On postponement strategies for product families with multiple points of differentiation

On postponement strategies for product families

with multiple points of di�erentiation

AMIT GARG1 and CHRISTOPHER S. TANG2

1IBM, Thomas J. Watson Research Center, Yorktown Heights, NY 10598, USA2Anderson Graduate School of Management, University of California, Los Angeles, Los Angeles, CA 90024, USA

Received July 1995 and accepted January 1997

Several researchers have studied the bene®ts of product and process design that calls for delaying the di�erentiation of products.Previous research has focused on products having only one point of di�erentiation. However, in reality most product families haveseveral points of di�erentiation. In this paper, we develop two models to study products with more than one point of di�eren-tiation. In each model, we examine the bene®ts of delayed di�erentiation at each of these points, and derive the necessaryconditions when one type of delayed di�erentiation is more bene®cial than the other. Our analysis indicates that demand vari-abilities, correlations and the relative magnitudes of the lead times play an important role in determining which point of di�er-entiation should be delayed.

1. Introduction

To increase their market share, many companies haveextended their product lines. However, product line ex-tension has increased the complexity of the system be-cause of a proliferation of products. Product proliferationhas also made it very di�cult to obtain accurate forecastsfor each end-product. Competitive pressure to providehigh service levels, coupled with inaccuracies in theforecasts of individual products, have resulted in highinventory levels throughout the system. Companies arenow realizing that increased market share does notnecessarily translate into greater pro®tability ([1] and[2]).Product and process design is an e�ective means of

reducing the costs of complexity when dealing with highproduct variety. In this paper, we study the impact ofpostponement, a product and process design principle thatis increasingly being applied by some companies [3].Products belonging to a family usually share commoncomponents and/or processes. In their initial stages ofproduction, these products are in a common (generic)form that is customized progressively into speci®c end-products over several stages by inserting specializedcomponents and/or by performing special processes. Thestages at which di�erent customizations occur are calledpoints of product di�erentiation. The principle of post-ponement calls for redesigning the products and/or theprocesses to delay these points of di�erentiation.Delayed product di�erentiation bene®ts the manufac-

turers in two ways: it increases their ¯exibility by enabling

them to commit their work-in-process to a particular end-product at a later time, and it decreases their costs ofcomplexity by reducing the variety of components andprocesses within the system.This research is motivated by the authors' work in the

area of product and process design with di�erent com-panies. We found that in most instances a product familyhas many attributes that di�erentiate products belongingto it. Because the bene®ts of delaying product di�erenti-ation can be di�erent at each point of di�erentiation, it isimportant to study the impact of these decisions. Ouranalysis studies the impact of postponement on inven-tories, and would be useful in identifying the points ofdi�erentiation that one should focus on ®rst. To simplifythe exposition of the model, we analyze the di�erences inthe bene®ts due to postponements by focusing on productfamilies having only two points of di�erentiation. In thiscase we shall concentrate on the impact of postponementsat each point of di�erentiation on inventories. The workin this paper can easily be extended to product familieswith more than two points of di�erentiation.The impact of postponement in product families with

multiple points of di�erentiation is very relevant to mostpractitioners and to our knowledge has not been analyzedby researchers. Research in this area (for example, [4±6])has focused on models of product families with only onepoint of di�erentiation. These models cannot easily beextended to the case of multiple points of di�erentiation.In addition to developing models for examining the im-pact of delaying product di�erentiation at di�erentpoints, we contribute to the current research by extending

0740-817X Ó 1997 ``IIE''

IIE Transactions (1997) 29, 641±650

an inventory model of a two-stage distribution system toa more general case.In this paper we present two di�erent models. The ®rst

model is based on a centralized control policy in whichonly ®nished goods inventory is kept. The second modelis based on a decentralized control policy in which in-ventory is kept after each stage of the production process.For each model we examine conditions under which aparticular postponement strategy is more desirable andidentify factors that have signi®cant impact on the in-ventory level.This paper is organized as follows. In Section 2 we

de®ne the product structure and the production processon which our models are based. Section 3 presents themodel for a system operating under a centralized controlpolicy, and Section 4 describes the model for a systemoperating under a decentralized control policy. The paperis concluded in Section 5.

2. The product structure

Consider a production process that has three manufac-turing stages and two di�erentiating attributes, and hencetwo points of di�erentiation. At the ®rst stage all prod-ucts are in generic form. The family di�erentiation point(i.e., the ®rst point of di�erentiation) occurs at the be-ginning of the second stage, where speci®c componentsare used to di�erentiate the generic products into di�er-ent families. We de®ne the strategy of delaying di�eren-tiation at the family di�erentiation point as earlypostponement. The product di�erentiation point (i.e., thesecond point of di�erentiation) occurs at the beginning ofthe third stage, where speci®c components are used tocustomize semi®nished products of a speci®c family intodi�erent end-products of that family. We de®ne thestrategy of delaying di�erentiation at the product di�er-entiation point as late postponement. A schematic of thisproduction process appears in Fig. 1.We assume that the generic-form stage (stage 1) has a

lead time of T1 periods, the family stage (stage 2) has alead time of T2 periods, and the end-product stage (stage 3)has a lead time of T3 periods. We also assume that themanufacturing lead times for customizing the productsinto di�erent families are the same, and that the manu-facturing lead times for customizing di�erent end-prod-ucts of di�erent families are the same. However, it ispossible to extend our analysis to the case of unequal leadtimes for customizing products at each stage. Therefore,the total lead time for manufacturing a product isT1 � T2 � T3 periods.We shall consider the following product structure. Let

indices I and J refer to families of products, and let RI bea set of end-products that belongs to family I . We assumethat there are F families of products, each familyI ; I � 1; . . . ; F , comprising of a set of end-products RI .

