Make-to-order, Make-to-stock, or Delay Product Differentiation? – ACommon Framework for Modeling and Analysis

Diwakar Gupta1 • Saifallah BenjaafarUniversity of Minnesota

Department of Mechanical EngineeringMinneapolis, MN 55455

Second revision, October 2000

Abstract

Widespread increase in product variety and the simultaneous emphasis on shorter order-deliverytimes, and lower costs, is forcing manufacturing firms to consider alternatives to both make-to-stock and make-to-order modes of production. One such alternative is delayed productdifferentiation. It is a hybrid strategy that reconciles the needs of high variety and quickresponse. A common product platform is built to stock and then differentiated into differentproducts once demand is known. In this paper, we examine the costs and benefits of delayeddifferentiation in environments where capacity is finite, and order lead-times are stochastic andload-dependent. We show that the advantage of delayed differentiation is affected by factorssuch as capacity utilization, product variety, and order delay requirement. In particular, thedesirability of delayed differentiation is found to be sensitive to the utilization of the productionsystem; its benefits diminishing with increasing utilization. In situations where order-delayrequirements are tight, we show that delaying differentiation can be more expensive thanproducing in a make-to-stock fashion. However, when utilization is in the mid-range, the benefitsof delayed differentiation can be significant. As intuition suggests, the benefits of delayeddifferentiation are also significant when product variety is high. In fact, the marginal benefitfrom delayed differentiation is, by and large, increasing in the number of products. If the point ofdifferentiation can be affected, we find that, where feasible, differentiating early by postponingmore work content until demand is realized is desirable, leading ultimately to a make-to-ordersystem. If the cheaper early differentiation is not possible due to tight delivery requirements, weshow that we should assign proportionally more capacity to the make-to-order stage.

1 Corresponding author. Telephone: (612) 625-1810, Fax: (612) 625-6069, Email: guptad@me.umn.edu.

2

1. Introduction

In today's business environment, a manufacturing firm that has the ability to fulfill customer

orders quickly, as well as offer a large variety of products, enjoys a competitive advantage.

However, the objectives of high product-variety and quick-response time are often conflicting [8,

15, 17]. Traditionally, industries where quick-response time is important have focused on

producing a limited portfolio of products, which are stocked ahead of demand and shipped

immediately upon receipt of a customer order. Producing to stock becomes costly and

unpractical when the number of products is high. It is also risky when demand is stochastic or

negatively correlated among products. Traditionally, a significant increase in product variety has

meant shifting from a make-to-stock to a make-to-order mode of production, where production is

not initiated until a customer order is received. While this strategy eliminates finished inventories

and reduces firm’s exposure to financial risk, it usually spells long customer lead times and large

order backlogs. An alternative to both make-to-order and make-to-stock that has recently gained

in popularity is delayed differentiation [7, 16, 21]. Delayed differentiation is a hybrid strategy

where a common product platform is built to stock. This common platform is differentiated, by

assigning to it certain product-specific features and components, only after demand is realized.

Hence, manufacturing occurs in two stages, a make-to-stock stage where the single

undifferentiated platform is produced and stocked, and a make-to-order stage where product

differentiation takes place in response to specific customer orders.

Delayed differentiation carries several benefits. Maintaining stocks of semi-finished goods

reduces order fulfillment delay relative to a pure make-to-order system. It also reduces risk

associated with holding finished goods inventory for which demand might not materialize. This

risk-pooling has long been known in the inventory literature to lower overall inventories while

still guaranteeing the same service level [6]. By holding semi-finished instead of finished-goods

inventories, delayed differentiation also reduces unit-holding costs, since less value is associated

with the stocked inventory. There is also the benefit of learning, realized from having better

demand information before committing generic semi-finished products to unique end-products.

Learning can arise when demand in consecutive periods is correlated [1]. Additional benefits

from delayed differentiation include a significant streamlining of the make-to-stock segment of

the manufacturing process and simplification of production scheduling, sequencing, and raw

material purchasing. However, implementing delayed differentiation also carries costs. These

3

include the costs of product and process redesign necessary to increase product commonality, use

of extra materials (when common designs are made possible by having redundant or more

expensive parts), and less efficient processing (when common processing leads to the use of a

less specialized production equipment or greater yield losses). Therefore, in assessing the value

of a delayed differentiation strategy, these costs have to be carefully balanced against the

associated benefits.

Recent literature showcases several examples, in industries ranging from consumer

electronics to apparels to automotive manufacturing, where delayed differentiation has been

successfully used to control inventory costs while maintaining high service levels [4, 7, 8, 10,

21]. For example, Feitzinger and Lee [7] describe how delayed differentiation, through either

operation reversal or component sharing, helped the Hewlett-Packard printer division customize

its products more cost effectively. Swaminthan and Tayur [21] describe how IBM exploited

component commonalities in personal computers to design a common platform, or a vanilla box,

from which end-products are differentiated based on customer orders. Fisher et al. [8] discuss

how component sharing is used by several major automotive companies to standardize braking

systems. Bruce [4] reports on the well known case of Benetton who, by reversing the order in

which yarn is dyed and knitted, was able to successfully delay color selection until the season’s

fashion preference become more established. Finally, Graman and Magazine [10] describe how

delayed packaging is used by a manufacturer of household cleaning products to reduce its

finished goods inventory and improve service levels.

Several of these studies also present quantitative models that assess the costs and benefits of

delayed differentiation (see, for example, Lee [15], Lee and Tang [17], Garg and Tang [9],

Swaminathan and Tayur [20, 21], and Aviv and Federgruen [1], among others). Most of these

models focus on the tradeoff between the benefits of inventory pooling (via statistical economies

or from having a shared inventory location), learning benefits, and product/process redesign

costs. The majority use order-up-to-level inventory models in which order lead times are not

affected by either order size, the number of pending orders, or the number of end-products. If

limited production capacity is modeled, order lead times are assumed constant which ignores any

congestion at the production facility. In doing so, these models ignore the interaction between

utilization of the production facility, processing time variability, and order delays.

In this paper, we extend the existing literature by explicitly modeling the effect of congestion

at both the make-to-stock and make-to-order stages of the production process. We show that

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there is a complex relationship between capacity utilization, order delay, optimal inventory

levels, and the degree to which differentiation should be delayed. In particular, we show that the

benefits of delayed differentiation are sensitive to the utilization levels at the production facility

with delayed differentiation having little value when utilization is high. Similarly, we show that

delayed differentiation carries little value when order delay requirements are tight. However, in

some other cases, the benefits of delayed differentiation can be significant. Notably, this happens

when utilization is in the mid-range, or product variety is high. In fact, the marginal benefit from

delayed differentiation is generally increasing with the number of products. If there is flexibility

in choosing the point of differentiation, it is clear that a firm should try to move as much toward

a make-to-order system as possible. However, the degree to which a firm should postpone work

until after demand is known is largely driven by the tightness of the order-delay requirement. In

cases, where there is flexibility in assigning capacity, we find that assigning proportionally more

capacity to the make-to-order stage tends to be more desirable.

