of the advantages of this

framework is the ability to analyze the ®rst two momentsof all the performance metrics.Like Graves and LB, we also assume end-product

demands to be stationary and normally distributed.However, the LB model was not applicable at LEM be-cause of non-availability of several types of input data.For example, the LB model requires the user to inputdelay at the downstream production site due to non-availability of parts. We found that the LEM did notmaintain this type of data for its supply chain. Therefore,we developed approximate analytical expressions forthese input data.Another input the LB model requires is the capacity at

a node allocated to each product ¯owing through thatnode. This input e�ectively allows one to decompose thenode into sub-nodes, one for each product. One can thenevaluate the performance of the entire network by usingthe results of a single-site inventory model for eachproduct-node combination. In practice, however, manyproducts usually share common resources: manufacturinglines, storage space, etc. It is very di�cult to pre-deter-mine the capacity allocated to each product ¯owingthrough a node.In fact, managers at the LEM were also interested in

capacity requirements at each site in the supply chain andin studying the implications of capacity on lead times andinventories. This type of analysis cannot be done usingmost previous periodic-review models of inventory net-works. In addition, SCMAT explicitly incorporates thecongestion e�ects due to capacity limitations at eachnode, and the interference e�ect because of multi-product¯ows through each node. The models developed byGraves [8], Cohen and Lee [9], and LB, do not considerthese e�ects.Queueing models allow one to capture the above-

mentioned e�ects explicitly. Our model incorporatesqueueing approximations within the inventory modelingframework. Considerable research has been done on an-alytical models of general queueing networks. Broadcategories of approaches taken by researchers are asfollows: decomposition methods, di�usion approxima-tions, mean value analysis, and operational analysis.We refer the reader to Bitran and Tirupati [11] for a

description of, and for details on, these approaches. De-composition approaches assume the nodes in the networkto be ``stochastically independent''. Therefore, one canevaluate the performance of the entire network by ag-gregating the results of a performance analysis of eachnode in the network. Whitt [12], Buzacott and Shan-thikumar [13], and Bitran and Tirupati [11] are someauthors who use this approach. Since it is consistent withthe disaggregation method of solving the production-inventory network, we have incorporated queueing ap-proximations arising in the decomposition approach intoour inventory model. These approximations are describedin more detail in Section 4.

418 Garg

3. Problem description

A new line of products, code-named George, was to beintroduced by the LEM. George is a high-technologyproduct in a market that is facing price pressures due toincreased competition. The product design team com-prising engineers from software, hardware and mechani-cal design, manufacturing, marketing, and distributionwere aware of the impact of design on total cost of theproduct and were interested in some analyses to selectamong various product and process design alternatives.However, this initiative was very new to the LEM. Wewere part of a corporate manufacturing department teamworking closely with the design team to support thisinitiative.Based on the discussions we had with members of the

design team, the questions they wanted the model toaddress can be summarized as follows:

· Where should the main components be manufac-tured?

· Where and how should the components be inte-grated?

· Where should the ®nal packaging be performed: atone of the factories or as a part of the distributionoperations?

· What kind of packaging is the best for this product?

The project consisted of three main parts: inventorymodeling, cost analysis and packaging design. In thispaper we will focus on the inventory modeling part of theproject.Our ®rst objective was to identify the feasible set of

product and process design alternatives to consider. Inorder to identify the feasible set of alternatives to study,we developed a Process Diagram and a Di�erentiationMap. The Process Diagram mapped processes in Geor-ge's supply chain. However, developing the process dia-gram for a product that had not been introducedpresented some challenges. Fortunately for us, all of thecomponents and processes to be used in George wereeither common with or were similar to those in existingproduct lines within the division. We used characteristicsof the similar or identical processes to construct theprocess diagram.The Di�erentiation Map depicted the course of prod-

uct proliferation along the Process Diagram within theGeorge family. At each point of di�erentiation, it high-lighted

· The source of di�erentiation ± product-driven orprocess-driven.

· The e�ect of the di�erentiation on product ± internalor external. (Our de®nition of di�erentiations in thisdimension is based on whether the e�ect of stan-dardization of parts, and/or processes, would beapparent to end users. For example, standardization

of integrated circuit chips across two models wouldnot be apparent to users, and therefore, di�erentchips result in an internal di�erentiation. However,standardization of the plastic housings across mod-els would be visible to users, and hence, the plastichousings are an external di�erentiation.)

· The number of new sub-families or products thatresult.

Classi®cation of di�erentiations into product- versusprocess-driven and internal versus external helped usnarrow the set of possibilities. For example, we foundthat changing processes for many process-driven di�er-entiations would not be cost-e�ective because it woulda�ect many other product lines. Similarly, it was relat-ively easy to standardize internal di�erentiations becausethey would allow us to retain the functionality acrossproduct lines without a�ecting the ``look and the feel'' ofthe product. External di�erentiations were usually cos-metic and were dictated by marketing. These and othertechnical considerations were used to determine the set ofscenarios we analyzed. Figures 1±3 depict simpli®edprocess diagrams for three of the scenarios.Board 1 and Board 2 are assembled along with other

parts into modules of type A. Type B modules are man-ufactured at a di�erent site. The Marry Station in Sce-narios 1 and 3 loads special software that enables type Aand type B modules to work together. However, by re-designing these modules, this operation can be elimina-

Fig. 1. Scenario 1 process diagram.