Hence, the total number of end-products is equal toPFI�1 � kRIk. In this case, RI \ RJ � / for I 6� J . Also,

let indices i and j refer to each type of end-product. De-®ne nit to be the demand of end-product i in period t,where nit � N�li; r

2i �; 8 i 2 RI ; 8 I (nit � N�li; r2i � de-

notes that nit is a normally distributed random variatewith mean, li, and variance, r2i .). We assume that nit isindependent over time. Let qij be the coe�cient of cor-relation between the demands end-products i and j ineach time period. Throughout this paper, we shall use thefollowing notation. Denote the mean and the variance ofthe total demand of the products that belong to family Ito be lI �

Pi2RI

li and r2I �P

i;j2RI�r2i �

Pj6�i qijrirj�,

respectively. De®ne the mean and the variance of the totaldemand of all products to equal l �PI lI and

r2 �PI�r2I �P

J ;J 6�I

Pi2RI

Pj2RJ

qijrirj�, respectively.

The sum of the standard deviations of all end-productsthat belong to family I is denoted by rI , whererI �

Pi2RI

ri. Also, the sum of the standard deviations ofall families of products is denoted by ~r, where ~r �PI rI .On the basis of the de®nitions of r, ~r, and rI , it is easy toshow that

r � ~r �X

I

rI : �1�

3. The centralized control model

In a centrally controlled system, a single decision-makermonitors the in-process inventory and ®nished goodsinventories throughout the system and decides on orderquantity. In this model, we assume that only ®nishedgoods inventories are kept. This centralized control policyis best explained in the context of a distribution system.Suppose we view the family di�erentiation point as a`central depot' and view each product di�erentiationpoint as a `distribution center'. Then we can view the

Fig. 1. Schematic of the production process.

642 Garg and Tang

production process as a three-stage distribution system.This three-stage system has a central depot supplyingdistribution centers (d.c.'s) which in turn supply retailerswithin their territories. The depot and the d.c.'s do nothold any inventories. Inventories are held only as ®nishedgoods by the retailers. Such a distribution system arises insituations when the depot and the d.c.'s are the break-bulk and the consolidation points within the system. Thedepot and the d.c.'s also function as agencies for re-allocating stock to their downstream sites on the basis ofthe latest information.Under this centralized control policy, the production

system depicted in Fig. 1 can be modeled as a three-stagedistribution system. In this case, we can extend the two-stage inventory model developed by Eppen and Schrage[7] to determine the average inventory level of each end-product and the average inventory level of the entiresystem. Whereas Eppen and Schrage assumed that thedemands of the end-products are independent, our ex-tension deals with the case when the demands of the end-products are correlated. Our extension of Eppen andSchrage's model is based on the following assumptions.We assume that the system operates under a periodicreview, base-stock policy. Demands at the retailers arenormally distributed and are independent and identicallydistributed (i.i.d.) across periods. However, within aperiod, demands at di�erent retailers are correlated.There are no ®xed ordering costs; therefore, the retailers,the d.c.'s, and the depot place orders every period.We now need to specify how the depot allocates orders

to its d.c.'s and how the d.c.'s allocate orders to theirretailers. We assume that when orders arrive at the depotand at the d.c.'s, on the basis of their downstream in-ventory positions, the depot and the d.c.'s allocate thenewly arrived order so that the probability of stock-out ateach of their downstream sites is the same. This is calledan equal-fractile allocation policy. Our model uses theequal-fractile allocation policy at the depot and at the d.c.As in our model, Eppen and Schrage [7] and Erkip

et al. [8] have also used the equal-fractile allocation policyin their models for a two-stage system. However, thisallocation policy may not be achievable in every period,and at each site. The equal-fractile allocation assumptionhas been shown to be reasonable when the service level ishigh. We o�er two major studies that support this allo-cation policy. First, Eppen and Schrage [7] have gener-ated various simulation experiments that showed that theequal-fractile allocation policy is feasible most of the timewhen the service level is high. Next, Erkip et al. [8] ex-tended Eppen and Schrage's model of a two-stage dis-tribution system to the case in which the demands arecorrelated. On the basis of various simulation experi-ments, Erkip et al. have shown that, in addition, theequal-fractile allocation policy is feasible most of the timeeven when the underlying demands are correlated (posi-tively or negatively correlated).