Throughout the analysis, we use our models to answer several fundamental questions that

arise when implementing delayed differentiation (DD). For example, under what conditions is

DD superior to a pure make-to-stock strategy? How is the value of DD affected by system

parameters such as capacity utilization and number of products? How should production capacity

be allocated between the differentiated and undifferentiated stages? How much undifferentiated

inventory should be kept between the two stages? What is the optimal point of differentiation,

given service level requirements and possible product/process redesign costs? The models we

present are intended for use as strategic tools that allow operations managers to rapidly examine

key tradeoffs from different DD configurations. Consistent with this spirit, we deliberately keep

the models simple and relevant for gaining managerial insights.

The remainder of this article is organized as follows. In section 2, we discuss our target

application and various modeling assumptions. In our first model presented in section 3, we

evaluate the advantage of delayed differentiation relative to a pure make-to-stock system when

commonalities arise naturally among products. In section 4, we consider the issue of identifying

the optimal point of differentiation when varying degrees of delayed differentiation can be

achieved at the cost of redesigning the products or the process. The effect of using several

partially differentiated products, instead of a single undifferentiated platform common to all

products, is explored in section 5. Finally, in section 6, we summarize our findings and discuss

various managerial implications.

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2. Target Application and Modeling Assumptions

In this paper, we focus our attention on systems in which the manufacturing process is

primarily component assembly with product-invariant assembly time. This is the case, for

example, in the computer industry, where all products undergo the same assembly process (e.g.,

all products require a microprocessor, a hard disk, a memory card, and a power supply, among

other components). The assembly time of these components does not usually vary from product

to product. For example, assembly time is the same for 500 and 700 Mhz processors (see chapter

6 of Wright [23] for a discussion of computer manufacturing and assembly). In these systems,

delayed differentiation is enabled by either (1) taking advantage of existing commonalities

among products, (2) standardizing components across products or (3) reorganizing operations so

that those that are common are performed first [17]. In all three cases, delayed differentiation

does not significantly affect the overall product assembly times.

In component assembly, delaying differentiation results in the consolidation, or pooling, of

three types of inventory: component inventory, work-in-process (WIP) inventory, and finished or

semi-finished goods inventory – see Figure 1. The pooling of component inventory is due to the

standardization of components. Depending on the ordering policy, pooling could reduce the

amount of total inventory needed to meet the same service level. The pooling of WIP inventory

is due to the undifferentiated nature of WIP produced using common components. In contrast to

component inventory, the pooling of WIP does not affect the size of this inventory since both the

total demand and assembly time are not affected (assuming, as we do, that raw material is

released immediately after each order arrives). The pooling of semi-finished inventory is due to

the undifferentiated nature of products that are stored in the intermediate buffer between the

make-to-stock and make-to-order stages. This pooling could result in a reduction in inventory

relative to a pure make-to-stock system. However, as we shall see later in the paper, this is

largely dependent on system parameters.

In evaluating the costs and benefits of delayed differentiation, we shall be concerned

primarily with the effect of delayed differentiation on the intermediate buffer of undifferentiated

inventory and how the size and placement of this inventory affects its holding cost. We are also

interested in examining how order-fulfillment delay requirements affect the desirability of

delayed differentiation and the associated placement and size of the undifferentiated inventory.

6

CA1 CB1

CA2 CB2

A

B

CAB3

CAB4

CA5 CB5

CA6 CB6

A

B

A

B

CA7 CB7

T1T2 T3 T4

CA8 CB8

A

B

Component inventory

WIP inventory

Finished goods inventory

Figure 1(a) Early differentiation: Products A and B are differentiated early since for their first assembly task T1 they requiredifferent components, CA1 and CA2 for product A and CB1 and CB2 for product B.

CAB1

CAB2

AB

CAB3

CAB4

CA5 CB5

CA5 CB5

ABA

B

CA7 CB7

T1T2 T3 T4

CA8 CB8

A

B

Figure 1(b) Component standardization without delayed differentiation: Components for task T1 are standardized. Thisleads to the pooling of component inventory for task T1 and WIP pooling following tasks T1 and T2.

CAB2

CAB1

AB

CAB3

CAB4

CA5 CB5

CA6 CB6

AB A

B

CA7 CB7

T1T2 T3 T4

CA8 CB8

AB

Figure 1(c) Component standardization with delayed differentiation:The differentiation tasks are postponed until demand is realized.Undifferentiated inventory is held in intermediate buffer AB instead of differentiated finished goods inventory.

Figure 1 - The effect of component standardization and delayed differentiation oninventory pooling

7

Although delayed differentiation could result in reducing the amount of component inventory

needed, we assume in this article that component availability is primarily the responsibility of the

supplier. In fact, the applications we have in mind are those where components are often

delivered just-in-time and little or no inventory is held on-site. Therefore, the benefits of

component standardization are mostly realized by the supplier [19]. These benefits can be

quantified by applying to the supplier operation a similar analysis to the one that we describe

here. We do however account for the additional cost that might come from using standardized

components or from redesigning the process to enable further delay of the differentiation point.

Since we assume that assembly times are product-invariant, delayed differentiation does not

affect total work content. For similar reasons, we assume that there is no setup time in switching

from assembling one product to another (this is certainly the case in computer assembly).

Therefore, delayed differentiation does not affect changeover times. We also assume that all

products carry the same priority and are quoted the same lead-time. Therefore, we use a first-

come first-served scheduling policy. Our measure of cost includes both inventory and process

redesign costs while our measure of service is customer order delay. Manufacturing managers

often set and strive to achieve explicit delivery time goals, which they may measure either as the

average order fulfillment time, or as the proportion of orders that exceed a critical delivery time

target (e.g., a quoted lead time). The models we present can treat either of these points of view

and capture the manner in which order delay depends on how workload and capacity are

assigned to the undifferentiated and differentiated stages. Our choice of order delay as the

measure of service stems from the observation that most applications of DD arise in situations

where quick response to customer orders is central to the competitiveness of the firm. However,

alternative measures of customer service are possible by including, for example, an inventory

backordering cost, or placing a constraint on the probability of backorders exceeding some

threshold. Both could be accommodated in our models, after suitable modifications.

In our analysis of systems with delayed differentiation, we shall consider three cases. In the

first case, we consider systems where some commonalities between products arise naturally (e.g.,

the first set of operations/components are naturally common among all products). The question

then is “do we take advantage of these existing commonalities and delay differentiation until

demand is realized or do we continue to produce each product in a make-to-stock fashion?” In

the second case, we consider systems where there is flexibility in choosing the point in the

manufacturing process at which differentiation takes place. The question here is “what is the

8

optimal point of differentiation and how is it affected by system parameters?” Since a change in

the point of differentiation results in a change in workload allocation among the make-to-order

and make-to-stock stages, an equally important question is “how should capacity be allocated

between the two stages?” In the third case, we consider systems where, instead of building a

single common platform across all products, which can be expensive, products could be more

cheaply partially differentiated into several product family platforms. Each platform is

differentiated into individual products once demand is realized. The question then is “what is the

relative advantage of full delayed differentiation (a single platform) as compared to partial

differentiation (multiple platforms)? Put differently, “are there cases for which partial

differentiation is nearly as good as full differentiation?”