An application of designing products and processes for supply chain management 419

ted, so that in the ®nal product any module of type A canoperate with any module of type B. This is an option theLEM was interested in moving toward in the long term.The Accessory site represents the internal stockpile of

accessories required to support production at factories.These accessories include batteries, carrying cases, power

cords and adaptors, manuals, etc. Some accessories arecommon to each type of model belonging to George.These accessories are packed along with the type Amodule or type A-B modules' kit at the Generic Packingstation. The distribution center performs the ®nal pack-ing where model-speci®c accessories are included in thepackage.Di�erences among the three scenarios stem from dif-

ferences in the sequence of operations and in the numberof processes in each supply chain. The sequence in whichthe Generic Packing operation is performed is one causeof di�erences among the scenarios. A change in the se-quence of the Generic Packing operation entailed rede-sign of packaging. Changing the process sequence is apowerful means of e�ecting postponement, and reducingthe cost of work-in-process inventory [14]. Another dif-ference among the scenarios is the existence of a MarryOperation. A redesign of modules of types A and Bwould obviate this stage in Scenario 2.

4. The supply chain modeling and analysis tool

We ®rst give an overview of our model and then derivethe analytical expressions for various performance mea-sures. The limitations of the assumptions and approxi-mations we make in our model are also discussed in thissection.

4.1. Model overview

Production sites and distribution centers in a supplychain are modeled as nodes in SCMAT. As a convention,we de®ne all inputs into a node to be parts and all outputsfrom a node to be SKUs. Therefore at a node, parts fromupstream nodes and/or external vendors are transformedinto SKUs. These SKUs may then become parts at somedownstream nodes. In general, parts and SKUs ¯owingthrough a node are distinct from each other. However, ifa node represents a distribution center, parts and SKUsare identical because parts are not transformed by anyproduction activity at the node.SCMAT requires four main types of input data: end-

product demand and service requirements, Bill of Mater-ials, capacity availability at a node, and the SKU-nodedata (see Section 4.2 for more details). Bill of Materialsand SKU-node data de®ne the routing of parts and SKUsin the supply chain, and are used to determine the de-mand and service requirements for each part and SKU atevery node. We now outline the methodology used inSCMAT to obtain the performance characteristics of asupply chain (Fig. 4). SCMAT consists of two sub-models: the Inventory Network sub-model, and theQueueing Network sub-model. The Inventory Networksub-model is used to model the supply chain, while theQueueing Network sub-model computes some inputs



420 Garg

required by the Inventory Network sub-model. The basicidea of the solution methodology is to aggregate results ofmultiple single-site, single-item inventory models.The main output from SCMAT is the base-stock level

for each SKU ¯owing through every node. The base-stock level for a SKU at a node is a function of the meanand variance of its demand, the mean and variance of itsreplenishment lead time, and the service-level or the ®ll-rate requirement. Given the base-stock level and thevalues of its three drivers, one can compute various per-formance measures useful for obtaining managerial in-sights. These performance measures include the mean andvariance of on-hand inventory, the mean and variance ofthe backorder level, the mean and variance of the re-sponse time, and the capacity utilization at each node inthe network. The relationship among these three driversof the base-stock level, and computation of various per-formance measures are detailed in Section 4.3.2.The replenishment lead time for a SKU-node combi-

nation is a sum of the the material lead time, the lead timerequired to receive and store input parts, the delay inproduction of the SKU due to non-availability of parts,and the production lead time. Of the four components ofreplenishment lead time listed above, the material leadtime and the lead time required to receive and store inputparts are given as input data; the delay due to non-availability of parts is determined by the response time ofthe upstream node where the parts are stocked; and, the

production lead time is estimated using queueing ap-proximations that form a part of the Queueing Networksub-model (Section 4.4). Note that if the node representsa distribution center, there is no production activity;therefore, the replenishment lead time comprises theother three elements only. The response times of the up-stream nodes (or the delay due to non-availability ofparts) are a function of the ®ll rates at the upstream nodeswhere parts are stocked. Therefore, the replenishmentlead time is a function of the ®ll rates at upstream nodes.The base-stock level for a SKU-node combination

cannot be determined by solving a single-site inventoryproblem independently because the replenishment leadtime at a site is a function of the ®ll rates at upstream siteswhich stock the parts. As a result, the base-stock levelsfor all SKU-node combinations are obtained simultan-eously by solving a non-linear programming problem inwhich the objective is to minimize network-wide inven-tory holding costs subject to ®ll-rate constraints on theend-products. Unit holding costs of parts and SKUs ateach node were obtained from a cost analysis. Unit costsinclude material cost, labour cost, and overhead costs.SCMAT transforms the non-linear programming

problem described above to one in which the decisionvariables are the ®ll rates for all SKU-node combina-tions. This transformation can be derived by noting thatgiven the mean and variance of the demand, the base-stock level for a SKU-node combination is a function ofits ®ll-rate requirements, and its replenishment lead time.And, the replenishment lead time for the SKU-nodecombination is in turn a function of the ®ll rates at up-stream sites that stock parts used to manufacture theSKU. This non-linear program is similar to that solvedby Deuermeyer and Schwarz [15] for optimizing ®ll rates.Since the number of SKU-node combinations can bequite large, one would need a considerable amount ofcomputation time to solve this optimization problem. Inorder to reduce the computation time, the search algo-rithm in SCMAT determines the vector of ®ll rates for allSKU-node combinations by searching over a grid ofdiscrete values of ®ll rates for each SKU-node combina-tion. See Gill et al. [16] for more details. Therefore, thedecision-maker would need to trade-o� the granularity ofthe grid with the computation time. A ®ner grid wouldresult in a better solution while requiring greater com-putation time.