However, the results of Eppen and Schrage and Erkipet al. for a two-stage distribution system cannot be ex-tended to systems with three or more stages unless someadditional assumptions are made. In our model, we makean additional assumption which we call the correlationassumption. This correlation assumption requires that

r2I =r2I � b2; �2�

8 I , where b is a constant. The rationale for the correla-tion assumption is as follows. From its de®nition, one cansee that r2I is the covariance of demands at d.c. I if thedemands at its retailers are perfectly correlated. There-fore, the ratio r2I =r

2I , or b2, can be viewed as an aggregate

measure of the level of correlation between the demandsof retailers within the territory of d.c. I . Our correlationassumption thus requires that the aggregate measure ofthe level of correlations between the demands at retailerswithin each territory be the same. This is similar to,though weaker than, the covariance assumption of Erkipet al. Notice that b2 � 1, and b decreases as qij increases;i.e., as the demands at the retailers become more highlycorrelated. We shall present three cases in which thecorrelation assumption holds in Section 3.2.The allocation and the correlation assumptions can be

used to derive expressions for the average on-hand in-ventory levels (see Appendix A). For a three-stage dis-tribution system with lead time vector �T1; T2; T3�, theaverage on-hand inventory at retailer i, I i, for i 2 RI , 8 I ,as observed in (A8), can be written as

I i � 12li � zri �T3 � 1� � T2

r2Ir2I� T1

r2

~r2r2Ir2I

� �12

; �3�

where z � Uÿ1�p=�p � h��. Notice that z corresponds tothe safety factor that is based on the service requirements.We now analyze the impact of early and late postpone-ments on a system operating under this policy.

3.1. Early and late postponements

Rede®ne the average on-hand inventory of product i, I i,as I i�T1; T2; T3�, 8 i 2 RI ;8 I . The expression for this in-ventory level is given by equation (3). Assume that thetotal lead time to manufacture all the products is in-variant, and is equal to T1 � T2 � T3. Incremental earlypostponement will result in a lead-time vector of�T1 � 1; T2 ÿ 1; T3�, whereas incremental late postpone-ment will result in the lead-time vector of �T1; T2 � 1;T3 ÿ 1�. In this paper we shall use the terms incrementalearly (late) postponement and early (late) postponementinterchangeably. De®ne D I ei to be the marginal bene®t ofearly postponement at retailer i; i 2 RI , where D I ei �I i�T1 � 1; T2 ÿ 1; T3� ÿ I i�T1; T2; T3�. De®ne D I e as thetotal marginal bene®t of early postponement, whereD I e �PI

Pi2RI

D I ei . Similarly, de®ne D I li to be themarginal bene®t of late postponement at retailer i, where

Postponement strategies for product families 643

D I il � I i�T1; T2 � 1; T3 ÿ 1� ÿ I i�T1; T2; T3�. Finally, let

D I l be the total marginal bene®t of late postponement.By comparing the average inventory level in (3) and thecorresponding average inventory level associated withearly postponement and late postponement, one canshow that both incremental early and late postponementsresult in inventory reduction, as follows.

Proposition 1. Assuming that the total lead time to manu-facture the end-products is unchanged as a result of post-ponement decisions, then the following are true:

(i) D I e � 0; and(ii) D I l � 0:

Proof: To prove (i), note that

D I ei � I i�T1 � 1; T2 ÿ 1; T3� ÿ I i�T1; T2; T3�: �4�D I ei � 0 is equivalent to

T3 � 1� �T2 ÿ 1� r2I

r2I� �T1 � 1� r

2I

r2I

r2

~r2

� �12

� T3 � 1� T2r2Ir2I� T1

r2Ir2I

r2

~r2

� �12

: �5�

Simplify (5), and we can see that to prove (i) it is su�cientto show that r2 � ~r2. The proof of (i) follows from thede®nitions of r2 and ~r2. Next, note that

D I li � I i�T1; T2 � 1; T3 ÿ 1� ÿ I i�T1; T2; T3�: �6�and D I li � 0 is equivalent to

T3 � �T2 � 1� r2I

r2I� T1

r2Ir2I

r2

~r2

� �12

� T3 � 1� T2r2Ir2I� T1

r2Ir2I

r2

~r2

� �12

: �7�

Equation (7) can be simpli®ed so that proving (ii) isequivalent to showing that r2I � r2I ; 8 I . This follows fromthe de®nitions of r2I and r2I . j

Proposition 1 shows the inventory savings realizedthrough delaying di�erentiation by one time period.However, it can easily be shown that these savings in-crease as we increase the amount of time by which thepoint of di�erentiation is delayed. We omit the details.By comparing the inventory levels associated with in-

cremental early and late postponement strategies, we es-tablish the condition under which incremental latepostponement is preferable. This condition is presented inthe following proposition.

