How To Know The Cost Of Delaed Dieentiation

Transcription

1 ae-to-ode, ae-to-stoc, o Dela Poduct Dieentiation? A Common Famewo o odeling and Analsis Diwaa Gupta Saiallah Benjaaa Univesit o innesota Depatment o echanical Engineeing inneapolis, N Second evision, Octobe 000 Abstact Widespead incease in poduct vaiet and the simultaneous emphasis on shote ode-delive times, and lowe costs, is ocing manuactuing ims to conside altenatives to both mae-tostoc and mae-to-ode modes o poduction. One such altenative is delaed poduct dieentiation. It is a hbid stateg that econciles the needs o high vaiet and quic esponse. A common poduct platom is built to stoc and then dieentiated into dieent poducts once demand is nown. In this pape, we examine the costs and beneits o delaed dieentiation in envionments whee capacit is inite, and ode lead-times ae stochastic and load-dependent. We show that the advantage o delaed dieentiation is aected b actos such as capacit utilization, poduct vaiet, and ode dela equiement. In paticula, the desiabilit o delaed dieentiation is ound to be sensitive to the utilization o the poduction sstem; its beneits diminishing with inceasing utilization. In situations whee ode-dela equiements ae tight, we show that delaing dieentiation can be moe expensive than poducing in a mae-to-stoc ashion. Howeve, when utilization is in the mid-ange, the beneits o delaed dieentiation can be signiicant. As intuition suggests, the beneits o delaed dieentiation ae also signiicant when poduct vaiet is high. In act, the maginal beneit om delaed dieentiation is, b and lage, inceasing in the numbe o poducts. I the point o dieentiation can be aected, we ind that, whee easible, dieentiating eal b postponing moe wo content until demand is ealized is desiable, leading ultimatel to a mae-to-ode sstem. I the cheape eal dieentiation is not possible due to tight delive equiements, we show that we should assign popotionall moe capacit to the mae-to-ode stage. Coesponding autho. Telephone: , Fax: , [email protected].

2 . Intoduction In toda's business envionment, a manuactuing im that has the abilit to ulill custome odes quicl, as well as oe a lage vaiet o poducts, enjos a competitive advantage. Howeve, the objectives o high poduct-vaiet and quic-esponse time ae oten conlicting [8, 5, 7]. Taditionall, industies whee quic-esponse time is impotant have ocused on poducing a limited potolio o poducts, which ae stoced ahead o demand and shipped immediatel upon eceipt o a custome ode. Poducing to stoc becomes costl and unpactical when the numbe o poducts is high. It is also is when demand is stochastic o negativel coelated among poducts. Taditionall, a signiicant incease in poduct vaiet has meant shiting om a mae-to-stoc to a mae-to-ode mode o poduction, whee poduction is not initiated until a custome ode is eceived. While this stateg eliminates inished inventoies and educes im s exposue to inancial is, it usuall spells long custome lead times and lage ode baclogs. An altenative to both mae-to-ode and mae-to-stoc that has ecentl gained in populait is delaed dieentiation [7, 6, ]. Delaed dieentiation is a hbid stateg whee a common poduct platom is built to stoc. This common platom is dieentiated, b assigning to it cetain poduct-speciic eatues and components, onl ate demand is ealized. Hence, manuactuing occus in two stages, a mae-to-stoc stage whee the single undieentiated platom is poduced and stoced, and a mae-to-ode stage whee poduct dieentiation taes place in esponse to speciic custome odes. Delaed dieentiation caies seveal beneits. aintaining stocs o semi-inished goods educes ode ulillment dela elative to a pue mae-to-ode sstem. It also educes is associated with holding inished goods invento o which demand might not mateialize. This is-pooling has long been nown in the invento liteatue to lowe oveall inventoies while still guaanteeing the same sevice level [6]. B holding semi-inished instead o inished-goods inventoies, delaed dieentiation also educes unit-holding costs, since less value is associated with the stoced invento. Thee is also the beneit o leaning, ealized om having bette demand inomation beoe committing geneic semi-inished poducts to unique end-poducts. Leaning can aise when demand in consecutive peiods is coelated []. Additional beneits om delaed dieentiation include a signiicant steamlining o the mae-to-stoc segment o the manuactuing pocess and simpliication o poduction scheduling, sequencing, and aw mateial puchasing. Howeve, implementing delaed dieentiation also caies costs. These

3 include the costs o poduct and pocess edesign necessa to incease poduct commonalit, use o exta mateials when common designs ae made possible b having edundant o moe expensive pats, and less eicient pocessing when common pocessing leads to the use o a less specialized poduction equipment o geate ield losses. Theeoe, in assessing the value o a delaed dieentiation stateg, these costs have to be caeull balanced against the associated beneits. Recent liteatue showcases seveal examples, in industies anging om consume electonics to appaels to automotive manuactuing, whee delaed dieentiation has been successull used to contol invento costs while maintaining high sevice levels [4, 7, 8, 0, ]. Fo example, Feitzinge and Lee [7] descibe how delaed dieentiation, though eithe opeation evesal o component shaing, helped the Hewlett-Pacad pinte division customize its poducts moe cost eectivel. Swaminthan and Tau [] descibe how IB exploited component commonalities in pesonal computes to design a common platom, o a vanilla box, om which end-poducts ae dieentiated based on custome odes. Fishe et al. [8] discuss how component shaing is used b seveal majo automotive companies to standadize baing sstems. Buce [4] epots on the well nown case o Benetton who, b evesing the ode in which an is ded and nitted, was able to successull dela colo selection until the season s ashion peeence become moe established. Finall, Gaman and agazine [0] descibe how delaed pacaging is used b a manuactue o household cleaning poducts to educe its inished goods invento and impove sevice levels. Seveal o these studies also pesent quantitative models that assess the costs and beneits o delaed dieentiation see, o example, Lee [5], Lee and Tang [7], Gag and Tang [9], Swaminathan and Tau [0, ], and Aviv and Fedeguen [], among othes. ost o these models ocus on the tadeo between the beneits o invento pooling via statistical economies o om having a shaed invento location, leaning beneits, and poduct/pocess edesign costs. The majoit use ode-up-to-level invento models in which ode lead times ae not aected b eithe ode size, the numbe o pending odes, o the numbe o end-poducts. I limited poduction capacit is modeled, ode lead times ae assumed constant which ignoes an congestion at the poduction acilit. In doing so, these models ignoe the inteaction between utilization o the poduction acilit, pocessing time vaiabilit, and ode delas. In this pape, we extend the existing liteatue b explicitl modeling the eect o congestion at both the mae-to-stoc and mae-to-ode stages o the poduction pocess. We show that 3

4 thee is a complex elationship between capacit utilization, ode dela, optimal invento levels, and the degee to which dieentiation should be delaed. In paticula, we show that the beneits o delaed dieentiation ae sensitive to the utilization levels at the poduction acilit with delaed dieentiation having little value when utilization is high. Similal, we show that delaed dieentiation caies little value when ode dela equiements ae tight. Howeve, in some othe cases, the beneits o delaed dieentiation can be signiicant. Notabl, this happens when utilization is in the mid-ange, o poduct vaiet is high. In act, the maginal beneit om delaed dieentiation is geneall inceasing with the numbe o poducts. I thee is lexibilit in choosing the point o dieentiation, it is clea that a im should t to move as much towad a mae-to-ode sstem as possible. Howeve, the degee to which a im should postpone wo until ate demand is nown is lagel diven b the tightness o the ode-dela equiement. In cases, whee thee is lexibilit in assigning capacit, we ind that assigning popotionall moe capacit to the mae-to-ode stage tends to be moe desiable. Thoughout the analsis, we use ou models to answe seveal undamental questions that aise when implementing delaed dieentiation DD. Fo example, unde what conditions is DD supeio to a pue mae-to-stoc stateg? How is the value o DD aected b sstem paametes such as capacit utilization and numbe o poducts? How should poduction capacit be allocated between the dieentiated and undieentiated stages? How much undieentiated invento should be ept between the two stages? What is the optimal point o dieentiation, given sevice level equiements and possible poduct/pocess edesign costs? The models we pesent ae intended o use as stategic tools that allow opeations manages to apidl examine e tadeos om dieent DD coniguations. Consistent with this spiit, we delibeatel eep the models simple and elevant o gaining manageial insights. The emainde o this aticle is oganized as ollows. In section, we discuss ou taget application and vaious modeling assumptions. In ou ist model pesented in section 3, we evaluate the advantage o delaed dieentiation elative to a pue mae-to-stoc sstem when commonalities aise natuall among poducts. In section 4, we conside the issue o identiing the optimal point o dieentiation when vaing degees o delaed dieentiation can be achieved at the cost o edesigning the poducts o the pocess. The eect o using seveal patiall dieentiated poducts, instead o a single undieentiated platom common to all poducts, is exploed in section 5. Finall, in section 6, we summaize ou indings and discuss vaious manageial implications. 4