4.2. Data speci®cations

We ®rst de®ne the notation and conventions used in ourmodel. Let indices k and g denote sites, and i and j partsand SKUs.

Site dataCk = Capacity in hours per week at site k.qk = Capacity utilization at site k.

Fig. 4. Outline of the solution methodology in SCMAT.


MTBFk = Mean time between failures in weeks atsite k.

ld�k�= Average duration of down time in weeksat site k.

md�k�= Variance of down time in weeks2 at site k

Part dataFi = Target ®ll rate for SKU i.Ai = Material ®ll rate for part i.Ri = Review period in weeks for SKU i.si = Target response time in weeks for SKU i.bi = E�ective response time in weeks for

SKU i.lM �i�= Average material lead time in weeks for

part i.mM �i�= Variance of the material lead time in

weeks for part i.lD�i�= Average delay due to non-availability of

part i in weeks given that the part is out ofstock.

mD�i�= Variance of delay due to non-availabilityof part i in weeks2, given that the part isout of stock.

Bill of materials data

dij =1 if part j is used in SKU i;0 otherwise.

�nij = Number of parts i needed to produce one

unit of SKU j.

Part-site datal�i; k�= Average demand per week in units for

SKU i at site k.m�i; k�= Variance of the demand per week in units

for SKU i at site k.lS�i; k�= Average material storage and receiving

lead time in weeks for parts required toproduce SKU i at site k.

mS�i; k�= Variance of material storage and receivinglead time in weeks2 for parts required toproduce SKU i at site k.

lPL�i; k�= Average production lot size in units forSKU i at site k.

mPL�i; k�= Variance of production lot size in units2

for SKU i at site k.lFLOW�i; k�= Average ¯ow time in weeks for SKU i at

site k.mFLOW�i; k�= Variance of ¯ow time in weeks2 for SKU i

at site k.lTL�i; k�= Average transfer lot size in units for SKU i

at site k to its downstream sites.mTL�i; k�= Variance of transfer lot size in units2 for

SKU i at site k to its downstream sites.lP �i; k�= Average production lead time in weeks for

SKU i at site k.

mP �i; k�= Variance of the production lead time inweeks2 for SKU i at site k.

lR�i; k�= Average replenishment lead time in weeksfor SKU i at site k.

mR�i; k�= Variance of the replenishment lead time inweeks2 for SKU i at site k.

S�i; k�= Order-up-to point for SKU i at site k.I�i; k�= Average inventory level of SKU i at site k.

K�i; k�= Safety stock factor for SKU i at site k.lr�i; k�= Average response time in weeks to de-

mands of SKU i at site k, given that i isbackordered.

mr�i; k�= Variance of the response time in weeks2 todemands of SKU i at site k, given that i isbackordered.

Transit time datalT �k; g�= Average transit time in weeks from site k

to site g.mT �k; g�= Variance of transit time in weeks2 from

site k to site g.

4.3. Inventory network sub-model

4.3.1. Demand and service transmission

Nodes in SCMAT are connected by two types of ¯ows:the ¯ow of information and the ¯ow of physical items(Fig. 5). There are two types of information ¯ows:Demand Transmission, and Service Transmission. Thedirection of information ¯ows is from the downstreamend-product stages to the upstream raw material stages.Of course, the physical ¯ows originate at the upstreamraw material stages and terminate at the downstream end-product stages.The Demand Transmission process enables us to de-

termine the ®rst two moments of the demands of everySKU-node combination in the network. Demands forSKUs at a node are translated into demands for eachpart, and hence SKUs at upstream nodes in the system,through the Bill of Materials. SCMAT assumes that thedemand for each end-product is stationary, uncorrelated,and a normally distributed random variable. Like Graves[8] and LB, we also assume that the Demand Transmis-sion process causes the demand of each SKU at every

Fig. 5. Schematic of ¯ows in SCMAT.