Proposition 2. D I l ÿ D I e � 0; i.e., incremental late post-ponement is preferred, if

2ÿ r=~r� �2h i

� b2: �8�

Proof: Note that D I l ÿ D I e � 0 if D I li ÿ D I ei � 0;8 i 2 RI ; 8 I . It follows from (5), (7), and (2) that we cansimplify D I li ÿ D I ei to

2ÿ r=~r� �2h i

� rI=rI� �2� b2: j

Through Proposition 2 we have shown that incrementallate postponement is preferable to incremental earlypostponement if the condition in (8) is satis®ed. To in-terpret Proposition 2, notice from (2) that b increases asqij decreases. This observation implies that condition (8)is more likely to be satis®ed when the demands are `morenegatively' correlated. In this case we can conclude thatlate postponement is preferable to early postponementwhen the demands are more negatively correlated.Therefore, the condition in (8) depends on the variabil-ities and the correlations of the demands. This resultsuggests that it is important to understand the nature ofdemand variabilities while determining a postponementstrategy.

Numerical example 1We now present a numerical example to illustrate theresults of Proposition 2. Consider a product familycomprising two subfamilies denoted by J � I, II respec-tively. Each subfamily consists of two end-products 1 and2. Assume that the demand for the end-products thatbelong to subfamily II is a times the correspondingdemands for products belonging to subfamily I. Withoutloss of generality, let a � 1.Let nJ ;i be the demand of product i belonging to sub-

family J , for i � 1; 2. In this case, the above scenario canbe represented mathematically as follows: let nI;1 �N�l1; r21�, nI;2 � N�l2; r22�, Cov�nI;1; nI;2� � qr1r2, nII;1�N�al1; a

2r21�, nII;2 � N�al2; a2r22�, while Cov�nII;1;

nII;2� �a2qr1r2. All other covariances are 0.For this scenario it is easy to show that the covariance

assumption (2) is valid. In this case we can use Proposi-tion 2 to express the condition under which late post-ponement is preferred to early postponement as:

2ÿ 1� a2

�1� a�2" #

� �r1 � r2�2r21 � r22 � 2qr1r2

:

From the above equation we can see that the right-handside increases as q decreases. Hence, late postponementyields greater bene®ts than early postponement as thecorrelation between the demands of the two end-productsdecreases.From (3), the average system-wide inventory per period

can be expressed as

I � 12l� z �T3 � 1�b2 � T2

ÿ �~r2 � T1r2

� �12; �9�

where z � Uÿ1�p=�p � h��. It follows from the de®nitionsof D I e and D I l that the relative inventory savings due to

644 Garg and Tang

early and late postponements can be measured in terms ofÿD I e=z and ÿD I l=z respectively, where

�ÿD I e=z� � �T3 � 1�b2 � T2ÿ �

~r2 � T1r2� �1

2

ÿ �T3 � 1�b2 � �T2 ÿ 1�ÿ �~r2 � �T1 � 1�r2� �1

2;

�10�

�ÿD I l=z� � �T3 � 1�b2 � T2ÿ �

~r2 � T1r2� �1

2

ÿ T3b2 � �T2 � 1�ÿ �

~r2 � T1r2� �1

2: �11�The sample data set used to construct the numerical ex-ample assumes that l1 � l2 � 1, r1 � 0:35, r2 � 0:4,a � 0:5, and Ti � 5, i � 1; . . . ; 3. By varying q from ÿ1through 1, we compute the relative inventory savings givenby (10) and (11). Fig. 2 summarizes the relative savingsassociated with early and late postponements. Notice fromthis ®gure that both early and late postponements result ininventory savings; however, early postponement ispreferred when the demand correlation, q, is high. As thedemand correlation decreases, condition (8) is more likelyto make late postponement more desirable. Theseobservations verify the results of Proposition 2.

3.2. Special cases

Proposition 2 can be used to answer a question of interestto many practitioners. Most product lines have many

points of di�erentiation and one would like to determinedelaying di�erentiation at which level would result inlargest reduction of ®nished goods inventories. To gainfurther insights into this problem, we now present threeexamples in which the correlation assumption (2) holds.

Example 1 (Two market segments)Consider the case of a company manufacturing twofamilies of products for two market segments, I and II.These market segments could comprise the private labeland major brand products. The products are customizedinto families for each market segment at the factory be-fore shipping to the distribution center. Products be-longing to each family are customized into two end-products, products 1 and 2 at the distribution center.Assume that the size of the market for each end-productin segment II is a �a � 1� times the size of the market forthat product in segment I. Fig. 3 depicts this scenario.Let nJ ;i be the demand of product i distributed through

d.c. J , for i � 1; 2 and for J � I, II. In this case, the abovescenario can be represented mathematically as follows: letnI;1 � N�l1; r21�, nI;2 � N�l2; r22�, Cov�nI;1; nI;2� � qr1r2,nII;1 � N�al1; a

2r21�, nII;2 � N�al2; a2r22�, and Cov

�nII;1; nII;2� � a2qr1r2. All other covariances are 0.In this scenario, early postponement can occur if we

standardize some family-speci®c components, whereaslate postponement can be e�ected by standardizing someproduct-speci®c components.

Fig. 2. Bene®ts of early/late postponements.