5 . Taget Application and odeling Assumptions In this pape, we ocus ou attention on sstems in which the manuactuing pocess is pimail component assembl with poduct-invaiant assembl time. This is the case, o example, in the compute indust, whee all poducts undego the same assembl pocess e.g., all poducts equie a micopocesso, a had dis, a memo cad, and a powe suppl, among othe components. The assembl time o these components does not usuall va om poduct to poduct. Fo example, assembl time is the same o 500 and 700 hz pocessos see chapte 6 o Wight [3] o a discussion o compute manuactuing and assembl. In these sstems, delaed dieentiation is enabled b eithe taing advantage o existing commonalities among poducts, standadizing components acoss poducts o 3 eoganizing opeations so that those that ae common ae peomed ist [7]. In all thee cases, delaed dieentiation does not signiicantl aect the oveall poduct assembl times. In component assembl, delaing dieentiation esults in the consolidation, o pooling, o thee tpes o invento: component invento, wo-in-pocess WIP invento, and inished o semi-inished goods invento see Figue. The pooling o component invento is due to the standadization o components. Depending on the odeing polic, pooling could educe the amount o total invento needed to meet the same sevice level. The pooling o WIP invento is due to the undieentiated natue o WIP poduced using common components. In contast to component invento, the pooling o WIP does not aect the size o this invento since both the total demand and assembl time ae not aected assuming, as we do, that aw mateial is eleased immediatel ate each ode aives. The pooling o semi-inished invento is due to the undieentiated natue o poducts that ae stoed in the intemediate bue between the mae-to-stoc and mae-to-ode stages. This pooling could esult in a eduction in invento elative to a pue mae-to-stoc sstem. Howeve, as we shall see late in the pape, this is lagel dependent on sstem paametes. In evaluating the costs and beneits o delaed dieentiation, we shall be concened pimail with the eect o delaed dieentiation on the intemediate bue o undieentiated invento and how the size and placement o this invento aects its holding cost. We ae also inteested in examining how ode-ulillment dela equiements aect the desiabilit o delaed dieentiation and the associated placement and size o the undieentiated invento. 5

6 WIP invento C A C B C AB3 C A5 C B5 C A7 C B7 A A A A T T T 3 T 4 B B B B C A C B C AB4 C A6 C B6 C A8 C B8 Finished goods invento Component invento Figue a Eal dieentiation: Poducts A and B ae dieentiated eal since o thei ist assembl tas T the equie dieent components, C A and C A o poduct A and C B and C B o poduct B. C AB C AB3 C A5 C B5 C A7 C B7 A A T AB T AB T 3 T 4 B B C AB C AB4 C A5 C B5 C A8 C B8 Figue b Component standadization without delaed dieentiation: Components o tas T ae standadized. This leads to the pooling o component invento o tas T and WIP pooling ollowing tass T and T. C AB C AB3 C A5 C B5 C A7 C B7 AB AB A T T AB T 3 T 4 B C AB C AB4 C A6 C B6 C A8 C B8 Figue c Component standadization with delaed dieentiation:the dieentiation tass ae postponed until demand is ealized. Undieentiated invento is held in intemediate bue AB instead o dieentiated inished goods invento. Figue - The eect o component standadization and delaed dieentiation on invento pooling 6

7 Although delaed dieentiation could esult in educing the amount o component invento needed, we assume in this aticle that component availabilit is pimail the esponsibilit o the supplie. In act, the applications we have in mind ae those whee components ae oten deliveed just-in-time and little o no invento is held on-site. Theeoe, the beneits o component standadization ae mostl ealized b the supplie [9]. These beneits can be quantiied b appling to the supplie opeation a simila analsis to the one that we descibe hee. We do howeve account o the additional cost that might come om using standadized components o om edesigning the pocess to enable uthe dela o the dieentiation point. Since we assume that assembl times ae poduct-invaiant, delaed dieentiation does not aect total wo content. Fo simila easons, we assume that thee is no setup time in switching om assembling one poduct to anothe this is cetainl the case in compute assembl. Theeoe, delaed dieentiation does not aect changeove times. We also assume that all poducts ca the same pioit and ae quoted the same lead-time. Theeoe, we use a istcome ist-seved scheduling polic. Ou measue o cost includes both invento and pocess edesign costs while ou measue o sevice is custome ode dela. anuactuing manages oten set and stive to achieve explicit delive time goals, which the ma measue eithe as the aveage ode ulillment time, o as the popotion o odes that exceed a citical delive time taget e.g., a quoted lead time. The models we pesent can teat eithe o these points o view and captue the manne in which ode dela depends on how woload and capacit ae assigned to the undieentiated and dieentiated stages. Ou choice o ode dela as the measue o sevice stems om the obsevation that most applications o DD aise in situations whee quic esponse to custome odes is cental to the competitiveness o the im. Howeve, altenative measues o custome sevice ae possible b including, o example, an invento bacodeing cost, o placing a constaint on the pobabilit o bacodes exceeding some theshold. Both could be accommodated in ou models, ate suitable modiications. In ou analsis o sstems with delaed dieentiation, we shall conside thee cases. In the ist case, we conside sstems whee some commonalities between poducts aise natuall e.g., the ist set o opeations/components ae natuall common among all poducts. The question then is do we tae advantage o these existing commonalities and dela dieentiation until demand is ealized o do we continue to poduce each poduct in a mae-to-stoc ashion? In the second case, we conside sstems whee thee is lexibilit in choosing the point in the manuactuing pocess at which dieentiation taes place. The question hee is what is the 7

8 optimal point o dieentiation and how is it aected b sstem paametes? Since a change in the point o dieentiation esults in a change in woload allocation among the mae-to-ode and mae-to-stoc stages, an equall impotant question is how should capacit be allocated between the two stages? In the thid case, we conside sstems whee, instead o building a single common platom acoss all poducts, which can be expensive, poducts could be moe cheapl patiall dieentiated into seveal poduct amil platoms. Each platom is dieentiated into individual poducts once demand is ealized. The question then is what is the elative advantage o ull delaed dieentiation a single platom as compaed to patial dieentiation multiple platoms? Put dieentl, ae thee cases o which patial dieentiation is neal as good as ull dieentiation? 3. ae-to-stoc vesus Delaed Dieentiation In this section, we conside a poduction sstem whee commonalities between poducts aise natuall. Ou objective is to investigate conditions unde which we should tae advantage o these commonalities and dela dieentiation instead o poducing in a pue mae-to-stoc ashion. As discussed in the pevious section, poduction is caied out in two stages. The ist stage consists o opeations that ae common to all poducts while the second stage includes opeations that esult in dieentiated end-poducts. Since the application we have in mind is component assembl, all poducts equie the same assembl tass in both steps. Dieentiation comes om the use o dieent components in step. In the pue mae-to-stoc sstem TS, inished goods inventoies ae stoced o each poduct while o a sstem with DD, poductspeciic assembl is postponed until demand mateializes. A gaphical depiction o both sstems is shown in Figue. We conside a sstem with poducts. Extenal demand o poduct i, i,,..., occus accoding to a Poisson pocess o ate λ i, with Λ i λi denoting the total demand ate. Fo the pue TS sstem, demand is satisied om bue stoc unless the coesponding bue is empt. In that case, it is immediatel baclogged. Fo the sstem with DD, each demand aival eleases an undieentiated item om the intemediate invento bue, which then joins the queue o jobs, i an, that ae waiting to be pocessed at stage- whee dieentiation taes 8

9 Extenal demand o individual poducts b b b 3 Finished poducts Stage- Common opeations Stage- Dieentiated opeations Dedicated bues o inished goods b Wo elease tigge a Pue mae-to-stoc sstem TS Extenal demand o all poducts Finished poducts Stage- Common opeations Single Bue o undieentiated invento Stage- Dieentiated opeations Wo elease tigge b Sstem with delaed dieentiation DD Figue - ae-to-stoc vesus delaed dieentiation 9