3. Make-to-Stock versus Delayed Differentiation

In this section, we consider a production system where commonalities between products arise

naturally. Our objective is to investigate conditions under which we should take advantage of

these commonalities and delay differentiation instead of producing in a pure make-to-stock

fashion. As discussed in the previous section, production is carried out in two stages. The first

stage consists of operations that are common to all products while the second stage includes

operations that result in differentiated end-products. Since the application we have in mind is

component assembly, all products require the same assembly tasks in both steps. Differentiation

comes from the use of different components in step 2. In the pure make-to-stock system (MTS),

finished goods inventories are stocked for each product while for a system with DD, product-

specific assembly is postponed until demand materializes. A graphical depiction of both systems

is shown in Figure 2.

We consider a system with M products. External demand for product i, i = 1, 2, ..., M occurs

according to a Poisson process of rate λi, with ∑ ==Λ Mi i1

λ denoting the total demand rate. For

the pure MTS system, demand is satisfied from buffer stock unless the corresponding buffer is

empty. In that case, it is immediately backlogged. For the system with DD, each demand arrival

releases an undifferentiated item from the intermediate inventory buffer, which then joins the

queue of jobs, if any, that are waiting to be processed at stage-2 where differentiation takes

9

Externaldemand forindividualproducts

Work release trigger

b2

Finishedproducts

Dedicated buffersof finished goods

Stage-1Common operations

Stage-2Differentiated operations

b3

b1

bM

(a) Pure make-to-stock system (MTS)

Externaldemand forall products

Work release trigger

Finishedproducts

Single Buffer ofundifferentiated

inventory

Stage-1Common operations

Stage-2Differentiated operations

(b) System with delayed differentiation (DD)

Figure 2 - Make-to-stock versus delayed differentiation

10

place. However, if the intermediate buffer is empty, then the demand is backlogged for

processing at stage-1. For both systems, inventories are managed according to a base stock

policy, where each demand arrival triggers the placing of an order with the production system.

Each order results in the immediate release of a new raw material kit to the queue of kits at

stage-1. For both systems, we assume that raw material kits are always available. The base stock

level for the system with DD is denoted as bd, whereas the base stock level for the system with

finished goods is called bf(i) for product i.

In each system, the total work content at stage-1 is T1 and at stage-2 is T2. The average rate at

which a unit of work content is processed in stage-i is denoted by Ri. Hence, the average rate at

which items are processed is µi = Ri/Ti. For stability, we require that Λ/µi = ρi < 1 for all i, where

ρi is also the utilization at stage-i. We assume that the unit processing times at each stage are

exponentially distributed. This helps simplify analysis considerably and represents the practical

worst case for benchmarking production system performance (see Hopp and Spearman [12] for a

discussion of this assumption). Finally, we let λi = Λ/M for all i, which leads to bf(i) = bf for all i.

This is necessary in order to carry out fair comparisons between systems with different levels of

product variety. It is, however, straightforward to extend the analysis to systems with asymmetric

demand.

For both systems, our objective is to minimize average inventory costs subject to a service

level constraint. We specify service level in terms of an upper bound on average order fulfillment

time. Order fulfillment time is the total time elapsed from the moment a demand arrives to the

moment the finished product is supplied to the customer. It is possible to use alternative

measures of service level, such as the percentage of orders filled within a quoted lead time or the

percentage of orders that are backordered. We defer discussion of these measures to section 4.

Since we impose an order fulfillment delay requirement on both systems, the two systems can be

compared in terms of the size and cost of inventories held either in the finished goods or as

undifferentiated items in the intermediate buffer. Note that WIP level is not affected since

products require the same operations regardless of system configuration.

For both systems, our design objective can be formulated as follows: Minimize average

inventory cost, subject to average order fulfillment delay ≤ α, where α is the maximum allowed

average delay. Average inventory cost for the pure make-to-stock system is given by zf =

M[hfIf(bf)], where hf is the holding cost per unit of finished goods per unit time and If(bf) is the

11

average finished goods inventory for each end product for a base stock level of bf. Similarly,

average inventory cost for the system with DD is given by zd = hdId(bd) where hd is the holding

cost per unit of undifferentiated inventory per unit time and Id(bd) is the average finished goods

inventory for each end product given a base stock level of bd. We use the notation Ff(bf) and

Fd(bd) to refer, respectively, to average order delay for the pure MTS system and system with

DD.

Treating each stage as a single server queueing system, average inventory and average order

fulfillment delay for the pure MTS system can be obtained. Exact analysis, is however, tedious

and does not lead to closed-form expressions. Therefore, what we present below is a close

approximation which has been found to work well in most cases. Detailed mathematical

development of these expressions can be found in Appendix A.

∑

∑

= +−−

+−+−−

+−

−

−−

=

==+−−

+−

−

=

]

(1)

otherwise,0 1)1)1((

11

1)2)1((

12)[(

)12(

2)21)(11(

0

,21 if 2))1((

)1)((

32)1(

)(

f

f

bk kMM

k

kMM

k

kfbM

bk

kMM

kkkfb

M

fbfI

ρ

ρ

ρ

ρ

ρρ

ρρ

ρρρρ

ρ

ρ

+−

−

−++−+

−

−

+−−−

+−= +−−

+−−−−−++

∞=

−=

−= −−−−

−−−++−

==+−

= +−−−

∞=

−=

−= −−−−

−−−++−

=

∑

∑ ∑ ∑

∑

∑ ∑ ∑

otherwise.(

])[())(.(

)1

()1

(

(2)

)1)(1]()2(

][)[2/1.(

)1

()1

(

. ))2/11(

2)1)2/1(1)(21)(11(

0)2/1(1

)2/1()2/1(11

121

0 )!()!1()!(!

)!1()!(

,21 if 20 )1(

12

11

0 )!()!1()!(!

)!1()!(

)(

ρρ

ρρρρρ

ρρ

ρρρρ

ρ

ρρρ

ρ

µµ

ρρµµ

rr

ybkr

rryfb

brkrkr

bkkr

y ykrfbfbky

kykfbryfbk

rybk

r

brkrkr

bkkr

y ykrfbfbky

kykfbryfbk

fbfF

f

f f

f

f f

k

M

M

M

rr

M

M

M

l 1l

1ll

l ll

12

Evaluating performance measures for the model with delayed differentiation is a little more

complicated. Although the output process of a single server queue with Poisson arrivals and

exponential processing is also a Poisson process [3], the stage-2 input process is Poisson only in

two special cases: bd = 0 and bd = ∞. The former instance results in two M/M/1 queues in tandem

whose steady state probabilities are known to have a product-form structure [13]. Similarly,

when buffer size is very large, the two stages are completely decoupled and behave like two

independent M/M/1 queues. Using notation Ai to denote inter-arrival time at stage-i, and 2iAC its

squared coefficient of variation, it is possible to show that 2

2AC ≈ 1 (see appendix B for proof).

That is, treating stage-2 queue as a M/M/1 queue is a reasonable approximation for estimating

delay at this stage. In fact, 2

2AC is always less than one. Thus, our results provide an upper

bound on the true delay at stage-2. In this sense, ours is a conservative viewpoint and values of

bd chosen by the model will always be sufficient to guarantee that order fulfillment requirement

will be met. Treating each production stage as an M/M/1 queue, we can obtain expressions for

average inventory and order fulfillment delay as follows (see appendix C for proof):

1

11

1

])(1[)(

ρρρ

−−

−=db

ddd bbI , (3)

and

)1()1()(

2

2

1

11

ρρ

ρρ

−Λ+

−Λ=

+db

dd bF . (4)

It is not too difficult to show that for both pure MTS and DD systems, average inventory is

increasing in the base stock level, bf or bd, and average order delay is decreasing in the same.