For this example we can derive expressions for rI , rI ,8 I , and show that the correlation assumption, (2), isvalid.From (8), and the expressions for r2 and ~r2, the sim-

pli®ed condition when late postponement is more bene-®cial than early postponement can be written as

2ÿ 1� a2

�1� a�2" #

� �r1 � r2�2r21 � r22 � 2qr1r2

: �12�

From the above equation we can see that the right-handside increases as q decreases. Hence, late postponementyields greater bene®ts than early postponement as thecorrelation between the demands of the two productsdecreases. The left-hand side increases as the ratio of theregional market sizes increases. Therefore, early post-ponement becomes more attractive as the two marketsegments become more comparable in size.

Example 2. (PC- and Macintosh-compatibleprinter design)Let us now revisit the example of the printer manufac-turer. There are two product families: PC-compatibleand Macintosh (Mac)-compatible printers. Within eachfamily, a printer can be color or black-and-white. Let aIbe the proportion of the demand of PC-compatibleprinters due to color printers. Similarly, let aII be theproportion of the demand of Mac-compatible printersdue to color printers. Fig. 4 depicts this scenario.The mathematical representation for this scenario is as

follows: let EI , for I � PC, Mac denote printer demandper period. In this case, EPC � N�m1; s21�;EMac �N�m2; s22�, and Cov�EPC;EMac� � qs1s2. Therefore, aIEPC

and �1ÿ aI�EPC are the random variables denoting thedemands of color and black-and-white PC-compatibleprinters, respectively. Similarly, aIIEMac and �1ÿ aII�EMac

are the random variables denoting the demands of colorand black-and-white Mac-compatible printers, respec-

tively. We can easily verify that the correlation assump-tion, (2), is satis®ed for this scenario.Observe that r2 � s21 � s22 � 2qs1s2 and ~r2 � �s1 � s2�2.

It follows from (8) of Proposition 2 that late postpone-ment is preferable to early postponement when

2ÿ s21 � s22 � 2qs1s2�s1 � s2�2

" #� 1:

However, this is impossible because the left-hand side isalways greater than 1. In this case, early postponement isalways preferable.

Example 3. (watch manufacturing)Finally, consider the case of a watch manufacturer sellingsport and dress watches for men and women. The processstarts with a generic mechanism that is customized intosport and dress models by adding certain model-speci®ccomponents such as integrated circuits. Each model isthen customized into men's or women's watches by as-sembling the mechanism into di�erent types of casing. Inthis case, early postponement can be achieved when somemodel-speci®c components are standardized, whereas latepostponement can be achieved by standardizing the cas-ings for men's and women's watches.We assume that the demands for men's and women's

watches are independent, whereas there are correlationsin the demands of sport and dress watches. Assume thatthe size of the men's market is a�a � 1� times the size ofthe women's market. Fig. 5 depicts this scenario.The mathematical representation of the scenario out-

lined above is as follows: let nI ;i be the demand of productfor gender i and style I , for i � w, m (denoting womenand men respectively) and for I � Sport, Dress. LetnSport;w � N�l1; r21�, nSport;m � N�al1; a

2r21�, nDress;w �N�l2;r22�, nDress;m � N�al2; a

2r22�, Cov�nSport;w; nDress;w�� qr1r2 and Cov�nSport;m; nDress;m� � a2qr1r2, all othercovariances being 0.

Fig. 3. Two-market segment scenario.Fig. 4. PC- and Mac-compatible printer design scenario.

646 Garg and Tang

Again, one can easily verify that the correlation as-sumption, (2), is satis®ed for this senario. Notice thatr2 � �1� a2��r21 � r22 � 2qr1r2� and ~r2 � �1� a2� �r1�r2�2. It follows from (8) of Proposition 2 that late post-ponement is preferable to early postponement when

2ÿ r21 � r22 � 2qr1r2�r1 � r2�2

" #� �1� a�2

1� a2: �13�

From the above equation we can see that the left-handside increases as q decreases. Hence, early postponementis preferred when demands between sport and dresswatches are more negatively correlated. The right-handside increases in a. Therefore, late postponement becomesmore attractive as the markets for men's and women'swatches become more comparable in size.

4. The decentralized control policy model

Last section described a system operating under a cen-tralized control policy in which only ®nished goods in-ventories are kept. This section focuses on a systemoperating under a decentralized control policy in whichboth work-in-process inventories as well as ®nished goodsinventories are stored at the corresponding bu�ers. Spe-ci®cally, under the decentralized control policy, eachproduction stage is responsible for planning its ownproduction quantity and order quantity at the beginningof each period. To perform its planning function, theinformation that each site needs are the lead time of itsimmediate predecessor, and the distribution of the de-mands and their correlations from its immediate succes-sor(s). Because each production stage is liable for itsinventory holding and back-order costs, it utilizes theabove information to optimize its base-stock level inde-pendently of any other site. Therefore, we call this type ofoperating policy a decentralized control policy. In thiscase, the optimal system-wide inventory cost is just the

sum of the optimal inventory costs at each site in thesystem.Under this decentralized control policy, we assume that

every inventory stockpile is managed under a periodicreview pull system. This implies that the demand of theend-products are pulled through the system at eachperiod. Suppose that we impose a very high service-levelrequirement such that the stock-out probability at each ofthe bu�ers is very small. (High service level is desirablewhen the back-order cost is high or when the manage-ment is concerned about customer service). In this case,the entire production process can be decomposed intosingle-stage systems. Each single-stage system has its ownbu�er that keeps the output inventory of the corre-sponding stage. Although the production planning deci-sion is controlled locally at each production stage, thecompany would be responsible for the inventory in theentire system. In addition, the company would be inter-ested in examining the impact of early and late post-ponement strategies on inventories in the entire system.Let z be the safety factor that corresponds to the servicelevel. Before any form of postponement has taken place,it is easy to check that the average total inventory for theentire system can be expressed as