10 place. Howeve, i the intemediate bue is empt, then the demand is baclogged o pocessing at stage-. Fo both sstems, inventoies ae managed accoding to a base stoc polic, whee each demand aival tigges the placing o an ode with the poduction sstem. Each ode esults in the immediate elease o a new aw mateial it to the queue o its at stage-. Fo both sstems, we assume that aw mateial its ae alwas available. The base stoc level o the sstem with DD is denoted as b d, wheeas the base stoc level o the sstem with inished goods is called b i o poduct i. In each sstem, the total wo content at stage- is T and at stage- is T. The aveage ate at which a unit o wo content is pocessed in stage-i is denoted b R i. Hence, the aveage ate at which items ae pocessed is i R i /T i. Fo stabilit, we equie that Λ/ i i < o all i, whee i is also the utilization at stage-i. We assume that the unit pocessing times at each stage ae exponentiall distibuted. This helps simpli analsis consideabl and epesents the pactical wost case o benchmaing poduction sstem peomance see Hopp and Speaman [] o a discussion o this assumption. Finall, we let λ i Λ/ o all i, which leads to b i b o all i. This is necessa in ode to ca out ai compaisons between sstems with dieent levels o poduct vaiet. It is, howeve, staightowad to extend the analsis to sstems with asmmetic demand. Fo both sstems, ou objective is to minimize aveage invento costs subject to a sevice level constaint. We speci sevice level in tems o an uppe bound on aveage ode ulillment time. Ode ulillment time is the total time elapsed om the moment a demand aives to the moment the inished poduct is supplied to the custome. It is possible to use altenative measues o sevice level, such as the pecentage o odes illed within a quoted lead time o the pecentage o odes that ae bacodeed. We dee discussion o these measues to section 4. Since we impose an ode ulillment dela equiement on both sstems, the two sstems can be compaed in tems o the size and cost o inventoies held eithe in the inished goods o as undieentiated items in the intemediate bue. Note that WIP level is not aected since poducts equie the same opeations egadless o sstem coniguation. Fo both sstems, ou design objective can be omulated as ollows: inimize aveage invento cost, subject to aveage ode ulillment dela α, whee α is the maximum allowed aveage dela. Aveage invento cost o the pue mae-to-stoc sstem is given b z [h I b ], whee h is the holding cost pe unit o inished goods pe unit time and I b is the 0

11 aveage inished goods invento o each end poduct o a base stoc level o b. Similal, aveage invento cost o the sstem with DD is given b z d h d I d b d whee h d is the holding cost pe unit o undieentiated invento pe unit time and I d b d is the aveage inished goods invento o each end poduct given a base stoc level o b d. We use the notation F b and F d b d to ee, espectivel, to aveage ode dela o the pue TS sstem and sstem with DD. Teating each stage as a single seve queueing sstem, aveage invento and aveage ode ulillment dela o the pue TS sstem can be obtained. Exact analsis, is howeve, tedious and does not lead to closed-om expessions. Theeoe, what we pesent below is a close appoximation which has been ound to wo well in most cases. Detailed mathematical development o these expessions can be ound in Appendix A. ] othewise, 0 [ 0, i 3 b b b b b I othewise. ] [. ] ][ [./. / / 0 / / / 0!!!!!!, i 0 0!!!!!! b b b b b b b b b b b b b b b b F l l l l l l l

12 Evaluating peomance measues o the model with delaed dieentiation is a little moe complicated. Although the output pocess o a single seve queue with Poisson aivals and exponential pocessing is also a Poisson pocess [3], the stage- input pocess is Poisson onl in two special cases: b d 0 and b d. The ome instance esults in two // queues in tandem whose stead state pobabilities ae nown to have a poduct-om stuctue [3]. Similal, when bue size is ve lage, the two stages ae completel decoupled and behave lie two independent // queues. Using notation A i to denote inte-aival time at stage-i, and C its A i squaed coeicient o vaiation, it is possible to show that C see appendix B o poo. A That is, teating stage- queue as a // queue is a easonable appoximation o estimating dela at this stage. In act, C is alwas less than one. Thus, ou esults povide an uppe A bound on the tue dela at stage-. In this sense, ous is a consevative viewpoint and values o b d chosen b the model will alwas be suicient to guaantee that ode ulillment equiement will be met. Teating each poduction stage as an // queue, we can obtain expessions o aveage invento and ode ulillment dela as ollows see appendix C o poo: and b d [ ] I d bd bd, 3 b d F d b d Λ Λ. 4 It is not too diicult to show that o both pue TS and DD sstems, aveage invento is inceasing in the base stoc level, b o b d, and aveage ode dela is deceasing in the same. Theeoe, the optimal values o b and b d ae alwas the smallest values that satis the sevice level constaint. Fo sstem DD, the optimal base stoc level can be ound b solving F d b d α, which ields the ollowing: ln[ Λ Λ α / ] b d, 5 ln whee x epesents the intege ceiling o x. Substituting b d in the expession o aveage invento, we can calculate the optimal cost o sstem DD. Fo the pue TS sstem, a closed om expession o b is diicult to obtain. Howeve, since F b is stictl deceasing in b, b

13 can be easil obtained though a simple numeical seach. Note that since F d b d /Λ- when b d and F b /Λ- /Λ- when b d 0, the value o α is meaningul onl i /Λ- α /Λ- /Λ-. I α < /Λ-, then DD is not a easible option and a pue TS sstem must be adopted. On the othe hand, i α > /Λ- /Λ-, thee is no need o holding invento. In that case, a pue mae-to-ode sstem is optimal. In compaing pue TS and DD sstems, we shall assume thoughout that /Λ- α /Λ- /Λ- is satisied. In ode to examine the eect o delaing dieentiation on cost, we caied out a seies o numeical expeiments in which we evaluated the eect o vaious opeating paametes, such as the numbe o poducts, utilization, capacit allocation, and holding costs. Repesentative esults ae shown in Figues 3-6. We ist examined the eect o poduct vaiet and utilization on the elative desiabilit o DD. To this eect, we evaluated the atio z / z d o dieent values o and dieent levels o utilization. To isolate the eect o, we limited ouselves initiall to cases whee, λ i Λ/ and Λ is ixed. In these compaisons, it is impotant to ensue that a sstem with DD is alwas easible. Fo that eason, we set α ε, whee ε > 0 and is ept the same acoss all utilization scenaios the eect o vaing ε is examined late. As we can see om Figue 3, the elative cost advantage o a sstem with delaed dieentiation is geneall inceasing in. This means that the beneits o DD tend to be moe signiicant when poduct vaiet is high. The esult can be explained b the invento pooling common bue eect that taes place with DD and which is paticulal signiicant when the numbe o poducts is lage. It is not, howeve, too diicult to ind instances whee a pue TS stateg is supeio. This is paticulal pevalent when poduct vaiet is low. In this case, the need to meet the sevice level tends to oce the sstem with DD to hold moe invento than a pue TS sstem. Because the numbe o poducts is small, the pooling eect is not suicient to oset the inceased equiement o invento. Although this patten is moe appaent when the numbe o poducts is small, it can occu egadless o the amount o poduct vaiet i ode dela equiement is suicientl shot o utilization is suicientl high. In act, as we can see om Figue 4, the elative advantage o DD is highl sensitive to utilization, with the advantage o DD diminishing with inceases in utilization. At ve high utilization, DD ceases to be desiable since the cost o meeting ode dela equiement becomes 3

14 Z /Zd Numbe o poducts Figue 3 The eect o numbe o poducts on the elative beneit o delaed dieentiation h, ε Z /Zd Utilization Figue 4 The eect o utilization on the elative beneit o delaed dieentiation h, ε