Therefore, the optimal values of bf and bd are always the smallest values that satisfy the service

level constraint. For system DD, the optimal base stock level can be found by solving Fd(bd) = α,

which yields the following:

−

−Λ−−Λ= 1

)ln(

))]1(/)(1(ln[

1

221*

ρρραρ

db , (5)

where x represents the integer ceiling of x. Substituting *db in the expression of average

inventory, we can calculate the optimal cost for system DD. For the pure MTS system, a closed

form expression of *fb is difficult to obtain. However, since Ff(bf) is strictly decreasing in bf,

*fb

13

can be easily obtained through a simple numerical search. Note that since Fd(bd) = ρ2/Λ(1- ρ2)

when bd → ∞ and Ff(bf) = ρ1/Λ(1- ρ1) + ρ2/Λ(1- ρ2) when bd = 0, the value of α is meaningful

only if ρ2/Λ(1- ρ2) ≤ α ≤ ρ1/Λ(1- ρ1) + ρ2/Λ(1- ρ2). If α < ρ2/Λ(1- ρ2), then DD is not a feasible

option and a pure MTS system must be adopted. On the other hand, if α > ρ1/Λ(1- ρ1) + ρ2/Λ(1-

ρ2), there is no need for holding inventory. In that case, a pure make-to-order system is optimal.

In comparing pure MTS and DD systems, we shall assume throughout that ρ2/Λ(1- ρ2) ≤ α ≤

ρ1/Λ(1- ρ1) + ρ2/Λ(1- ρ2) is satisfied.

In order to examine the effect of delaying differentiation on cost, we carried out a series of

numerical experiments in which we evaluated the effect of various operating parameters, such as

the number of products, utilization, capacity allocation, and holding costs. Representative results

are shown in Figures 3-6. We first examined the effect of product variety and utilization on the

relative desirability of DD. To this effect, we evaluated the ratio *d

*f zz / for different values of

M and different levels of utilization. To isolate the effect of M, we limited ourselves initially to

cases where ρ1 = ρ2 = ρ, λi = Λ/M and Λ is fixed. In these comparisons, it is important to ensure

that a system with DD is always feasible. For that reason, we set α = ρ2(1 – ρ2) + ε, where ε > 0

and is kept the same across all utilization scenarios (the effect of varying ε is examined later).

As we can see from Figure 3, the relative cost advantage of a system with delayed differentiation

is generally increasing in M. This means that the benefits of DD tend to be more significant when

product variety is high. The result can be explained by the inventory pooling (common buffer)

effect that takes place with DD and which is particularly significant when the number of

products is large. It is not, however, too difficult to find instances where a pure MTS strategy is

superior. This is particularly prevalent when product variety is low. In this case, the need to meet

the service level tends to force the system with DD to hold more inventory than a pure MTS

system. Because the number of products is small, the pooling effect is not sufficient to offset the

increased requirement of inventory. Although this pattern is more apparent when the number of

products is small, it can occur regardless of the amount of product variety if order delay

requirement is sufficiently short or utilization is sufficiently high.

In fact, as we can see from Figure 4, the relative advantage of DD is highly sensitive to

utilization, with the advantage of DD diminishing with increases in utilization. At very high

utilization, DD ceases to be desirable since the cost of meeting order delay requirement becomes

14

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

160.0

180.0

200.0

0 5 10 15 20 25 30

Number of products (M )

Zf*

/Zd*

ρ = 0.4ρ = 0.5ρ = 0.6ρ = 0.7ρ = 0.8ρ = 0.9

Figure 3 – The effect of number of products on the relative benefit of delayeddifferentiation (h = 1, ε = 0.0001)

0.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

160.00

180.00

200.00

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

Utilization (ρ )

Zf*

/Zd*

M = 30

M = 25

M = 20

M = 15

M = 10

M = 5

M = 2

Figure 4 – The effect of utilization on the relative benefit of delayed differentiation(h = 1, ε = 0.0001)

15

prohibitively expensive. Note that in our comparisons, we always guarantee that DD is feasible.

However, if α remained fixed and utilization were allowed to increase, DD could become

quickly infeasible. In fact, in that case, there is a finite range of utilization given by αΛ/(2 + α)

≤ ρ ≤ αΛ/(1 + α) where comparing the two systems is meaningful. For ρ > αΛ/(1 + α), DD is

infeasible and a pure MTS system must be adopted while for ρ < αΛ/(2 + α), a pure make-to-

order system is optimal. Note that the width of the feasibility interval approaches 0 as α → 0.

Similarly, as α → ∞, DD is a candidate only for systems that have very little excess capacity. In

all other cases, we can use the cheaper make-to-order mode.

The effect of varying service level is illustrated in Figure 5. Here, in order to carry out a fair

comparison we let α = ρ(1 – ρ) + ε, and vary ε. Thus, increasing α means increasing the order

delay slack available to the system with DD. As we can see, a higher service level (i.e., a smaller

order delay slack) tends to diminish the benefit of DD since it requires the system to maintain

higher inventories. This effect is particularly pronounced when utilization is high. The fact that

DD is more sensitive to changes in α than a pure MTS is due to two factors: (a) there is generally

more buffer slack in pure MTS (more about this in the next paragraph) and (b) DD must always

maintain sufficient inventory to meet the required service level while carrying out the

differentiation tasks post-demand realization.

Although our previous observations generally hold true, we should warn that the integrality

of inventory and number of products could cause the ratio *d

*f zz / to be non-monotonic in M or

ρ (see Figure 6). Specifically, the integrality of inventory could result in selecting a larger *fb

than what is exactly needed to meet the order delay requirement, an effect that is compounded

with multiple products. Consequently, within a limited range, small increases in utilization could

result in a lower average inventory while still satisfying the order delay requirement. Managers

need to be mindful of these peculiarities when comparing specific systems. However, wherever

possible, we recommend use of analysis to determine the desirability of one configuration over

another.

In examining the effect of utilization, we have assumed so far that both systems are balanced

with equal utilization at both stages. In section 4, we examine the effect of unequal capacity and

workload allocation between the two stages. However, let us just note here that the effect of

asymmetry in utilization on the MTS system is different from its effect on the DD system. For an

16

0.00

100.00

200.00

300.00

400.00

500.00

600.00

700.00

5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50

Service level (α )

Zf*

/Zd*

M = 20

M = 15

M = 10

M = 5

Figure 5 – The effect of order delay requirement on the relative benefit of delayeddifferentiation (h = 1, ρ1 = ρ2 = 0.85)

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

0 5 10 15 20Number of products (M )

Zf*

ρ2 = 0.4ρ2 = 0.5ρ2 = 0.8

ρ2 = 0.9

Figure 6 – The effect of order delay requirement on the relative benefit of delayeddifferentiation (h = 1, ρ1 = ρ2 = 0.85)

17

MTS system, interchanging the value of ρ1 and ρ2 has no effect on performance. More generally,

if we start with ρ1 = ρ2, then increasing the utilization of stage-1 (while maintaining the

utilization of stage-2 constant) has the same effect as increasing the utilization of stage-2 (while

maintaining the utilization of stage-1 constant). In contrast, the DD system is very sensitive to

which of the two stages has the higher utilization. In fact, a higher utilization at stage-2 may not

even be feasible if that makes the order delay larger than the minimum required α. More

generally as ρ2 approaches α/(1 + α), the cost of the system with DD starts to escalate at a faster

rate than the cost of the MTS system. For ρ2 ≥ α/(1 + α), only the MTS system is feasible.