I�T1; T2; T3� � 32l� z

��T1 � 1

pr� z

��T2 � 1

p~r

� z��T3 � 1

p XF

I�1rI : �14�

4.1. Early and late postponement decisions

The average total inventory per period resulting fromincremental early postponement is equal to

I�T1 � 1; T2 ÿ 1; T3� � 32l� z

��T1 � 2

p� �r� z

��T2p

~r

� z��T3 � 1

p� �XF

I�1rI ; �15�

and as a result of incremental late postponement, theaverage total inventory per period is

I�T1; T2 � 1; T3 ÿ 1� � 32l� z

��T1p

r� z��T2 � 2

p� �~r

�z��T3

p XF

I�1rI : �16�

By comparing the average total inventory before post-ponement, early postponement, and late postponement(i.e., I�T1; T2; T3�; I�T1 � 1; T2 ÿ 1; T3�; and I�T1; T2 � 1;T3 ÿ 1�, respectively), we present the conditions underwhich the average total inventory would decrease as aresult of either early or late postponement in the follow-ing proposition. In addition, we develop necessary con-ditions for incremental early (or late) postponementstrategy to be optimal.

Fig. 5. Watch-manufacturing scenario.


Proposition 3. By comparing the total average inventoryper period, one can show the following statements:

(i) If T2 < T1, then I�T1 � 1;T2 ÿ 1; T3� � I�T1; T2; T3�;i.e., early postponement is bene®cial.

(ii) If T3 < T2, thenI�T1; T2 � 1; T3 ÿ 1� �I�T1; T2; T3�;i.e., late postponement is bene®cial.

(iii) If T2 � minfT1; T3g, then early postponement strat-egy is optimal.

(iv) If T2 � maxfT1; T3g, then late postponement str-ategy is optimal.

Proof: First, let us compare I�T1 � 1; T2 ÿ 1; T3� andI�T1; T2; T3�. I�T1 � 1; T2 ÿ 1; T3� � I�T1; T2; T3� if T2 �T1 � 1. This is because r � ~r (from (1)) and because thefunction

��x� 1p ÿ ��

xp

is decreasing in x. We can use thesame approach to prove the second statement. To provethe third statement, let us compare the average total in-ventory associated with early and late postponementstrategies; i.e., we compare I�T1 � 1; T2 ÿ 1; T3� withI�T1; T2 � 1; T3 ÿ 1�. In this case, it is easy to show thatearly postponement dominates late postponement strat-egy if the following inequality holds;

~r��T2 � 2

pÿ ��

T2p� �

� r��T1 � 2

pÿ

��T1 � 1

p� ��XF

I�1rI

��T3 � 1

pÿ

��T3

p� �: �17�

Recall from (1) that r � ~r �PFI�1 rI , and that

��x� 1p

ÿ ��xp

is decreasing in x; it is enough to show that theabove inequality holds if T2 is su�ciently smaller than T1and T3; i.e., when T2 � minfT1; T3g. In this case, the totalaverage inventory associated with early postponement islower than that of the late postponement if T2 �minfT1; T3g. Combining this observation with the ®rststatement of the proposition �i.e., I�T1; T2; T3� >I�T1 � 1; T2 ÿ 1; T3� when T2 < T1�, we can conclude thatthe early postponement strategy is optimal. For thefourth statement, we can apply the same approach toshow that the late postponement strategy is optimal ifT2 � maxfT1; T3g. j

Proposition 3 has the following interpretation. The ®rsttwo statements provide simple conditions for determin-ing whether it is bene®cial to postpone product di�er-entiation or not. These two statements imply that bothincremental early and late postponement will be bene-®cial if T1 > T2 > T3. The third and the fourth state-ments provide necessary conditions under which eitherearly or late postponement is optimal. Speci®cally, thethird statement states that early postponement is opti-mal when T2 is su�ciently smaller than T1 and T3. Fi-nally, the fourth statement suggests that the latepostponement is optimal when T2 is su�ciently largerthan T1 and T3. In summary, the above propositionimplies that the lead time of each stage plays an im-

portant role in determining which type of postponementwe should focus on.

Numerical example 2We now present a numerical example to illustrate theresults of Proposition 3. We construct an example basedon the scenario described in numerical example 1. We usethe same data set as given in numerical example 1, exceptthat we set q � 0:5; T1 � T3 � 5, and we shall vary T2from 1 through 20.For this scenario, the expressions for variances are as

follows. ~r2 � �1� a�2�r21� r22 � 2qr1r2�; r2 � �1� a2��r21 � r22 � 2qr1r2�; and

PI rI � �1� a��r1 � r2�.