15 pohibitivel expensive. Note that in ou compaisons, we alwas guaantee that DD is easible. Howeve, i α emained ixed and utilization wee allowed to incease, DD could become quicl ineasible. In act, in that case, thee is a inite ange o utilization given b αλ/ α αλ/ α whee compaing the two sstems is meaningul. Fo > αλ/ α, DD is ineasible and a pue TS sstem must be adopted while o < αλ/ α, a pue mae-toode sstem is optimal. Note that the width o the easibilit inteval appoaches 0 as α 0. Similal, as α, DD is a candidate onl o sstems that have ve little excess capacit. In all othe cases, we can use the cheape mae-to-ode mode. The eect o vaing sevice level is illustated in Figue 5. Hee, in ode to ca out a ai compaison we let α ε, and va ε. Thus, inceasing α means inceasing the ode dela slac available to the sstem with DD. As we can see, a highe sevice level i.e., a smalle ode dela slac tends to diminish the beneit o DD since it equies the sstem to maintain highe inventoies. This eect is paticulal ponounced when utilization is high. The act that DD is moe sensitive to changes in α than a pue TS is due to two actos: a thee is geneall moe bue slac in pue TS moe about this in the next paagaph and b DD must alwas maintain suicient invento to meet the equied sevice level while caing out the dieentiation tass post-demand ealization. Although ou pevious obsevations geneall hold tue, we should wan that the integalit o invento and numbe o poducts could cause the atio z / z to be non-monotonic in o see Figue 6. Speciicall, the integalit o invento could esult in selecting a lage b than what is exactl needed to meet the ode dela equiement, an eect that is compounded with multiple poducts. Consequentl, within a limited ange, small inceases in utilization could esult in a lowe aveage invento while still satising the ode dela equiement. anages need to be mindul o these peculiaities when compaing speciic sstems. Howeve, wheeve possible, we ecommend use o analsis to detemine the desiabilit o one coniguation ove anothe. In examining the eect o utilization, we have assumed so a that both sstems ae balanced with equal utilization at both stages. In section 4, we examine the eect o unequal capacit and woload allocation between the two stages. Howeve, let us just note hee that the eect o asmmet in utilization on the TS sstem is dieent om its eect on the DD sstem. Fo an d 5

16 Z/Zd Sevice level α Figue 5 The eect o ode dela equiement on the elative beneit o delaed dieentiation h, Z Numbe o poducts Figue 6 The eect o ode dela equiement on the elative beneit o delaed dieentiation h,

17 TS sstem, intechanging the value o and has no eect on peomance. oe geneall, i we stat with, then inceasing the utilization o stage- while maintaining the utilization o stage- constant has the same eect as inceasing the utilization o stage- while maintaining the utilization o stage- constant. In contast, the DD sstem is ve sensitive to which o the two stages has the highe utilization. In act, a highe utilization at stage- ma not even be easible i that maes the ode dela lage than the minimum equied α. oe geneall as appoaches α/ α, the cost o the sstem with DD stats to escalate at a aste ate than the cost o the TS sstem. Fo α/ α, onl the TS sstem is easible. Hence, the TS sstem tends to be moe stable in the ace o luctuations in utilization since additional invento bueing is alwas suicient to meet a tighte ode dela. This, howeve, ma come at a high pice. Italso means that in ode to enable delaed dieentiated suicient capacit investments should be made at stage-, the mae-to-ode stage. In summa, esults o this section show that although DD can ield signiicant cost savings, it is not alwas desiable even i no pocess o poduct edesign is needed. anages should be paticulal cautious about implementing DD i poduct vaiet is low, utilization is high, o 3 allowable slac in ode lead-time is small. anages should also pa close attention to the elative loading o the dieentiated and undieentiated stages since that might aect the desiabilit o one sstem ove anothe. 4. The Optimal Point o Dieentiation In this section we exploe the economics o aecting the point o dieentiation POD, given that delaing dieentiation is desiable o a base case. The base case ees to the situation studied in section 3 whee we simpl exploe existing commonalities and maintain a common bue o undieentiated items at the point sepaating the common and custom opeations. Fo this pupose, we set ET ET T, whee T is a constant and conside the implication o changing t ET. Such a change could be made possible b esequencing o assembl steps putting common opeations ist, o edesigning assembl opeations to incease the numbe o common steps, o though edesign o the poducts so that the could be assembled om common components. 7

18 Thus, the e issue now is to detemine the value o the optimal point o dieentiation t ET. Since a change in t coesponds to a change in the woload assigned to each poduction stage, this change is usuall accompanied b a eallocation o capacit. Seveal capacit allocation schemes ae possible. We limit ouselves to two cases. In the ist case, we assume that capacit is allocated popotionall to woload so that utilizations, and, emain unchanged. This means that i we choose to incease wo content o the TS stage, its pocessing speed R is also inceased popotionall. Although we do not put an limits on the amount b which pocessing speed might be aected, an incease in capacit incus an acquisition and owneship cost. We assume that a eduction in capacit educes this cost, although it is possible to include a one-time downsizing cost e.g., lao o equipment disposal costs. This omulation allows us to isolate the eect o alteing the point o dieentiation om those that come om changing utilization. This is impotant since the eect o utilization can easil oveshadow those caused b changes in point o dieentiation. In the second case, we let the amount o available capacit be ixed and then ca out an optimal allocation o this capacit based on woload assignment. Thus, we assume that R R R, whee R is a constant, and detemine optimal R o R based on ou choice o optimal t. In this case, b, t and R ae all decision vaiables. Fo both cases, we shall use the ollowing notation: h i t Cost pe unit pe unit time o holding invento at stage-i, C t Cost o poduct and pocess edesign, C R Cost o capacit R, and C 3 b Cost o maintaining an invento bue o size b. All o the cost unctions ae assumed to be dieentiable, positive, inceasing and convex in thei aguments. Each cost unction is nomalized to 0 when its agument is 0. In the case o C t, we also equie that C t 0 o t t 0, whee t 0 is the point o dieentiation in the base case, since dieentiating ealie simpl equies moving the bue o undieentiated invento ealie in the poduction pocess without a need o edesigning eithe the poduct o the pocess. In identiing the optimal point o dieentiation, ou objective is to meet the speciied delive time peomance while minimizing the sum o invento, edesign, and capacit costs. Ou optimization model can be omulated as ollows: 8

19 inimize Zt, b, R h tit, b, R, R h tewip t, b, R, R C t C R C 3 b 6 Subject to Ft, b, R, R α. 7 8 /- α 9 t T 0 t, b, R 0 Note that with t being a decision vaiable, we must include the cost o WIP in stage- in the objective unction since the holding cost o this WIP is aected b ou choice o point o dieentiation. Expessions o I and F have been deived in section 3. Expected WIP at stage-, b vitue o Little s law, is /-. Constaint 8 ensues stabilit o stage-; constaint 9 guaantees that the sevice level constaint is easible; and constaint ensues that the wo content assigned to stage- does not exceed total wo content. Note that subscipt d is dopped om notation since all sstems being compaed utilize delaed dieentiation. We also limit ouselves to cases whee a pue mae-to-ode sstem is not easible. Theeoe, we implicitl assume that α < /- /-. Although ou sevice level measue is speciied in tems o aveage ode dela, it is possible to use altenate measues such as the pobabilit o being able to delive within a quoted lead time, aveage numbe o bacodes, o the pobabilit that bacodes exceed some citical value. Expession o each o these metics ae given b the ollowing o the sae o bevit, the poos ae omitted and can be ound in Gupta and Benjaaa []: F x b, t, R, R Pobode dela x b [ Λ / ] x [ / ] [ / ] Λ x Λ x e e e i, b [ Λ / ] x Λx - e othewise, and b,,, aveage numbe o bacodes S b t R R, 3 9

20 b x x x i S,,, PobS x b t R R x x x- b x othewise., 4 4. The Case o Fixed Utilization Since capacit is assigned popotionall to wo content, the values o and ae unaected b ou choice o t, R and R. Consequentl, I, EWIP, F and C 3 ae all unaected b t. Given the choice o t, R is given b: R R R Λt/ ΛΤ t/ Λ{t - T }/. 5 Theeoe, capacit is uniquel detemined b t. Since ode dela is deceasing in b but the objective unction is inceasing in b, the optimal base stoc level b is still given b expession 5. Substituting b in zt, b, R, R, the objective unction educes to a unction in the single vaiable t. The optimal value o t can be obtained b seaching in the ange 0 t T. Howeve, as we shall discuss next, this is necessa onl when >. Since delaing dieentiation beond the base value t 0 inceases o at least does not decease the holding costs h and h and the edesign cost C, additional delaed dieentiation is desiable onl i the capacit cost C is educed as a esult. This means that the tadeo in deciding on a point o dieentiation is between additional holding costs vesus educed capacit. Using the act that R i ΛΕT i / i, it can be easil conimed that R R R R R R Λ t t /. 6 I t > t 0, the dieence R - R 0 is negative onl i <. That is, uthe dela o the point o dieentiation educes equied capacit onl i the utilization o the mae-to-stoc stage is geate than that o the mae-to-ode stage. Theeoe, i, we should neve dela dieentiation beond its base value o t 0. In geneal, we can distinguish thee cases.. : In this case, we have a balanced sstem. Capacit emains constant egadless o t. Since both holding costs and edesign costs ae inceasing in t, it is neve optimal to postpone dieentiation beond t 0. In act, in this case, the soone we dieentiate the bette. That is, choosing t < t 0 is alwas desiable. 0