Hence, the MTS system tends to be more stable in the face of fluctuations in utilization since

additional inventory buffering is always sufficient to meet a tighter order delay. This, however,

may come at a high price. Italso means that in order to enable delayed differentiated sufficient

capacity investments should be made at stage-2, the make-to-order stage.

In summary, results of this section show that although DD can yield significant cost savings,

it is not always desirable even if no process or product redesign is needed. Managers should be

particularly cautious about implementing DD if (1) product variety is low, (2) utilization is high,

or (3) allowable slack in order lead-time is small. Managers should also pay close attention to the

relative loading of the differentiated and undifferentiated stages since that might affect the

desirability of one system over another.

4. The Optimal Point of Differentiation

In this section we explore the economics of affecting the point of differentiation (POD),

given that delaying differentiation is desirable for a “base” case. The base case refers to the

situation studied in section 3 where we simply explore existing commonalities and maintain a

common buffer of undifferentiated items at the point separating the common and custom

operations. For this purpose, we set E(T1) + E(T2) = T, where T is a constant and consider the

implication of changing t = E(T1). Such a change could be made possible by resequencing of

assembly steps (putting common operations first), or redesigning assembly operations to increase

the number of common steps, or through redesign of the products so that they could be

assembled from common components.

18

Thus, the key issue now is to determine the value of the optimal point of differentiation t =

E(T1). Since a change in t corresponds to a change in the workload assigned to each production

stage, this change is usually accompanied by a reallocation of capacity. Several capacity

allocation schemes are possible. We limit ourselves to two cases. In the first case, we assume

that capacity is allocated proportionally to workload so that utilizations, ρ1 and ρ2, remain

unchanged. This means that if we choose to increase work content of the MTS stage, its

processing speed R1 is also increased proportionally. Although we do not put any limits on the

amount by which processing speed might be affected, any increase in capacity incurs an

acquisition and ownership cost. We assume that a reduction in capacity reduces this cost,

although it is possible to include a one-time downsizing cost (e.g., layoff or equipment disposal

costs). This formulation allows us to isolate the effect of altering the point of differentiation from

those that come from changing utilization. This is important since the effect of utilization can

easily overshadow those caused by changes in point of differentiation.

In the second case, we let the amount of available capacity be fixed and then carry out an

optimal allocation of this capacity based on workload assignment. Thus, we assume that R1 + R2

= R, where R is a constant, and determine optimal R1 (or R2) based on our choice of optimal t. In

this case, b, t and R1 are all decision variables. For both cases, we shall use the following

notation:

hi(t) = Cost per unit per unit time of holding inventory at stage-i,

C1(t) = Cost of product and process redesign,

C2(R) = Cost of capacity R, and

C3(b) = Cost of maintaining an inventory buffer of size b.

All of the cost functions are assumed to be differentiable, positive, increasing and convex in their

arguments. Each cost function is normalized to 0 when its argument is 0. In the case of C1(t), we

also require that C1(t) = 0 for t ≤ t0, where t0 is the point of differentiation in the base case, since

differentiating earlier simply requires moving the buffer of undifferentiated inventory earlier in

the production process without a need for redesigning either the product or the process.

In identifying the optimal point of differentiation, our objective is to meet the specified

delivery time performance while minimizing the sum of inventory, redesign, and capacity costs.

Our optimization model can be formulated as follows:

19

Minimize Z(t, b, R) = h1(t)I(t, b, R1, R2) + h2(t)E(WIP2(t, b, R1, R2)) + C1(t) + C2(R) + C3(b) (6)

Subject to

F(t, b, R1, R2) ≤ α. (7)

ρ1 ≤ 1 (8)

ρ2/(1- ρ2) ≤ α (9)

t ≤ T (10)

t, b, R ≥ 0 (11)

Note that with t being a decision variable, we must include the cost of WIP in stage-2 in the

objective function since the holding cost of this WIP is affected by our choice of point of

differentiation. Expressions for I and F have been derived in section 3. Expected WIP at stage-2,

by virtue of Little’s law, is ρ2/(1- ρ2). Constraint 8 ensures stability of stage-1; constraint 9

guarantees that the service level constraint is feasible; and constraint 11 ensures that the work

content assigned to stage-1 does not exceed total work content. Note that subscript d is dropped

from notation since all systems being compared utilize delayed differentiation. We also limit

ourselves to cases where a pure make-to-order system is not feasible. Therefore, we implicitly

assume that α < ρ1/(1- ρ1) + ρ2/(1- ρ2).

Although our service level measure is specified in terms of average order delay, it is possible

to use alternate measures such as the probability of being able to deliver within a quoted lead

time, average number of backorders, or the probability that backorders exceed some critical

value. Expression for each of these metrics are given by the following (for the sake of brevity,

the proofs are omitted and can be found in Gupta and Benjaafar [11]):

(12)

otherwise, ))-(11(

, if ))())(1(

(

)delay Prob(order),,,(

]/)1([1

12]/)1([]/)1([

12

112]/)1([ 112222

21

Λ+

≠−−

−+=

≥=

−Λ−−

−Λ−−Λ−+

−Λ−

xb

xxb

x

ex

eee

xRRtbxF

ρρ

ρρρρρρ

ρρ

ρρρρρρ

,11

backorders ofnumber average),,,(1

11

2

221 ρ

ρρ

ρ−

+−

=≡+b

RRtbS (13)

and

20

(14)

otherwise. xb)-x(1

,12 if )12

)(12

)21(11(2) Prob(S),,,( 21

++

≠−−−+

+=≥=

ρρρ

ρρρρρρρρ

ρ

x

xxb

xxRRtbxS

4.1 The Case of Fixed Utilization

Since capacity is assigned proportionally to work content, the values of ρ1 and ρ2 are

unaffected by our choice of t, R1 and R2. Consequently, I, E(WIP2), F and C3 are all unaffected

by t. Given the choice of t, R is given by:

R = R1 + R2 = Λt/ρ1 + Λ(Τ − t)/ρ2 = Λ{t(ρ2 - ρ1) + Tρ1}/ρ1ρ2. (15)

Therefore, capacity is uniquely determined by t. Since order delay is decreasing in b but the

objective function is increasing in b, the optimal base stock level b* is still given by expression

(5). Substituting b* in z(t, b, R1, R2), the objective function reduces to a function in the single

variable t. The optimal value of t can be obtained by searching in the range 0 ≤ t ≤ T. However,

as we shall discuss next, this is necessary only when ρ1 > ρ2.