It follows from the de®nitions of D I e and D I l that therelative inventory savings due to early and late post-ponements can be measured in terms of ÿD I e=z andÿD I l=z, respectively. By substituting the above expres-sions into (14)±(16), we get

ÿD I e=z� � � r��T1 � 1

pÿ

��T1 � 2

p� �� ~r

��T2 � 1

pÿ ��

T2p� �

;

�18��ÿD I l=z� � ~r

��T2 � 1

pÿ

��T2 � 2

p� ��X

I

rI

��T3 � 1

pÿ

��T3

p� �: �19�

Fig. 6 examines the impact of varying T2 on the relativeinventory savings associated with early and late post-ponements given by (18) and (19). Notice from this ®gurethat early postponement is not bene®cial (i.e., has nega-tive relative inventory savings) when T2 is small. Thisveri®es the ®rst two statements of Proposition 3. Next,observe that late postponement is optimal when T2 islarger than T1 and T3. This veri®es the last two statementsof Proposition 3. The sample data for these plots assumethat a � 0:5; q � 0:5; and T1 � T3 � 5.

5. Concluding remarks

We have developed two models for examining the im-pact of early and late postponement strategies on in-ventories. To obtain some basic insights related to earlyand late postponements, we have made various as-sumptions that enable us to simplify our analysis. Thus,the models presented in this paper have the followinglimitations:

1. Demands at the retailers are normally distributedand are independent and identically distributed. In real-ity, the demands could be correlated over time.2. The correlation assumption (i.e., condition (2))

holds. This requires that the aggregate measure of thelevel of correlations between the demands at retailerswithin each territory be the same. In general, the corre-lation assumption may not hold.

648 Garg and Tang

3. There is ample production capacity at each stage.One needs to address the capacity issue for system withhigh utilization.4. The total manufacturing lead time (i.e., T1 � T2 � T3)

is invariant and would not be a�ected by the postpone-ment decisions. In general, depending on the design of theproduct and the production process, lead times at eachstage could be a�ected by postponement decisions. Thishas not been captured in the current model.5. Our analyses have focused solely on the impact of

early and late postponements on inventories. There is aneed for a cost model for evaluating di�erent postpone-ment strategies that incorporate other costs such as de-sign cost, production cost, transportation cost, anddistribution cost.

References

[1] Ho, T. and Tang, C. (1995) Product line extension: the right way?.Working paper, Anderson Graduate School of Management,UCLA.

[2] Simison, R.L. and Suris, O. (1995) Racing Cars Ð Alex Trotman'sgoal: To make Ford no.1 in world auto sales; but some fear theChairman may slight pro®tability; GM says it isn't worried; the artof ®nding a balance. The Wall Street Journal (1995) July 18, p. A1.

[3] Lee, H.L. (1993) Design for supply chain management: methodsand examples, in Perspectives in Operations Management Sarin, R.(ed.), Kluwer, Boston, MA. pp. 45±66.

[4] Garg, A. (1995) Product and process design strategies for e�ectivesupply chain management. Ph.D. Dissertation, Stanford Univer-sity.

[5] Lee, H.L. (1996) E�ective management of inventory and servicethrough product and process redesign. Operations Research, 44(1),151±159.

[6] Lee, H.L. and Tang, C. (1997) Modeling the costs and bene®ts ofdelayed product di�erentiation. Management Science, 43(1), 40±53.

[7] Eppen, G. and Schrage, L. (1981) Centralized ordering policies ina multi-warehouse system with lead times and random demand, inMulti-Level Production/Inventory Systems: Theory and Practice,Schwarz, L.B. (ed.), North-Holland, Amsterdam, 51±69.

[8] Erkip, N., Hausman, W. and Nahmias, S. (1990) Optimal cen-tralized ordering policies in multi-echelon inventory systems withcorrelated demands. Management Science, 36, 381±392.

[9] Lee, H.L. (1992) Single and Multiple Location StochasticInventory Models, Lecture Notes, Department of IndustrialEngineering-Engineering Management, Stanford University,Stanford, CA.

Appendix A. Model for three-stage distribution systemwith centralized ordering

Let S be the system-wide order-up-to level. In period 1the depot places an order with the external supplier tobring the system-wide inventory position up to S. Thisorder will arrive at the depot in period T1. Let YI be theechelon inventory position at d.c. I after the depot makesthe allocation decision in period T1. (We omit the time

Fig. 6. E�ect of lead times on early and late postponements in decentralized systems.


subscripts from the notations for ease of exposition.) YI isrequired to provide protection against demands in peri-ods T1 through T1 � T2 � T3 � 1 at d.c. I . Therefore, wecan express YI as

YI � lI�T2 � T3 � 1� � KIrI

��T2 � T3 � 1

p; 8 I : �A1�

Because an equal-fractile allocation policy is used at thedepot, KI � K;8 I . Let V be the system-wide demands inperiods 1 through T1. Therefore, V � N�lT1; T1r2�. Thetotal stock available at the depot for allocation to thed.c.'s at the beginning of period T1 is S ÿ V �PI YI .Substituting this in (A1) and simplifying, we get