21 . < : In this case, capacit is inceasing in t. Theeoe, uthe dela is neve optimal. Hee again, since both capacit and holding costs ae deceasing in t, no change in POD is ecommended. 3. >. In this case, capacit is deceasing in t. Howeve, holding costs ae inceasing in t. Consequentl, we need to tade-o the savings om capacit eduction against the incease in holding costs. Geneall, this would equie seaching o the optimal t ove the entie ange 0, T. Howeve, in the case o linea holding, capacit, and edesign costs, it is eas to show that uthe dela o dieentiation is desiable onl i b C Cmin { h b [ d d ]/ h / C Λ Othewise, the ealie we dieentiate the bette. The value o C min is inceasing in h, h and C and deceasing in the dieence -. It is clea that o dieent cost unctions, it is possible to obtain an optimal POD that is dieent om the exteme cases o eithe maximum o minimum dela. } The Case o Fixed Capacit Beoe we examine the geneal case whee we jointl choose t, R, R R R R and b, we shall conside the ollowing two special cases: capacit assignment is ixed and wo-content allocation is lexible, and capacit assignment is lexible but wo content allocation is ixed. In the ist case, R and R ae ixed but t is vaiable while in the second t is ixed but R and R ae vaiable. Distinguishing the two cases allows us to gain geate insight into the elationship between delaed dieentiation and capacit assignment. It is also useul in developing a solution pocedue o the geneal case. When capacit allocation is ixed, ou optimization poblem educes to selecting t and b. It is not diicult to see that constaint 7 is still binding and, theeoe, b is given b expession 5. Substituting b in the objective unction, zt, b, R, R becomes again a unction o the single vaiable t. Noting that constaints 8 and 9 can be ewitten, espectivel, as t < R /Λ and T R R / Λ /α < t, the optimal t can be obtained b seaching in the ange max{0, T R R / Λ /α} < t < min{t, R /Λ}. 8 Examining the objective unction we can see that edesign costs, unit holding costs, and expected WIP in stage- ae all inceasing in t. Howeve, aveage invento in the intemediate bue as

22 well as the size o this bue is not necessail monotonic in t. Fo example, educing t leads to aste eplenishment o the intemediate stoc but longe delas at stage-. Depending on the value o t, this leads to a choice o b that could eithe incease o decease aveage invento. In geneal, eal dieentiation is moe desiable as long as the associated wo allocation does not signiicantl incease lead-time in stage-. This would be the case when the capacit assigned to stage- is elativel lage. Invesel, when little capacit is assigned to stage-, delaed dieentiation tends to become moe attactive. This is illustated in Figue 7 o an example sstem. As we can see, the optimal point o dieentiation is also aected b the total amount o available capacit R. Speciicall, an incease in total capacit tends to mae eal dieentiation moe desiable since assigning additional woload to stage-, without a signiicant penalt in highe invento, becomes possible. In addition to being aected b the available capacit, as well as the elative allocation o this capacit, the optimal point o dieentiation is detemined b the ode dela equiement α. Since / - < α is a necessa condition and the onl mechanism b which we contol is b inceasing t, a smalle α tends to avo late dieentiation. On the othe hand, a lage α maes eal dieentiation moe desiable since we can educe edesign costs while satising ode dela equiements. This eect is gaphicall illustated in Figue 8. Tuning to the second case, whee the point o dieentiation is ixed but capacit assignment is lexible, we can see that it too educes to a single decision vaiable optimization poblem in R. In contast to t, vaing R caies no penalties o the tpes o manuactuing sstems we model. Howeve, the eect o vaing R on invento levels is diicult to pedict. While assigning moe capacit to stage- educes eplenishment lead times to the intemediate bue, it inceases lead-time at stage-. The combined eect could lead to eithe a net incease o decease in total holding costs. In geneal, we should expect moe capacit to be assigned to stage- when dieentiation is caied out eal due to the associated highe woload. The need to assign moe capacit to stage- can also be exacebated when ode dela equiement is tight. In act, when α is suicientl small, moe capacit must be assigned to stage- to ensue easibilit. The elationship between optimal assignment, point o dieentiation and sevice level is illustated in Figues 9 and 0 o an example sstem.

23 .00 Optimal point o dieentiation - t R.5 R.6 R.7 R Capacit in stage - R Figue 7 The eect o capacit allocation on optimal point o dieentiation T, Λ, α, h t h t 5t, C t 0, C 3 b Optimal point o dieentiation - t R0.7 R R0.9 R.0 R Ode dela equiement - α Figue 8 The eect o ode dela equiement on optimal point o dieentiation T, R.5, Λ, h t h t 5t, C t 0, C 3 b 0 3

24 .60 Optimal capacit allocation - R R.3 R.4 R.5 R.6 R.0 R Point o dieentiation - t Figue 9 The eect o point o dieentiation on optimal capacit allocation T, Λ, α, h t h t 5t, C t 0, C 3 b Optimal capacit allocation - R t0.3 t0.5 t0.7 t Ode dela equiement - α Figue 0 The eect o ode dela equiement on optimal capacit allocation T, Λ, R, h t h t 5t, C t 0, C 3 b 0 4

25 Putting esults om these two special cases togethe, two pinciples emege: wheneve possible, we should dieentiate eal this becomes moe diicult as the ode dela constaint gets tighte, and we should assign capacit so that dieentiation could tae place ealie this means, wheneve easible, assigning popotionall moe capacit to stage-. These pinciples can be used b manages to quicl identi a ist-cut solution to the geneal poblem whee t, b, and R ae all decision vaiables. Howeve, due to the non-convex natue o the objective unction in the decision vaiables, these pinciples do not guaantee optimalit. In act, shot o an exhaustive seach ove the t and R dimensions, an optimal solution is had to obtain. Fotunatel, in pactice t and R can be vaied onl in discete steps. Theeoe, an exhaustive seach, even when the numbe o discete steps is lage, is alwas possible. 5. The Eect o Patial Delaed Dieentiation In the pevious two sections, we assumed that a platom common to all poducts can be built in stage-. In man industies, this is eithe not possible o too expensive. Instead, poducts ae oten gouped into amilies and a standadized platom is designed o each amil [9]. Thus, instead o building a single undieentiated poduct in stage-, two o moe patiall dieentiated items ae poduced and stoced in sepaate intemediate bues. These patiall dieentiated items ae then ull dieentiated in stage- once demand is ealized. In this section, we examine the elative advantage o ull delaed dieentiation DD elative to patial delaed dieentiation. We ae paticulal inteested in identiing conditions when patial dieentiation PD does not esult in signiicant deteioation in peomance. Note that i we ignoed edesign costs, a sstem with ull undieentiated poducts is alwas supeio. This obsevation is consistent with invento liteatue. Howeve, since the costs o designing a common platom that is shaed b all poducts can be signiicant especiall when the numbe o these poducts is high, thee is a need to tadeo the additional beneits o ull delaed dieentiation against cost savings ealized with patial dieentiation. A sstem with patiall dieentiated poducts is simila to the sstem with DD we consideed ealie. Howeve, instead o a single bue o undieentiated poducts, we have bues o patiall dieentiated poducts. In ode to isolate the eect o numbe o patiall dieentiated PD poducts, we let the demand associated with each PD poduct be λ Λ/. 5

26 Aveage total invento I pd and aveage ode dela, F pd, can then be obtained as ollows see Appendix D o poo: and whee b b pd pd b pd, 8 I pd b pd F b pd pd, 9 Λ Λ. 0 Since the component o ode dela due to stage- is not aected b, we can ewite the sevice level constaint as: whee b pd Λ αˆ, αˆ α. Λ The optimal base stoc level can then be obtained as: ln[ αˆ Λ ] b pd 3 ln Note that we assume αˆ < / λ. I the condition is not satisied, then a pue mae-to-ode sstem is optimal. As we did in section 3, let us examine the advantage o complete lac o dieentiation elative to patial dieentiation unde vaing conditions o utilization, poduct vaiet and ode dela equiement. We begin b examining the eect o utilization. Fist, note that the mae-to-ode segment o the poduction pocess is not aected b patial dieentiation. Theeoe, we estict ou attention to the utilization o the mae-to-stoc stage,. In contast to section 3, we ae able hee to analticall chaacteize seveal popeties elating the eect o 6