Since delaying differentiation beyond the base value t0 increases (or at least does not

decrease) the holding costs h1 and h2 and the redesign cost C1, additional delayed differentiation

is desirable only if the capacity cost C2 is reduced as a result. This means that the tradeoff in

deciding on a point of differentiation is between additional holding costs versus reduced

capacity. Using the fact that Ri = ΛΕ(Ti)/ρi, it can be easily confirmed that

)./())(()()( 211200

22011

0 ρρρρ −−Λ=−+−=− ttRRRRRR (16)

If t > t0, the difference R - R0 is negative only if ρ2 < ρ1. That is, further delay of the point of

differentiation reduces required capacity only if the utilization of the make-to-stock stage is

greater than that of the make-to-order stage. Therefore, if ρ2 ≥ ρ1, we should never delay

differentiation beyond its base value of t0.

In general, we can distinguish three cases.

1. ρ1 = ρ2: In this case, we have a balanced system. Capacity remains constant regardless of t.

Since both holding costs and redesign costs are increasing in t, it is never optimal to postpone

differentiation beyond t0. In fact, in this case, the sooner we differentiate the better. That is,

choosing t < t0 is always desirable.

21

2. ρ1 < ρ2: In this case, capacity is increasing in t. Therefore, further delay is never optimal.

Here again, since both capacity and holding costs are decreasing in t, no change in POD is

recommended.

3. ρ1 > ρ2. In this case, capacity is decreasing in t. However, holding costs are increasing in t.

Consequently, we need to trade-off the savings from capacity reduction against the increase

in holding costs. Generally, this would require searching for the optimal t over the entire

range (0, T). However, in the case of linear holding, capacity, and redesign costs, it is easy to

show that further delay of differentiation is desirable only if

}.)1/())1/(])(1[({)(

)(1122111

*1

21

21min2 ChbhCC db

d +−+−−−−Λ

=≥ ρρρρρρρ

ρρ (17)

Otherwise, the earlier we differentiate the better. The value of Cmin is increasing in h1, h2 and

C1 and decreasing in the difference (ρ1 - ρ2). It is clear that for different cost functions, it is

possible to obtain an optimal POD that is different from the extreme cases of either

maximum or minimum delay.

4.2 The Case of Fixed Capacity

Before we examine the general case where we jointly choose t, R1, R2 (R2 = R – R1) and b, we

shall consider the following two special cases: (1) capacity assignment is fixed and work-content

allocation is flexible, and (2) capacity assignment is flexible but work content allocation is fixed.

In the first case, R1 and R2 are fixed but t is variable while in the second t is fixed but R1 and R2

are variable. Distinguishing the two cases allows us to gain greater insight into the relationship

between delayed differentiation and capacity assignment. It is also useful in developing a

solution procedure for the general case.

When capacity allocation is fixed, our optimization problem reduces to selecting t and b. It is

not difficult to see that constraint (7) is still binding and, therefore, b* is given by expression (5).

Substituting b* in the objective function, z(t, b, R1, R2) becomes again a function of the single

variable t. Noting that constraints 8 and 9 can be rewritten, respectively, as t < R1/Λ and T – (R –

R1)/( Λ + 1/α) < t, the optimal t can be obtained by searching in the range

max{0, T – (R – R1)/( Λ + 1/α)} < t < min{T, R1/Λ}. (18)

Examining the objective function we can see that redesign costs, unit holding costs, and expected

WIP in stage-2 are all increasing in t. However, average inventory in the intermediate buffer (as

22

well as the size of this buffer) is not necessarily monotonic in t. For example, reducing t leads to

faster replenishment of the intermediate stock but longer delays at stage-2. Depending on the

value of t, this leads to a choice of b* that could either increase or decrease average inventory.

In general, early differentiation is more desirable as long as the associated work allocation

does not significantly increase lead-time in stage-2. This would be the case when the capacity

assigned to stage-2 is relatively large. Inversely, when little capacity is assigned to stage-2,

delayed differentiation tends to become more attractive. This is illustrated in Figure 7 for an

example system. As we can see, the optimal point of differentiation is also affected by the total

amount of available capacity R. Specifically, an increase in total capacity tends to make early

differentiation more desirable since assigning additional workload to stage-2, without a

significant penalty in higher inventory, becomes possible.

In addition to being affected by the available capacity, as well as the relative allocation of

this capacity, the optimal point of differentiation is determined by the order delay requirement α.

Since ρ2/(1 - ρ2)< α is a necessary condition and the only mechanism by which we control ρ2 is

by increasing t, a smaller α tends to favor later differentiation. On the other hand, a larger α

makes early differentiation more desirable since we can reduce redesign costs while satisfying

order delay requirements. This effect is graphically illustrated in Figure 8.

Turning to the second case, where the point of differentiation is fixed but capacity

assignment is flexible, we can see that it too reduces to a single decision variable optimization

problem in R1. In contrast to t, varying R1 carries no penalties for the types of manufacturing

systems we model. However, the effect of varying R1 on inventory levels is difficult to predict.

While assigning more capacity to stage-1 reduces replenishment lead times to the intermediate

buffer, it increases lead-time at stage-2. The combined effect could lead to either a net increase

or decrease in total holding costs. In general, we should expect more capacity to be assigned to

stage-2 when differentiation is carried out early (due to the associated higher workload). The

need to assign more capacity to stage-2 can also be exacerbated when order delay requirement is

tight. In fact, when α is sufficiently small, more capacity must be assigned to stage-2 to ensure

feasibility. The relationship between optimal assignment, point of differentiation and service

level is illustrated in Figures 9 and 10 for an example system.

23

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.30 0.50 0.70 0.90 1.10 1.30 1.50

Capacity in stage 1 - R 1

Opt

imal

poi

nt o

f di

ffer

enti

atio

n -

t*

R=1.5

R=1.6

R=1.7

R=1.8

Figure 7 – The effect of capacity allocation on optimal point of differentiation(T = 1, Λ = 1, α = 1, h1(t) = h2(t) = 5t, C1(t) = 0, C3(b) = 0)

0.45

0.55

0.65

0.75

0.85

0.95

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

Order delay requirement - α

Opt

imal

poi

nt o

f di

ffer

enti

atio

n -

t*

R1=0.7

R1=0.8

R1=0.9

R1=1.0

R1=1.1

Figure 8 – The effect of order delay requirement on optimal point of differentiation(T = 1, R = 1.5, Λ = 1, h1(t) = h2(t) = 5t, C1(t) = 0, C3(b) = 0)

24

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.00 0.20 0.40 0.60 0.80 1.00Point of differentiation - t

Opt

imal

cap

acit

y al

loca

tion

- R

1*

R=1.3

R=1.4

R=1.5

R=1.6

R=2.0

R=2.5

Figure 9 – The effect of point of differentiation on optimal capacity allocation(T = 1, Λ = 1, α = 2, h1(t) = h2(t) = 5t, C1(t) = 0, C3(b) = 0)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80

Order delay requirement - α

Opt

imal

cap

acit

y al

loca

tion

- R

1*

t=0.3

t=0.5

t=0.7

t=0.9

Figure 10 – The effect of order delay requirement on optimal capacity allocation(T = 1, Λ = 1, R = 2, h1(t) = h2(t) = 5t, C1(t) = 0, C3(b) = 0)

25

Putting results from these two special cases together, two principles emerge: (1) whenever

possible, we should differentiate early (this becomes more difficult as the order delay constraint

gets tighter), and (2) we should assign capacity so that differentiation could take place earlier

(this means, whenever feasible, assigning proportionally more capacity to stage-2). These

principles can be used by managers to quickly identify a first-cut solution to the general problem

where t, b, and R1 are all decision variables. However, due to the non-convex nature of the

objective function in the decision variables, these principles do not guarantee optimality. In fact,

short of an exhaustive search over the t and R1 dimensions, an optimal solution is hard to obtain.