422 Garg

node to be stationary, uncorrelated, and normally dis-tributed. However, this is an approximation because end-products may share common components, resulting inpositively correlated demands for the common compo-nents. Yao [17] has simulated such assembly systems andfound the e�ect of correlations on various performancemeasures of inventory systems such as lead times, back-order levels, etc., to be insigni®cant when service-levelrequirements on the end-products are high. In fact, thehigh service-level requirement is also necessary in derivingEquations (6) and (7).The impact of the Demand Transmission process can

be represented mathematically as follows. De®ne D�k; i�to be the set of downstream sites to which site k suppliespart i, and pj�g; k� be the proportion of the requirement ofpart j at downstream site k that is sourced from site g, dijto be the indicator variable that equals 1 if part j is used inSKU i, and is 0 otherwise, and nij is the number of parts oftype j required to manufacture one unit of SKU i. Then,

l�i; g� �X

k2D�g;i�

Xj

dijnijl�j�pi�g; k�; �1�

m�i; g� �X

k2D�g;i�

Xj

dijn2ijm�j�p2i �g; k�; �2�

where g is a site where i is an SKU.The decision maker speci®es service-level requirements

for each end-product. The availability of a SKU at a sitedepends upon response times and ®ll rates of its inputparts from the upstream sites. Therefore, service-levelrequirements on end-products drive the service-level re-quirements for every input part, and hence for each SKU-node combination. We call this the Service Transmissionprocess.In SCMAT service-level requirement is de®ned as a

combination of the SKU ®ll rate and its target responsetime. For example, a service-level requirement speci®ed asa ®ll rate of 95% with a target response time of 5 daysimplies that more than 95% of demands should be satis-®ed within 5 days of observing them. If the target re-sponse time is set to 0, then our service-level requirementreverts to the traditional o�-the-shelf ®ll-rate metric.Therefore, the speci®cations of service-level requirementsin SCMAT are more general than the traditional de®ni-tions of service-level requirements. Since service-level isspeci®ed as a combination of the target response time and®ll rate, we can see that the Service Transmission processcomprises Fill-Rate Transmission, and Response TimeTransmission. Note that Response Time transmission willa�ect upstream part-level inventories only if a SKU'sresponse time requirement is greater than its replenish-ment lead time.The expression for response time requirements at part

level is a result of Response Time transmission. If re-sponse time required of an SKU is greater than its re-plenishment lead time, then the SKU is make-to-order.

Under this condition, the upstream site supplying part jcould delay the start of its production by �bi ÿ lR�i; k��.In general, one could delay the start of production by�bi ÿ lR�i; k� ÿ j

��mR�i; k�

p ��, where j � 0 is a safety or a``fudge'' factor. SCMAT assumes j � 0, however, wecould be more conservative and set j > 0. I would like tothank one of the referees for suggesting this. However, ifpart j is used by multiple SKUs, maximum time by whichthe start of its production can be delayed ismin�bi ÿ lR�i; k��, i 2Kj, where Kj is the set of SKUsthat use part j. Independent of downstream responsetimes, users may also set a target response time on part jitself, sj. Therefore, response time requirements for part jcan be written as

bj � min sj;mini2Kj

bi ÿ lR�i; k��

: �3�

4.3.2. Determination of replenishment lead time andperformance measures

The single-site inventory model is the basic building blockused to construct the Inventory Network sub-model. Inthis section we present expressions for computing two ofthe components of replenishment lead time, and the base-stock level in a single-site inventory model.The material lead time, a component of the replenish-

ment lead time, is the time to procure parts if they are tobe sourced from an external supplier. If the part ismanufactured internally, then the material lead time isthe lead time to transfer the part from its stockpile up-stream in the supply chain.Since lead times are also driven by lot sizes [18] the

material lead time is expressed as the sum of the transittime between sites k and g and the waiting time resultingfrom the time required to manufacture a transfer lot. Thiswaiting time depends on two random variables: thetransfer lot size for part i and its manufacturing ¯owtime. We assume that the transit time, the transfer lotsize, and the manufacturing ¯ow time are independent ofone another.Mean and variance of the waiting time can be derived

as follows. Let X , Y , and Z denote the waiting time, lotsize (with EY � lTL and Var Y � mTL), and manufacturing¯ow time (with EZ � lFLOW and Var Z � mFLOW) ran-dom variables respectively. The expression for E X j Y ; Z� �can be obtained by noting that the ®rst job in the lotwill wait for YZ time units, the second job �Y ÿ 1�Z timeunits, and so on. Therefore, the average wait givenunit processing time and given the lot size isE X j Y ;Z� � � Z�Y � 1�=2, and EX � EYEZE X j Y ; Z� �yielding Equation (4).

lM �i� � lT �k; g� � 12lFLOW�i; k� lTL�i; k� � 1� �: �4�

We apply the conditional variance formulas given belowrepeatedly to derive an expression for the variance of the


waiting time. Var X� � � E Var X j Y� �� Var E X j Y� �� and Var X j Y� � � EZ Var X j Y ; Z� �� VarZ E X j Y ;�� Z��.Using an argument similar to the case for E X j Y ; Z� �,we get E X 2 j Y ;Z� � � Z2Y �Y � 1��2Y � 1�= 6Y� �, andVar X j Y ; Z� � � Z2�Y 2 ÿ 1�=12. Note that VarZ E�X j Y ;�Z�� mFLOW�i; k��Y � 1�2=4, EZ Var�X j Y ; Z�� l2FLOW�Y 2 ÿ 1�=12, E�X j Y � � lFLOW�Y � 1�=2,