YI � �T2 � T3 � 1�lI � �S ÿ V ÿ �T2 � T3 � 1�l�rI=~r; 8 I ;

�A2�where r2I �

Pi;j2RI

ÿr2i �

Pj6�i qijrirj

�and ~r �PF

I rI :The stock allocated to d.c. I in period T1 arrives at the

d.c. in period T1 � T2. Let vI be the total demand observedat all the retailers supplied by d.c. I from periods T1 � 1through T1 � T2. Therefore, vI � N�lI T2; T2r

2I �. Let xi be

the inventory position at retailer i; i 2 RI , after the d.c.makes its allocation decision at the beginning of periodT1 � T2. Because xi is required to protect retailer i againstdemands in periods T1 � T2 through T1 � T2 � T3 � 1, wecan express it as

xi � li�T3 � 1� � kiri

��T3 � 1

p; 8 i 2 RI : �A3�

The allocation made by the depot to the d.c.'s is ®xed,therefore the total quantity available at d.c. I at the be-ginning of period T1 � T2 for allocation to its retailers isYI ÿ vI �

Pi2RI

xi. We can use (A2) and the above-men-tioned fact to rewrite (A3) as

xi � li�T3 �1� � lI T2 � �S ÿ V ÿ lh

��T2 � T3 � 1�� rI

~rÿ vI

i ri

�rI: �A4�

De®ne yi to be the inventory position at retailer i at thebeginning of period T1 � T2 � T3 after the order that wasdue that period has arrived but before any demands areobserved, and di to be the total demands at retailer i inperiods T1 � T2 through T1 � T2 � T3. Since yi � xi ÿ di,(A4) enables us to express yi as follows:

yi � li�T3 � 1� � lI T2 � �S ÿ l�T2 � T3 � 1�� rI

~r

h i ri

�rI

ÿ di � vIri

�rI� V

rI

~rri

�rI

� �: �A5�

De®ne

Si � li�T3 � 1� � lI T2 � �S ÿ l�T2 � T3 � 1�� rI

~r

h i ri

�rIand

gi � di � vIri

�rI� V

rI

~rri

�rI� ni;T1�T2�T3�1:

Si is a constant, and gi is a random variable such that

gi � N li�T3 � 1� � lIT2ri

�rI� lT1

rI

~rri

�rI

�; �T3 � 1�r2i

�T2r2Ir2i�r2I� T1r2

r2I~r2

r2i�r2I

�:

Ordering and allocation decisions made at the depot inperiod 1 a�ect costs at the retailers in period T1 � T2�T3 � 1 only, because orders are placed every period.Therefore, the total costs at the retailers can be minimizedby minimizing the single-period inventory holding andshortage costs. This expected shortage and penalty costfunction for retailer i; i 2 RI , is given by

hE�Si ÿ gi�� pE�gi ÿ Si��; �A6�where h and p are the unit holding and penalty costs perperiod. Minimization of this cost function results in asolution similar to the familiar newsboy problem:

USi ÿ li�T3 � 1� ÿ lI T2

ri�rIÿ lT1 rI

~rri�rI

ri �T3 � 1� � T2r2I�r2I� T1 r2

~r2r2I�r2I

h i12

8><>:9>=>; � p

p � h; �A7�

where U�� is the cumulative distribution function of astandard normal random variate. Alternatively, we canexpress Si, for i 2 RI , as

Si � li�T3 � 1� � lI T2ri

rI� lT1

rI

~rri

rI

� zri �T3 � 1� � T2r2Ir2I� T1

r2

~r2r2Ir2I

� �12

; �A8�

where z � Uÿ1�p=�p � h��. Because the correlation as-sumption holds, i.e., rI=rI � b;8 I , and S �PF

I

Pi2RI

Si,we can express S as

S�l�T1 � T2 � T3 � 1� � z �T3 � 1�b2 � T2ÿ �

~r2 � T1r2� �1

2:

�A9�

Biographies

Amit Garg is a Research Sta� Member at IBM's T.J. Watson ResearchCenter, Yorktown Heights, NY. His current research interests includestochastic inventory systems, supply chain management, product andprocess design, and promotional pricing for items in retail. He alsoworked at Bender Management Consultants, Arlington, VA, where hedeveloped scheduling and logistics applications for clients in the paperindustry. He has a Ph.D. in Industrial Engineering-Engineering Man-agement from Stanford University, an M.S. in Industrial Engineeringfrom SUNY at Bu�alo, and a B.Tech. in Mechanical Engineering fromthe Indian Institute of Technology±Kharagpur.

Christopher Tang is Professor of Decision Sciences, and Professor ofOperations and Technology Management, at the Anderson GraduateSchool of Management of UCLA. He is presently the co-faculty di-rector of the Executive Program at UCLA. He has been the co-chair ofthe Department of Decision Sciences at UCLA, and the Director of theCenter for Technology Management at UCLA. He has served as theDepartmental Editor for IIE Transactions.

650 Garg and Tang

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