Fortunately, in practice t and R1 can be varied only in discrete steps. Therefore, an exhaustive

search, even when the number of discrete steps is large, is always possible.

5. The Effect of Partial Delayed Differentiation

In the previous two sections, we assumed that a platform common to all products can be built

in stage-1. In many industries, this is either not possible or too expensive. Instead, products are

often grouped into families and a standardized platform is designed for each family [9]. Thus,

instead of building a single undifferentiated product in stage-1, two or more partially

differentiated items are produced and stocked in separate intermediate buffers. These partially

differentiated items are then fully differentiated in stage-2 once demand is realized. In this

section, we examine the relative advantage of full delayed differentiation (DD) relative to partial

delayed differentiation. We are particularly interested in identifying conditions when partial

differentiation (PD) does not result in significant deterioration in performance. Note that if we

ignored redesign costs, a system with fully undifferentiated products is always superior. This

observation is consistent with inventory literature. However, since the costs of designing a

common platform that is shared by all products can be significant (especially when the number

of these products is high), there is a need to tradeoff the additional benefits of full delayed

differentiation against cost savings realized with partial differentiation.

A system with M partially differentiated products is similar to the system with DD we

considered earlier. However, instead of a single buffer of undifferentiated products, we have M

buffers of partially differentiated products. In order to isolate the effect of number of partially

differentiated (PD) products, we let the demand associated with each PD product be λ = Λ/M.

26

Average total inventory Ipd and average order delay, Fpd, can then be obtained as follows (see

Appendix D for proof):

( )

−

−−= pdb

pdpdpd rr

rbMbI 1

1

1 11

)( , (18)

and

( ) )1(1)(

2

2

1

11

ρρ−Λ

+−Λ

=+

r

MrbF

pdb

pdpd , (19)

where

11

11 )1( ρρ

ρ+−

=M

r . (20)

Since the component of order delay due to stage-2 is not affected by M, we can rewrite the

service level constraint as:

( ) α̂1 1

11 ≤

−Λ

+

r

Mr pdb

, (21)

where

)1(ˆ

2

2

ρραα−Λ

−= . (22)

The optimal base stock level can then be obtained as:

[ ]

+−

Λ−=

11

1

1*

)1(ln

)(ˆln

ρρρ

µα

M

bpd (23)

Note that we assume )/(1ˆ 11 λµα −< . If the condition is not satisfied, then a pure make-to-order

system is optimal.

As we did in section 3, let us examine the advantage of complete lack of differentiation

relative to partial differentiation under varying conditions of utilization, product variety and

order delay requirement. We begin by examining the effect of utilization. First, note that the

make-to-order segment of the production process is not affected by partial differentiation.

Therefore, we restrict our attention to the utilization of the make-to-stock stage, ρ1. In contrast to

section 3, we are able here to analytically characterize several properties relating the effect of

27

utilization when it is either high, low, and in the mid-range (proofs of all properties can be found

in Appendix E).

Property 1: *d

*pd zz / → 1 when ρ1 → 1, where )**

pdpdpd (bhIz = .

Property 1 shows that the relative advantage of full standardization is insignificant when

utilization is high. In fact, in the limiting case, a system with M partially differentiated products

becomes equivalent to a system with a single undifferentiated product. This also means that

partial differentiation carries less of a penalty (relative to no differentiation) in highly loaded

systems, even though inventory levels are very large in both cases. Similarly, when utilization is

sufficiently low, full standardization carries little value since in that case a pure make-to-order

system is optimal. The limiting utilization is given by property 2.

Property 2: *d

*pd zz = = 0 when )ˆ1/(ˆmin1 Λ+Λ=≤ ααρρ .

The result follows from the fact that when )ˆ1/(ˆ1 Λ+Λ≤ ααρ , we have )/(1ˆ 1 Λ−≥ µα , which

means that a pure make-to-order system is feasible regardless of level of differentiation.

Although full delayed differentiation carries little value in the extreme cases of high and low

utilization, its benefits can be significant when utilization is in the midrange. This can be seen in

Figure 11. It can also be shown by considering the case where ρ1 is slightly greater than ρmin. As

we can see from property 3, the advantage of full standardization is at least M times that of

partial differentiation in this case.

Property 3: *d

*pd zz / → M(1 – r1)/(1 – ρ1) as ρ1 → ρmin

+.

The result follows from the fact that when ρ1 is slightly greater than ρmin, each individual buffer

in the partially differentiated system must maintain at least one unit of inventory in order to meet

the customer delay requirement. In contrast, the system with a standardized platform needs only

one unit of inventory to meet the same requirement. This result, due to the integrality of the base

stock level, leads to the observed discontinuity into the effect of utilization on performance.

Next, we examine the effect of M (the number of partially differentiated platforms) on the

ratio *d

*pd zz / . As shown in Figure 12, this effect is not monotonic. Hence, increasing the

number of partially differentiated products could either increase or decrease total cost. This non-

monotonicity is however limited to cases when M is relatively small. In fact, as we show in

property 4, for M ≥ Mmax cost increases linearly in M.

28

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8 1

Utilization ( ρ 1)

zpd*/z d*

N = 5

N = 10

N = 15

N = 20

M = 5M = 10M = 15M = 20

Figure 11 – The effect of utilization on the relative cost of PD(h = 1, α̂ = 1.5, µ1 = 1)

0

20

40

60

80

100

120

140

0 10 20 30 40 50

M

zpd

*

ρ = 0.3

ρ = 0.5

ρ = 0.7

ρ = 0.9

ρ = 0.95

ρ 1 = 0.3

ρ 1 = 0.5

ρ 1 = 0.7

ρ 1 = 0.9

ρ 1 = 0.95

Figure 12 – The effect of number of partially differentiated products on optimal cost(h = 1, α̂ = 1, µ1 = 1)

29

Property 4: If ]1)1(ˆ/)][1/([ 1111 −−Λ−=≥ ραρρρmaxMM , then *pdz = M(1 – r1).

Property 4 shows that when M is sufficiently large the benefits of standardization are increasing

in M. This result is in line with observations we made in section 3 where we found that the

marginal benefits of delayed differentiation are generally increasing in the number of end-

products. This result also shows that the benefits of delayed differentiation can be significantly

greater than those predicted by the inventory pooling literature, where inventory costs are shown

to be reduced by a factor of M when M products are pooled.

Finally, we consider the effect of the order delay requirement. Similar to the effect of

utilization, we can show that in the two extreme cases of very large or very small α̂ full delayed

differentiation is of little value and partial differentiation could be pursued with no significant

impact on performance. The ratio *d

*pd zz / is equal to 1 when α̂ ≥ 1/(µ1 – λ1) since a pure make-

to-order system is optimal in this case. As shown in property 5, *d

*pd zz / → Mln[ρ1]/ln[r1],

which tends to be a relatively small number, when α̂ approaches 0.