E Var�X j Y �� mFLOW�lTL � 1�2=4� l2FLOW�l2

TL ÿ 1�=12,and Var E�X j Y �� lFLOWmTL=2. Therefore,

mM �i� � mT �k; g� � l2FLOW�i; k�

12l2

TL�i; k� ÿ 1ÿ �� mFLOW�i; k�

4

� lTL�i; k� � 1� �2� l2FLOW�i; k�mTL�i; k�

4: �5�

Mean and variance of the production lead time are ob-tained from the Queueing Network sub-model (Section4.4). A problem related to computing delay in produc-tion due to non-availability of parts has been studied byYano [19] and Hopp and Spearman [20]. Exact expres-sions for modeling the replenishment process of SKUsthat require multiple parts with di�erent lead times re-sults in a complex non-linear programming problem. Wecompute the e�ect of delay due to the non-availability ofparts with the help of analytical approximations for thewaiting times.The delay, lD�j�, is computed as the expected waiting

time given that the item is backordered. The probabilityof a delay due to the non-availability of part j is �1ÿ Aj�and the expected delay is �1ÿ Aj�lD�j�. This assumptionis valid if the ®ll rates, Ajs, are close to 1.Given expressions for the ®rst two moments of each

component of the replenishment lead time, we can nowderive the ®rst two moments of the replenishment leadtime. We assume that the components of the replenish-ment lead time are independent of one another, and thatat most one part is delayed. The latter assumption isreasonable when service-level requirements are very high.

lR�i; k� �P

j dijlM�j�Pj dij

�X

j

dij 1ÿ Ajÿ �

lD�j�

� lP �i; k�; �6�mR�i; k� �max

jfdijmM�j�g �

Xj

dij�1ÿ Aj�mD�j�

�X

j

dijAj�1ÿ Aj�l2D�j� � mP �i�: �7�

We now want to give expressions for the operatingcharacteristics of a single-site system when a target serviceperformance is speci®ed for the SKU. De®ne lLR�i; k� tobe the demand of SKU i during the replenishment leadtime and review period, lLR�i; k� � l�i; k��lR�i; k� � Ri�,mLR�i; k; s� � �lR�i; k� ÿ s� Ri�m�i� � l�i�2mR�i; k� to bethe variance of demand during the residual lead time,where s is a target response time, therefore, (lR�i; k�ÿs� Ri) is the residual lead time. We omit subscripts i and

k from all variables to simplify the exposition in thissection. There are two possible cases:

Case 1: Make-to-Order (F � 0 or b � lR � R)Expressions for the base-stock level, the average inven-tory, etc. are given below.

S � 0;

I � ÿ lLR ÿ lR=2;

lr � lR � R=2;

mr � mR:

If this SKU is used as a part at a downstream site, themean and variance of the delay due to its non-availabilityare lD � lr and mD � mr respectively.

Case 2: Make-to-Stock (F � 0 and b < lR � R)In this case, since the response time is b weeks, the safety-stock necessary to provide the required service level needsto protect against demands over lR � Rÿ b weeks. Theprobability that the waiting time for a customer order in aperiod is greater than j review periods is the probabilitythat the demand in lR � Rÿ bÿ jR weeks is greater thanor equal to S. Demands are assumed to be normallydistributed; therefore, we have

1ÿ F � 1� exp2��2=p

pS ÿ l lR � Rÿ b� ��

mLR�b�p" #( )ÿ1

:

�8�Equation (8) can be used to derive expressions for thebase-stock level (Equation (9)) and the safety-stock factor(Equation (10)).

S � lLR ÿ bl� K��mLR�b�

p; �9�

where

K � 1

2

��p=2

pln

F1ÿ F

� �: �10�

The average on-hand inventory of SKU is given by

I � lR2� K

��mLR�b�

p: �11�

And for j 2 f0; . . . ;Ng de®ne

fj � 1� exp2��2=p

pS ÿ lLR � bl� ljR� ��

mLR�b� jR�p" #( )ÿ1; �12�

where the probability of the waiting time for an order ofSKU i at site k is greater than jÿ 1=2 review periods is�1ÿ fj�, and N � bmLR�0�=�Rm�c. One can show that themean and variance of the response time given that theSKU is out of stock is

lr �Rf0

XN

m�0fm ÿ R

2; �13�

424 Garg

mr � R2

4f0

XN

m�12m� 1� �fm ÿ l2

r : �14�

In this case, if the SKU is used as a part at a downstreamsite the mean and variance of the delay due its non-availability can be expressed as lD � lr and mD � mrrespectively.

4.4. Queueing network sub-model

The production lead time at each site is the only lead timethat explicitly re¯ects the interactions between di�erenttypes of SKUs ¯owing through a node. Like the transitlead times, this lead time is also driven by the productionlot sizes of each SKU. Other factors that a�ect this leadtime are capacity constraints, breakdown frequency andits duration, and variance of the input process. Varianceof the output process at a site is higher than that of itsinput process because of the e�ects of variability in theprocessing time and machine breakdowns. Since theoutput process of one node is the input process into adownstream node, its variability has greater impact onthe production lead times downstream.We use a node decomposition technique similar to that

implemented in QNA [12, 21] to compute the productionlead time at a node. QNA's decomposition techniqueextends the product-form results of Jackson-type net-works to more general systems. Besides assuming nodesto be stochastically independent, this approach also as-sumes that the two-moment approximations providereasonably good results.Inputs required for the Queueing Network sub-model