27 utilization when it is eithe high, low, and in the mid-ange poos o all popeties can be ound in Appendix E. Popet : pd z d pd pd bpd z / when, whee z hi. Popet shows that the elative advantage o ull standadization is insigniicant when utilization is high. In act, in the limiting case, a sstem with patiall dieentiated poducts becomes equivalent to a sstem with a single undieentiated poduct. This also means that patial dieentiation caies less o a penalt elative to no dieentiation in highl loaded sstems, even though invento levels ae ve lage in both cases. Similal, when utilization is suicientl low, ull standadization caies little value since in that case a pue mae-to-ode sstem is optimal. The limiting utilization is given b popet. Popet : pd z d z 0 when min αˆ Λ / αˆ Λ. The esult ollows om the act that when αˆ Λ / αˆ Λ, we have α ˆ / Λ, which means that a pue mae-to-ode sstem is easible egadless o level o dieentiation. Although ull delaed dieentiation caies little value in the exteme cases o high and low utilization, its beneits can be signiicant when utilization is in the midange. This can be seen in Figue. It can also be shown b consideing the case whee is slightl geate than min. As we can see om popet 3, the advantage o ull standadization is at least times that o patial dieentiation in this case. Popet 3: z pd / z d / as min. The esult ollows om the act that when is slightl geate than min, each individual bue in the patiall dieentiated sstem must maintain at least one unit o invento in ode to meet the custome dela equiement. In contast, the sstem with a standadized platom needs onl one unit o invento to meet the same equiement. This esult, due to the integalit o the base stoc level, leads to the obseved discontinuit into the eect o utilization on peomance. Next, we examine the eect o the numbe o patiall dieentiated platoms on the atio z pd / z d. As shown in Figue, this eect is not monotonic. Hence, inceasing the numbe o patiall dieentiated poducts could eithe incease o decease total cost. This nonmonotonicit is howeve limited to cases when is elativel small. In act, as we show in popet 4, o max cost inceases lineal in. 7

28 50 z pd /z d N 55 N 0 0 N 5 5 N Utilization Figue The eect o utilization on the elative cost o PD h, αˆ.5, 40 z pd Figue The eect o numbe o patiall dieentiated poducts on optimal cost h, αˆ, 8

29 Popet 4: I / ][ / αˆ Λ ], then max [ z pd. Popet 4 shows that when is suicientl lage the beneits o standadization ae inceasing in. This esult is in line with obsevations we made in section 3 whee we ound that the maginal beneits o delaed dieentiation ae geneall inceasing in the numbe o endpoducts. This esult also shows that the beneits o delaed dieentiation can be signiicantl geate than those pedicted b the invento pooling liteatue, whee invento costs ae shown to be educed b a acto o when poducts ae pooled. Finall, we conside the eect o the ode dela equiement. Simila to the eect o utilization, we can show that in the two exteme cases o ve lage o ve small αˆ ull delaed dieentiation is o little value and patial dieentiation could be pusued with no signiicant impact on peomance. The atio z pd / z d is equal to when αˆ / λ since a pue maeto-ode sstem is optimal in this case. As shown in popet 5, z pd / z d ln[ ]/ln[ ], which tends to be a elativel small numbe, when αˆ appoaches 0. Popet 5: z pd / z d ln[ ]/ln[ ] as αˆ 0. Howeve, when αˆ is in the midange, patial dieentiation could be costl. Fo example, when αˆ is in the ange speciied in popet 6, patial dieentiation is at least times moe costl than a stateg with complete delaed dieentiation. Popet 6: pd z d Λ z / / when /[ Λ ] αˆ. ˆ When α /{ Λ }, z pd / z d although not monotonic is geneall inceasing in αˆ. The eect o ode dela equiement ove the entie ange o possible αˆ values is illustated o an example sstem in Figue 3. In summa, we have shown that the advantage o delaed dieentiation elative to patial dieentiation is highl sensitive to the utilization o the poduction acilit, poduct vaiet, and ode dela equiement. Although the beneits o delaed dieentiation can be signiicant when utilization is in the midange, this advantage tends to diminish when utilization is eithe ve high o ve low. Similal, thee is a limited band o ode dela values within which delaed dieentiation caies signiicant beneits. Fo values outside this band, the beneits ae eithe 9

30 limited o non-existent. These esults suggest that manages should be paticulal cautious in edesigning thei poducts o thei pocesses to suppot a single common platom when eithe utilizations ae high o ode lead time equiements ae tight. Remaabl, because the maginal beneits o a single platom ae inceasing in the numbe o poducts, the economics o a single platom ae moe advantageous when the numbe o poducts is high. Theeoe, manages should be moe willing to invest in a common platom when thei poduct potolio is lage. This, o example, might explain the success o compute maes with the vanilla box concept since individual compute coniguations can be numeous. The esults o this section ae in line with those we obseved numeicall in section 3 egading the desiabilit o delaed dieentiation. The also paallel esults ecentl obtained b one o the co-authos o this aticle egading the eect o invento pooling in integated manuactue-etaile suppl chains [] N 5 N 0 0 N 5 5 N z pd /z d α Figue 3 The eect o ode dela equiement on the elative cost o PD h, 0.9, 30

31 6. Concluding Remas Table povides a summa o ou e indings and oes manages boad guidelines as to when delaed dieentiation is most valuable elative to a stateg o eithe pue mae-to-stoc o patial dieentiation. Ou esults ae consistent with the existing invento liteatue with egad to the desiabilit o delaed dieentiation when poduct vaiet is high, edesign costs ae low, o inished goods invento is expensive. Ou esults shed additional light on the eects o capacit, congestion, and ode dela equiements. In paticula, we show that delaed dieentiation, and moe geneall invento pooling, is signiicantl less valuable when utilization is high. This is an impotant esult since most actoies tend to opeate nea ull capacit. Theeoe, analsis based on pue invento models would lead to ove-estimating the beneits o delaed dieentiation. Futhemoe, we ound that unless ve little wo-content is postponed, delaed dieentiation can be an expensive mechanism o achieving quic esponse. In act, in envionments whee ode dela equiements ae tight, delaing dieentiation until demand is ealized could equie holding signiicantl moe invento than in a pue mae-tostoc sstem. This esult shows again that using pue invento models could lead to oveestimating the value deived om delaed dieentiation. In envionments whee the point o dieentiation can be aected we ound that dieentiating eal b postponing moe o the wo until demand is ealized is geneall moe desiable. In act, wheneve possible a pue mae-to-ode sstem should be adopted. Obviousl, this becomes diicult to ealize when ode dela equiements ae tight. In that case, moe capacit, i possible, should be assigned to the mae-to-ode stage. Othewise we ae let with two choices, eithe to go bac to poducing in a mae-to-stoc ashion o to edesign poducts and pocesses to allow o ve late dieentiation. This latte option is easible onl i edesign costs ae not too high. Nevetheless, this option has become inceasingl popula in seveal industies whee customes demand shot lead times and cost pessues equie high utilization o poduction acilities. Fo example, in the compute indust, delaed dieentiation has been pushed up to the point o sale with etailes caing out the inal coniguation and assembl o each compute b quicl adding o emoving modula pats. Finall, we acnowledge that while ou models captue seveal o the tadeos that aise om delaed dieentiation, thee ae additional costs and beneits that all beond the scope o ou models. Fo example, DD caies beneits deived om steamlining scheduling, sequencing, 3

32 and aw mateial puchasing. Thee ae also beneits om standadizing components in the om o shote poduct design ccle and cheape new poduct development. On the othe hand, thee ae some ovelooed costs. Fo example, eling on a common platom and emphasizing the use o common components could inadvetentl educe the lexibilit o the im and its abilit to oe tul customized poducts. In tun, this ma lead to loss o maet shae. anages need to be awae o these additional tadeos and to account o them beoe adopting a stateg o delaed dieentiation. Table When is delaed dieentiation valuable? Low edium High Utilization Less valuable oe valuable Less valuable Poduct vaiet Less valuable odeatel valuable oe valuable Ode dela Less valuable oe valuable Less valuable equiement α Holding costs Less valuable odeatel valuable oe valuable Redesign costs oe valuable odeatel valuable Less valuable Acnowledgements: We would lie to than the Associate Edito and two anonmous eviewes o man useul comments on an ealie vesion o the pape. We ae thanul to Poesso Bill Coope o the Univesit o innesota o his help with developing a omal poo o popet. This eseach was suppoted in pat b the Natual Sciences and Engineeing Council o Canada, the gaduate school o the Univesit o innesota, and the National Science Foundation though eseach gants to the authos. 3