are similar to those required for the Inventory Networksub-model. This sub-model's main inputs include themeanand variance of the demand rates of all end-products, thebill of materials, part routing information, the productioncapacity at each node, and the mean and variance of theservice times at each node. The main outputs from thissub-model are the capacity utilization at each node, andthe mean and variance of the time spent at a node by eachSKU. Time spent by a SKU at a node is the productionlead time for that SKU. The mean and variance of thisproduction lead time are input into the InventoryNetworksub-model where they are used to compute the mean andvariance of the replenishment lead time.Because of the di�erent SKUs ¯owing through a node

we model the supply chain as a multi-class queueingnetwork with deterministic routings, where each SKU ata node represents a class. A modeling arti®ce for ana-lyzing the performance of multi-class queues is to focuson the three processes at each node.

· Superposition or merging.· Flow through a queue.· Splitting or decomposition.

Since each node has multiple classes of job arrivals, asuperposition process is used to determine the ®rst twomoments of an aggregate job based on all classes of jobsarriving at that node. Flow through a queue process thendetermines the performance characteristics of the nodebased on the aggregate job resulting from the superpo-sition process. However, we need to determine the per-formance characteristics of each class of jobs ¯owingthrough the node. As a result, we need the splitting or thedecomposition process to determine class-based perfor-mance metrics at the node.

4.4.1. Superposition process

Each SKU j processed at a node k is a job of a di�erentclass at that node. The arrival rate for each class at thenode is kjk with a squared coe�cient of variation denotedby cajk. The arrival rate, kjk, is equal to the mean demandrate for SKU j at site k because the system is assumed tobe stable. In this section, since we focus on a single node,we shall drop the subscript k for nodes from all notationto simplify the exposition.De®ne C to be the set of job classes processed at a

node, and

a �Xj2C

kj; �15�

ca �Xj2C

kj

acaj: �16�

The expression for the mean arrival rate of the aggregatejob is exact; however, there are no exact expressions forthe squared coe�cient of arrivals for the aggregate job.Equation (16) has been obtained by the asymptoticmethod. It can also be obtained via the stationary-inter-val method. It has been found that a considerable im-provement in accuracy is attained by using a convexcombination of approximations from the two methods.This modi®ed approximation is

ca0 � wca� 1ÿ w; �17�where w � �1� 4�1ÿ q�2�vÿ 1��ÿ1, vÿ1 �Pj k2j=a

2, andq is the utilization at the node.

4.4.2. Flow through the node

In this section, we ®rst evaluate the mean and variance ofprocessing time for an aggregate job ¯owing through thenode without the impact of breakdowns. The mean andvariance of the e�ective service time for the aggregate jobis then calculated by incorporating the impact of machinebreakdowns. Finally, the mean and variance of the timespent in the queue by the aggregate job are computedusing expressions similar to those in QNA.De®ne lFLOW�i�, mFLOW�i�, and cgi as the mean, vari-

ance, and the squared coe�cient of variation respectivelyof the processing time for a class i job. Therefore, we have


lFLOW �X

j

lFLOW�j�kj

a; �18�

cg �X

j

cgjkj

a; �19�

where lFLOW, mFLOW, and cg are the mean, variance, andsquared coe�cient of variation respectively of the ag-gregate job.In order to incorporate the e�ect of machine break-

downs we have assumed them to occur as a Poissonprocess. This is quite reasonable since we have found thatbreakdowns are a relatively rare event at the electronicmanufacturer's factories. Further, we assume break-downs to occur only when the machines are busy, and jobprocessing resumes from the same point after an inter-ruption in service. We also assume that downtimes are iid,and are independent of the type of job being processed.De®ne B to be the number of breakdowns that occur

during the processing of a job, and d to be the repair timeduring each breakdown, where Ed � ld and Var�d� � md .We can see that B is a stopping time for repair times,therefore, we can apply Wald's identity to determine theaverage total downtime during processing of a job, lDown,and its variance mDown. We have assumed thatB � Poi�lFLOW=MTBF �. Therefore, mean and variance ofdowntime are given by

lDown � EBld ; �20�mDown � md�EB�2 � EBl2

d : �21�Let cs denote the squared coe�cient of variation of thee�ective service time. The mean and variance of the ef-fective service time, lservice and mservice respectively, are

lservice � lFLOW � lDown; �22�mservice � mFLOW � mDown: �23�

We now need to compute the mean and variance of thewaiting time in the queue. Let W denote the waiting timein the queue. Then,

EW � lFLOWq�ca� cs�g�q; ca0; cs�2�1ÿ q� ; �24�

where the function g�q; ca0; cs� is based on an approxi-mation due to Kraemer and Langenbach-Belz [22] and isgiven by

g�q; ca0; cs� � exp ÿ 2�1ÿ q�3q

�1ÿ ca0�2ca0 � cs

" #ca0 < 1;

1 ca0 � 1:

8><>:In order to compute the variance of the waiting time inthe queue, de®ne D to be the conditional delay given theserver is busy, and cD the squared coe�cient of variationof D. As in QNA, the expression for cD is based on thatfor an M/G/1 queue instead of a GI/G/1 queue because

the conditional delay, D, depends on the service timedistribution and not on the inter-arrival time distribution.

cD � 2qÿ 1� 4�1ÿ q�d3s

3�1� cs�2 ; �25�

where d3s is the ratio of the third moments of the pro-

cessing time variable, i.e., d3s � E�s3�=�Es�3, where s is the

random variable denoting the e�ective service time. Sincewe are working with two-moment approximations only,we need to approximate d3

s .

d3s �

3cs�1� cs� cs � 1;�2cs� 1��cs� 1� cs < 1:

�The squared coe�cient of variation of waiting time in thequeue, cW , can be approximated by

cW � cD� 1ÿ rr

�26�where r � P�W > 0� � q� q�1ÿ q��ca0 ÿ 1�h�q; ca0; cs�and

h�q; ca0; cs� �1� ca0 � qcs

1� q�csÿ 1� � q2�4ca0 � cs� ca � 1

4qca0 � q2�4ca0 � cs� ca > 1

8>><>>:Now Var�W � � �EW �2cW ; therefore, we can now obtainexpressions for the time spent in system, Q, at the node.

EQ � EW � lservice; �27�Var�Q� � Var�W � � mservice: �28�

We have used results of Yao et al. [23] to incorporate thee�ects of lot sizes on production lead times.

4.4.3. Decomposition

De®ne cdj to be the squared coe�cient of variation of thedeparture process of job class j, cd to be the squaredcoe�cient of variation of the aggregate job, andpj � kj=a. The method used in QNA to split the outputstream from a node is equivalent to assuming that therouting is Markovian. However, this is an approximationbecause QNA uses deterministic routings. The resultingapproximation can be written as

cdj � pjcd � 1ÿ pj: �29�Bitran and Tirupati [11] derive several heuristics whichimprove upon the results of QNA. Bitran and Tirupati'sheuristics perform better than that in QNA because theyhave explicitly considered interactions between di�erentclasses of jobs ¯owing through a node. They express thesquared coe�cient of the output process as

cdj � pjcd � cnj: �30�The ®rst term in Equation (30) re¯ects the e�ect of thequeueing process. The second term on the right hand sideof Equation (30) is independent of the service process and

426 Garg

it captures the e�ect of the distribution of arrivals ofaggregate jobs between two arrivals of jobs of class j.Bitran and Tirupati developed three approximations forcnj. We use their Poisson approximation for cnj, which isless computationally intensive while giving relativelygood estimates of the departure-process squared coe�-cient of variation.The Poisson approximation is motivated by the fact

that superposition of a large number of independent re-newal processes can be approximated by a Poisson pro-cess. Assuming that the arrival process of each job class isan independent renewal process, then the arrival processof the aggregate job (composed of all classes except classj), can be assumed to follow a Poisson process. Bitranand Tirupati use this property to derive the probabilitydistribution of the arrival process of this aggregate jobclass. Given the probability distribution, one can thenderive expression for cnj. Due to a lack of space, we referthe reader to Bitran and Tirupati [11] for details of thesederivations and present the ®nal expression for cdj here.

cdj � pjcd � �1ÿ pj� pj � �1ÿ pj�caj� �

: �31�The Poisson approximation performs well when pj issmall, or when there are a large number of products, orwhen the arrivals of each job class are close to beingPoisson. Equation (31) implies that cdj � caj. cdj � cajwill be equal only if the arrivals are Poisson, i.e., caj � 1.The squared coe�cient of variation of the arrival process,caj, is a departure process from an upstream node, and isa result of the approximation in Equation (31). Therefore,we can see that variability increases progressively as onegoes downstream.

5. Results

5.1. Model validation

We validated the model in two phases. The ®rst phaseincluded program debugging and veri®cation of compu-tations and logic through manual calculations. The sec-ond phase of validation involved checking the results ofthe model with current operating conditions as input andthrough frequent interactions with users. We used thisphase to verify that the results of our model correspondedwith the users' expectations. This phase also includedverifying the accuracy of input data.

5.2. Evaluation of scenarios

We evaluated the three scenarios assuming 12 end-prod-ucts in each case. The bills of materials for these end-products di�ered due to variations in the type of eachmodule used and in the number of modules of type B inthe ®nal product. We considered three di�erent type Amodules, and six di�erent type B modules, one type each

of Boards 1 and 2. The number of sites considered in eachscenario is as depicted in Figs. 1±3. Our model took ap-proximately 60 seconds to produce results on an AppleMacintosh computer.In reality, the end-products in our analyses, repre-

sented sub-families within this line of products.End-products within each sub-family would result fromexternal di�erentiations in modules of types A and B, andthe accessories. These external di�erentiations, whichwere very di�cult to predict while the product family wasstill at a design stage, would manifest themselves as dif-ferent types of housings, displays, etc. The main impact ofsuch di�erentiations would be in the procurement ofparts, which could conceivably a�ect material lead timesat a node. Di�erences among end-products within a sub-family would not a�ect production lead times at any ofthe stages in the supply chain. However, the details of thebreadth of products within each sub-family were sketchyat best during the design stage. Another reason for con-ducting our analyses at the sub-family level was the trade-o� between the level of detail in the model and the timerequired to collect accurate input data and perform theanalysis.Figures 6 and 7 present a comparison of the three