33 Reeences [] Aviv, Y. and A. Fedeguen, The Beneits o Design o Postponement, in Quantitative odels o Suppl Chain anagement, S. Tau, R. Ganeshan, and. agazine, Editos, Kluwe Academic Publishes, , 999. [] Benjaaa, S. and J. S. Kim, Impact o Invento Decentalization on the Peomance o Poduction-Invento Sstems, Woing Pape, Univesit o innesota, 000. [3] Bue, P. J., The Output o a Queueing Sstem, Opeations Reseach, 4 956, [4] Buce, L. The Bight New Wolds o Benetton, Intenational anagement, Novembe 987, [5] Buzacott, J. A. and J. G. Shanthiuma, Stochastic odels o anuactuing Sstems, Pentice Hall, NJ, 993. [6] Eppen, G., Eects o Centalization on Expected Costs in ulti-location Newsbo Poblems, anagement Science, 5, , 979. [7] Feitzinge, E. and H. Lee, ass Customization at Hewlett Pacad: The Powe o Postponement, Havad Business Review, Janua-Febua 997, 6-. [8] Fishe,., K. Ramdas, and K. Ulich, Component Shaing in anagement o Poduct Vaiet, anagement Science, , [9] Gag, A. and C. S. Tang, On Postponement Stategies o Poduct Families with ultiple Points o Dieentiation, IIE Tansactions, 9 997, [0] Gaman, G. A. and. J. agazine, An Analsis o Pacaging Postponement, Poceedings o the 998 SO Coneence, School o Business Administation, The Univesit o Washington, Seattle, June 9-30, 998, [] Gupta, D. and S. Benjaaa, ae-to-ode, ae-to-stoc, o Dela Poduct Dieentiation? - A Common Famewo o odeling and Analsis, Woing pape, ichael G. DeGoote School o Business, caste Univesit, Ontaio, Canada, 999. [] Hopp, W. and. L. Speaman, Facto Phsics, Second Edition, Iwin/cGaw-Hill, NY, 000. [3] Jacson, J. R., Netwos o Waiting Lines," Opeations Reseach, 5 957, [4] Kleinoc, L., Queueing Sstems, Volume I: Theo, John Wile and Sons, 975. [5] Lee, H. L., Eective Invento and Sevice anagement Though Poduct and Pocess Redesign, Opeations Reseach, ,

34 [6] Lee, H. L.and C. Billington, Designing Poducts and Pocesses o Postponement, anagement o Design: Engineeing and anagement Pespectives, S. Dasu and C. Eastman Eds., Kluwe Academic Publishes, Boston, A, 994, 05-. [7] Lee, H. L., and C. S. Tang, odeling the Costs and Beneits o Delaed Poduct Dieentiation, anagement Science, , [8] Luenbege, D. G., Linea and Nonlinea Pogamming, Addison-Wesle Publishing Compan, Reading, A, 984. [9] agetta, J., The Powe o Vitual Integation: An Inteview with Dell Compute's ichael Dell, Havad Business Review, ach-apil 998, [0] Swaminathan, J.. and S. R. Tau, anaging Design o Assembl Sequences o Poduct Lines that Dela Poduct Dieentiation, IIE Tansactions, 3 999, [] Swaminathan, J.. and S. R. Tau, anaging Boade Poduct Lines Though Delaed Dieentiation Using Vanilla Boxes, anagement Science, , S6-S7. [] Whitt, W., The Queueing Netwo Analze, Bell Sstem Technical Jounal, 6 983, [3] Wight, P., st Centu anuactuing, Uppe Saddle Rive, Pentice Hall, NJ,

35 Appendix A - Peomance etics o a Pue ae-to-stoc Sstem Since demand is Poisson and pocessing times ae exponentiall distibuted the two stages o the poduction pocess behave lie // queues in tandem. Theeoe, π, the pobabilit that thee ae a total o jobs in pocess is π π π 0 [ / / ] i othewise. A. Conside the size o the aveage invento o tpe j inished goods. The invento bue is nonempt onl i the numbe o tpe j jobs in pocess is less than b. Thus, b deinition, the aveage invento o tpe j inished items can be witten as: I b, j b b j p, π, A. j 0 j whee p j, is the conditional pobabilit that j jobs ae pesent at stages and, given that thee ae a total o jobs. Futhemoe, owing to the equal aival ates assumption, we have j j! p,. j A.3 j! j! Due to smmet, the total aveage invento, I b, is simpl times I b. Upon substituting om A. and A.3 into A., and simpliing, we obtain the expession o I b shown in. Next, we deive an appoximate expession o the aveage ode dela expeienced b an abita tpe j poduct demand tagged custome aiving to the sstem. It expeiences dela onl i the total numbe o tpe j jobs in the two poduction stages exceeds b. Let thee be b tpe j jobs in the sstem at the moment o aival o the tagged custome. Notice that we have dopped the subscipt j on account o smmet. Then, the tagged aival will expeience a dela which equals b Y sevice completions, whee Y is a andom vaiable with suppot [0, - ]. It epesents all tpe l, l j, jobs that will have to be pocessed, on account o ist-in-ist-out j 35

36 sevice discipline, until the -b th tpe j job is pocessed. The latte will be used to satis the tagged custome. Then, using combinatoial aguments, the pobabilit that Y, denoted b p, can be witten as ollows: b! b!!! p. A.4!!!! b b! Let D denote the aveage inte-depatue time om the poduction sstem when thee ae jobs in it. Also, let Q i denote the stead-state queue lengths at stage-i. Since aivals ae Poisson, we have: T t i Q > 0, D A.5 T t t othewise. Fom ealie aguments, the conditional pobabilit that Q 0, given that Q Q, denoted b π,0, can be witten as ollows: i π,0 A.6 / [ / ] othewise. / Let F, j b denote the aveage waiting time expeienced b a tpe j tagged custome. Then, b F, j b p, p D l π. A.7 b b 0 l 0 Note that equation A.7 is an appoximation since it ignoes dependence among states o the sstem. Due to smmet, an abita tpe j custome expeiences the same aveage dela as an abita aival, iespective o class, i.e., F b F, j b. Theeoe, upon substituting om A., A.3, A.4, and A.5 into A.6, and simpliing, we obtain. 36

37 B - Input Pocess o Stage- in a Sstem with Delaed Dieentiation Lemma B.: The input pocess to poduction stage- can be chaacteized as ollows. A s Λ Λ Λ Λ s b d Λ s Λ s Λ s s Λ s i b d othewise. 0, whee As is the Laplace Stieltjes tansom o the andom vaiable A. B. Poo: I b d 0, the stage- input pocess coincides with the depatue pocess o stage-. Hee s Bue's theoem applies and theeoe A s Λ / Λ. Fo b d, let s conside the jth aival to stage- unde stead state opeations. It coincides eithe with an extenal aival this happens when the intemediate bue is not altogethe empt,o with the moment o a sevice completion at stage-. Let τ j denote the aival epoch o the jth aival, and A,j the inte-aival time. Cleal, A τ j τ j, j, j, whee we deine τ 0 0. Let j Q denote the stead state numbe o customes in stage- at the moment o jth aival to stage-, and Q denote the stead state numbe in stage- at an abita obsevation epoch. Since stage- queue is an // queue and each stage- aival epoch coincides eithe with a stage- aival epoch o with a stage- depatue epoch, it can be agued that j Q is identical in distibution to Q o all j. We deine thee andom vaiables, and. All thee ae exponentiall distibuted with paametes Λ, Λ and espectivel. Thus, is the time between consecutive extenal aivals and is the time between depatues om stage- when Q j > 0. Similal, is the time until next event which could eithe be an aival o a sevice completion at stage-. Conside, τ, j 0, the moment at which jth custome has just aived. The time until next j aival to stage-, A,j, can be witten as: 37

38 A, j Λ Λ Λ > > i i Q j i Q Q j < b j d > b, b d. d, B. In B., the notation > denotes the duation o given that >. The ollowing contingencies give ise to elationship B.: j. I at τ j, Q < bd, next aival to stage- will coincide with the next extenal aival. Since the owad ecuence time o is identical in distibution to, it ollows that A j wheneve Q j < b d. j. I at τ j, Q bd, next aival to stage- occus ate events tae place. The ist event happens when eithe an aival o a sevice completion occus. Owing to the memoless popet o the o the exponential distibution, the time until this event has distibution. The second event is eithe an aival o a sevice completion depending on whethe > o >. This explains the two tems, > and >, which epesent the patial distibutions o and espectivel. > ollows i the ist event is a sevice completion which occus with pobabilit / Λ. Similal, > ollows with pobabilit Λ / Λ. Q j j > 3. I at τ, b d, next aival to stage- coincides with the next depatue om stage-. Since the latte has a owad ecuence time o, it is clea that in this situation A,j equals. Let G a > denote the cumulative distibution unction o given that >. Using notation g Y to denote the pobabilit densit unction o a andom vaiable Y, we get 38