scenarios. Figure 6 compares the contribution each stageof the supply chain makes to the total inventory cost inthe three scenarios. We can see that Scenario 2 has thelowest inventory holding cost. A major cause of lowerinventories in Scenario 2 is the absence of the MarryStation. Scenario 1 has a lower total inventory holdingcost than Scenario 3, however, this di�erence is insignif-icant.If we examine the contribution of each stage in the

supply chain, we can see that the cost of Boards 1 and 2,modules of types A and B, and the Accessories is the sameacross the three scenarios. Di�erences among the sce-narios occur because of the presence (or the absence) ofthe Marry Station, and the sequence of Generic Packingprocess in the supply chain. A change in the sequence ofprocesses causes a change in the rate at which unit costsincrease down the supply chain. As a result, inventorycost at Marry Station in Scenario 1 is lower than that in

Fig. 6. Inventory costs across the scenarios.


Scenario 3. Di�erences in the material lead times at theFinal Packing stage contribute to the di�erences in theinventory costs among the scenarios. In fact, di�erencesin material lead times at the Final Packing stage com-pensate for (to some extent) the impact of di�erences inthe rate at which unit costs increase across the threescenarios.Figure 7 compares the e�ects of service requirements

on inventory costs of the three scenarios. The indepen-dent variable in this plot is �1ÿ fillrate�, an approximatemeasure of the probability of shortage. We can see thatinventory costs grow exponentially as the ®ll rate in-creases, and that Scenario 2 dominates the other twoscenarios at all service levels. Although percentage sav-ings across the scenarios do not appear to be very high,these percentages represent signi®cant dollar amounts.

6. Conclusions and future research

This paper describes SCMAT, a tool that can be used tostudy various issues in supply chain management. Weapplied SCMAT in designing products and processes at alarge electronics manufacturer. Our project comprisedthree main parts: inventory modeling, cost analysis, andpackaging design. However, this paper focused only oninventory implications of the project.We used SCMAT to analyze the inventory implications

of three product and process design scenarios. Thesescenarios di�ered in the number of stages in their supplychain and in the sequence of some of the processes.Process sequence a�ects the rate at which unit costs ofproducts increase down the supply chain, which in turnimpact total inventory costs. This result highlights thedi�erences in the inventory costs due to the GenericPacking stage in Scenarios 1 and 3. However, in our case,the net di�erence in inventory costs between the twoscenarios is insigni®cant because di�erences in lead timesat the Final Packing stage compensate for the impact ofdi�erences in unit costs.SCMAT extends previous work in this area by incor-

porating queueing approximations to model the conges-

tion e�ects at each site. Like the previous models, oursmakes several approximations in order to keep the modelanalytically tractable. Some of these approximationscould be tested more thoroughly and be improved uponwith the help of simulation. This is an area of futureresearch.The application of such techniques at the LEM resulted

in great bene®ts, both tangible and intangible. Tangiblebene®ts included a tool that can be used to analyze theimpact of various decisions on costs in the supply chain.Among the intangible bene®ts were an increased aware-ness of the implications of product and process design onthe entire supply chain. Engineers and managers involvedin the project realized that product is de®ned not by whatis shipped out of the factory but by what is received bythe end-customer. The design team also came to under-stand the important role played by packaging. This wasthe ®rst time that packaging design began so early in theproduct and manufacturing process design cycle. Previ-ously, packaging design was undertaken after manufac-turing test runs started.SCMAT provided senior managers with a means of

making more informed decisions by enabling them toperform what-if analyses and obtain insights into theproblem. We found Scenario 2 to yield the lowest in-ventory costs, and this was also the scenario managementwanted to move toward in the long-term. However, ingeneral the ®nal decision, based on a total cost analysiswhich includes the costs of product and process redesign,could be di�erent from the outcome favored by an in-ventory analysis.

Acknowledgments

This paper is based on a chapter of my dissertation.I would like to thank my advisor, Professor Hau Lee ofStanford University, for his support and comments onthis work. I would also like to thank engineers in theproduct design team and in the corporate manufacturingdepartment of the LEM for their help and their supportin this work. Finally, I would like to thank the depart-mental editor and the anonymous referees of this paperfor their comments. These comments have helped im-prove the paper considerably.

References

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Fig. 7. Comparison of the e�ect of service requirements oninventory levels.

428 Garg

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Biography

Amit Garg is the Vice President and Chief Scientist at Nonstop Solu-tions, in San Francisco, CA. Prior to joining Nonstop, he was aResearch Sta� Member at the IBM T.J. Watson Research Center inYorktown Heights, New York. His research interests lie in the areas ofsupply chain management, product and process design, and stochasticinventory systems. He is currently working on supply chain issues inthe retail and retail distribution industry. Before enrolling in the Ph.D.program, he worked as a consultant at Bender Management Consul-tants, Arlington, Virginia, where he focused in the areas of distributionand scheduling for clients in the process industry. He has a Ph.D. inIndustrial Engineering from Stanford University, an M.S. from SUNYat Bu�alo, and a B.Tech. in Mechanical Engineering from the IndianInstitute of Technology-Kharagpur.


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An application of designing products and processes for supply chain management