39 39 Pob and Pob a a G > > > > > 0 0 Pob and Pob d g d g a Λ Λ Λ 0 0 a a x x d e e d e dx e a a a e e e Λ Λ Λ. Upon dieentiating the above, we get the ollowing expession o the PDF o > : a a e e a g Λ Λ Λ >. B.3 Let π denote the pobabilit that stage- has 0 customes at an abita obsevation epoch. Since π, we can wite. } {, x g x g x g x g x g d d d j A b b b Λ Λ Λ > > B.4 Next, we tae the Laplace Tansoms on both sides o B.4 and simpli to obtain the ollowing elationship:., s s s s s s s A d d d b b b j Λ Λ Λ Λ Λ Λ Λ Λ Λ B.5 Since the distibution o A,j the RHS o B.5 is independent o the index j, we dop this subscipt and simpli B.5 to obtain:. s s s s s s A b d Λ Λ Λ Λ Λ Λ B.6

40 Λ Notice that A s as bd. This agees with intuition: when bue size is lage, Λ s the two stages opeate lie two independent // queues. Hence poved. Next we show that C and that the stage- can be appoximated b a // queue. A Fom B., it is eas to obtain the ollowing additional popeties o A : ' '' A A 0, E A A 0 i b d 0, and Λ Λ Λ b d i b. Similal, d C A b d i b d 0, and / i b. Thus, we have a // queue at stage- and a G// queue at stage-. The stead state queue length distibution at stage-, π, can be obtained as π x, o 0,,, whee x 0, is a solution to the x equation: x A x see, o example, Kleinoc [4], pp. 5. Since we need closed om expessions to be able to peom optimization, we set C A, i.e., we teat stage- queue as a // queue. The easonableness o this appoximation can be seen upon calculating [ C A / CA ] 00, o the pecent eo upon assuming A C. Notice that o b, the pecent eo is monotonicall deceasing as b. d Theeoe, we set b d and ind that the maximum eo is 7.7% and that it occus when In act o b d geate than, the eo quicl goes to zeo, as shown below. Fo most pactical situations, we expect b to be geate than in ode to eep esponse times ast. Also, the intended use o ou models lies in pe-design evaluation o bue size and location altenatives. Given these acts, we believe assuming C is a easonable appoximation. A close match was also obseved between exact mean ode ulillment times obtained b solving the G// queue, and the coesponding appoximate values obtained om assuming C A. Fo example, i we set b, Λ , and notice that this is a case in which ou appoximation is most inaccuate, we obseve that ou appoximation oveestimates the exact value o the mean ode ulillment time at stage- b.85%. d A 40

41 b d b d 5 b d 0 b d 0 b d 40 b d C - Peomance etics o a Sstem with Delaed Dieentiation Recall that Q i denotes the stead state numbe o in-pocess jobs at stage-i at an abita moment o obsevation. It is eas to see that the numbe o semi-inished items in the intemediate bue max0, b d - Q and that Q is independent o stage-. Theeoe, the aveage bue invento can be witten as: b d I b b π, C. d d 0 whee π Pob{ Q }, o 0. We obtain equation 3 upon simpliing the above sum i i ate substituting π. To obtain F, note that each aival must wait o pocessing at stage-, wheeas it d b d waits at stage- onl when the intemediate bue is empt. Futhemoe, since each extenal aival is a Poisson aival, it sees time aveage behavio. Thus, π b F d bd, C. Λ Λ b whee the second tem on the ight hand side o C. is the expected ode dela in a // queue. Simpliication o equation C. ields 4. d 4

42 D - Peomance etics o a Sstem with Patial Delaed Dieentiation The pobabilit, π, o having n i jobs o tpe i in pocess in stage- is given b []: i n i whee n i ni i n i n i π, D.. D. Aveage invento o each poduct i is given b: I b pd b pd j pd, i bpd bpd j pi j bpd j j 0 j 0 which, upon simpliication, leads to:, D.3 b pd. D.4 I pd i bpd b, pd Given the smmet in the sstem, total aveage invento is obtained b multipling D4 b which ields 8. Aveage ode dela can be obtained b ist noting that the aveage numbe o bacodes o each poduct is given b: B pd, i b pd b p Then aveage ode dela, b vitue o Little s law, is given b n i ni b pd b pd i n i b n i ni b pd Λ b pd ni D.5 B pd, i bpd F pd, i bpd. D.6 λ Λ Given smmet, we have F pd b pd F pd,i b pd which leads to 9. i 4

43 E - Popeties o a Sstem with Patial Delaed Dieentiation Popet : The atio z pd / z d as. Poo: Recognizing that / - /{ - }, the atio z pd / z d Noting that ln[ α ˆ ] ln! ln ln! [ α ] ln[ ] [ α ] ln and z pd / z d can be ewitten as: ln! ln [ α ˆ ] ln[ ] [ α ] [ ] ln! ln[ α ] ln ae ve lage numbe when, we can ignoe integalit. Then, taing advantage o the x ln a x act that a e, it is staightowad to show that Let lim [ α ˆ ] [ α ] ln[ ] α ˆ. E. ln! α ln / lim z pd zd!. E. ln 43

44 44 [ ] [ ] [ ]. ˆ and ; ln ln, ln ˆ ln 0 α α α! Then, we can ewite ou limit as:. lim / lim 0 0 z z d pd E.3 In ode to show that that / lim z d z pd, it is suicient to show that. lim lim lim 0 0 Fist, note that lim 0 E.4 and [ ] [ ] [ ] α ln ln ˆ ln lim lim, which, upon appling L Hospital ule, educes to [ ] α ] /[ / ˆ ln lim lim. E.5 Similal, we can show that. lim E.6 Consequentl, we have } lim { lim 0 0 E.7 and } / lim { lim 0 0. E.8

45 Finall, we have lim ln lim. ln Since both the numeato and denominato ae zeo as., we appl L Hospital s ule to obtain z pd z d Hence, lim /. pd z d lim / lim. /[ ] E.9 Popet : z when min αˆ Λ / αˆ Λ. Poo: The esult ollows om the act that when αˆ Λ / αˆ Λ, we have α ˆ / Λ. In this case, a pue mae-to-ode sstem is optimal. Theeoe, we have z pd z 0. d Popet 3: z pd / z d / as min. Poo: B the limit min, we mean that min ε, with ε ve small. Thus, even though ε is appoaching zeo, we have > min. Theeoe, a pue mae-to-ode sstem is not easible i.e., this is the smallest o which a pue-mae-to-ode is not possible and we have b pd b d. I we substitute b pd in z pd, we obtain z pd and i we do the same o z d, we obtain z d. Taing the atio leads to the desied esult. Note that this esult is due to the integalit o b so that i we slightl inceased utilization above min, we move om a pue mae-to-ode to a sstem with an intemediate bue. The sstem with poducts is oced to eep a bue o one unit o each poduct while the sstem with pooling can get awa with just one bue o unit size. Again, this is due to the act that b is an intege and not a continuous vaiable. Popet 4: I [ max ], then αˆ Λ Poo: Since the optimal bue size is given b: z pd. 45

46 b pd i o equivalentl ln[ αˆ Λ ] b pd, ln [ αˆ Λ ] ln, ln E.0 [ ]. E. αˆ Λ Substituting b pd in z pd leads to z pd. Popet 5: Poo: Noting that z pd / z d ln[ ]/ln[ ] as αˆ 0. ln! [ α ] ln and! ln [ α ] [ ] ln ae ve lage numbe when α! 0, we can ignoe integalit. Then, the limit can be witten as [ α ˆ ] ln! α ln lim / lim! z pd zd!!. E. α 0 α 0 ln! α [ α ] ln[ ] Since the limit o both numeato and denominato is ininit, we appl L Hospital s ule to obtain ln ln[ ] lim / lim! z pd zd!. E.3 α 0 α 0 ln ln[ ] 46

47 Popet 6: Poo: The atio z pd / z d / when pd z d αˆ. Λ Λ b pd bd z / / when. Since b pd b d we have b pd bd when o equivalentl when ln [ αˆ Λ ] ln[ ] αˆ Λ. The desied esult ollows b noting that a b pd and b d ae non-inceasing in αˆ and b pd b b o α ˆ > / Λ. d 47