Property 5: *d

*pd zz / → Mln[ρ1]/ln[r1] as α̂ → 0.

However, when α̂ is in the midrange, partial differentiation could be costly. For example, when

α̂ is in the range specified in property 6, partial differentiation is at least M times more costly

than a strategy with complete delayed differentiation.

Property 6: *d

*pd zz / = M(1 – r1)/(1 – ρ1) when )(1ˆ)]1(/[ 11

21 Λ−≤≤−Λ µαρρ .

When )},1(/{ˆ 121 ρρα −Λ≤ *

d*pd zz / (although not monotonic) is generally increasing in α̂ .

The effect of order delay requirement over the entire range of possible α̂ values is illustrated for

an example system in Figure 13.

In summary, we have shown that the advantage of delayed differentiation relative to partial

differentiation is highly sensitive to the utilization of the production facility, product variety, and

order delay requirement. Although the benefits of delayed differentiation can be significant when

utilization is in the midrange, this advantage tends to diminish when utilization is either very

high or very low. Similarly, there is a limited band of order delay values within which delayed

differentiation carries significant benefits. For values outside this band, the benefits are either

30

limited or non-existent. These results suggest that managers should be particularly cautious in

redesigning their products or their processes to support a single common platform when either

utilizations are high or order lead time requirements are tight. Remarkably, because the marginal

benefits of a single platform are increasing in the number of products, the economics of a single

platform are more advantageous when the number of products is high. Therefore, managers

should be more willing to invest in a common platform when their product portfolio is large.

This, for example, might explain the success of computer makers with the vanilla box concept

since individual computer configurations can be numerous. The results of this section are in line

with those we observed numerically in section 3 regarding the desirability of delayed

differentiation. They also parallel results recently obtained by one of the co-authors of this article

regarding the effect of inventory pooling in integrated manufacturer-retailer supply chains [2].

0

30

60

90

120

150

0 4 8α

zpd

*/z

d*

N = 5N = 10

N = 15N = 20

M = 5M = 10M = 15M = 20

Figure 13 – The effect of order delay requirement on the relative cost of PD(h = 1, ρ = 0.9, µ1 = 1)

31

6. Concluding Remarks

Table 1 provides a summary of our key findings and offers managers broad guidelines as to

when delayed differentiation is most valuable relative to a strategy of either pure make-to-stock

or partial differentiation. Our results are consistent with the existing inventory literature with

regard to the desirability of delayed differentiation when product variety is high, redesign costs

are low, or finished goods inventory is expensive. Our results shed additional light on the effects

of capacity, congestion, and order delay requirements. In particular, we show that delayed

differentiation, and more generally inventory pooling, is significantly less valuable when

utilization is high. This is an important result since most factories tend to operate near full

capacity. Therefore, analysis based on pure inventory models would lead to over-estimating the

benefits of delayed differentiation. Furthermore, we found that unless very little work-content is

postponed, delayed differentiation can be an expensive mechanism for achieving quick response.

In fact, in environments where order delay requirements are tight, delaying differentiation until

demand is realized could require holding significantly more inventory than in a pure make-to-

stock system. This result shows again that using pure inventory models could lead to over-

estimating the value derived from delayed differentiation.

In environments where the point of differentiation can be affected we found that

differentiating early by postponing more of the work until demand is realized is generally more

desirable. In fact, whenever possible a pure make-to-order system should be adopted. Obviously,

this becomes difficult to realize when order delay requirements are tight. In that case, more

capacity, if possible, should be assigned to the make-to-order stage. Otherwise we are left with

two choices, either to go back to producing in a make-to-stock fashion or to redesign products

and processes to allow for very late differentiation. This latter option is feasible only if redesign

costs are not too high. Nevertheless, this option has become increasingly popular in several

industries where customers demand short lead times and cost pressures require high utilization of

production facilities. For example, in the computer industry, delayed differentiation has been

pushed up to the point of sale with retailers carrying out the final configuration and assembly of

each computer by quickly adding or removing modular parts.

Finally, we acknowledge that while our models capture several of the tradeoffs that arise

from delayed differentiation, there are additional costs and benefits that fall beyond the scope of

our models. For example, DD carries benefits derived from streamlining scheduling, sequencing,

32

and raw material purchasing. There are also benefits from standardizing components in the form

of shorter product design cycle and cheaper new product development. On the other hand, there

are some overlooked costs. For example, relying on a common platform and emphasizing the use

of common components could inadvertently reduce the flexibility of the firm and its ability to

offer truly customized products. In turn, this may lead to loss of market share. Managers need to

be aware of these additional tradeoffs and to account for them before adopting a strategy of

delayed differentiation.

Table 1 – When is delayed differentiation valuable?

Low Medium High

Utilization Less valuable More valuable Less valuable

Product variety Less valuable Moderately valuable More valuable

Order delayrequirement (αααα)

Less valuable More valuable Less valuable

Holding costs Less valuable Moderately valuable More valuable

Redesign costs More valuable Moderately valuable Less valuable

Acknowledgements: We would like to thank the Associate Editor and two anonymousreviewers for many useful comments on an earlier version of the paper. We are thankful toProfessor Bill Cooper of the University of Minnesota for his help with developing a formal prooffor property 1. This research was supported in part by the Natural Sciences and EngineeringCouncil of Canada, the graduate school of the University of Minnesota, and the National ScienceFoundation through research grants to the authors.

33

References

[1] Aviv, Y. and A. Federgruen, “The Benefits of Design for Postponement,” in QuantitativeModels for Supply Chain Management, S. Tayur, R. Ganeshan, and M. Magazine, Editors,Kluwer Academic Publishers, 553-584, 1999.

[2] Benjaafar, S. and J. S. Kim, “Impact of Inventory Decentralization on the Performance ofProduction-Inventory Systems,” Working Paper, University of Minnesota, 2000.

[3] Burke, P. J., “The Output of a Queueing System,” Operations Research, 4 (1956), 699-704.

[4] Bruce, L. “The Bright New Worlds of Benetton,” International Management, November(1987), 24-35.

[5] Buzacott, J. A. and J. G. Shanthikumar, Stochastic Models of Manufacturing Systems,Prentice Hall, NJ, 1993.

[6] Eppen, G., “Effects of Centralization on Expected Costs in Multi-Location NewsboyProblems,” Management Science, 25, 498-501, 1979.

[7] Feitzinger, E. and H. Lee, “Mass Customization at Hewlett Packard: The Power ofPostponement,” Harvard Business Review, January-February 1997, 116-121.

[8] Fisher, M., K. Ramdas, and K. Ulrich, “Component Sharing in Management of ProductVariety,” Management Science, 45 (1999), 297-315.

[9] Garg, A. and C. S. Tang, “On Postponement Strategies for Product Families with MultiplePoints of Differentiation,” IIE Transactions, 29 (1997), 641-650.

[10] Graman, G. A. and M. J. Magazine, “An Analysis of Packaging Postponement,”Proceedings of the 1998 MSOM Conference, School of Business Administration, TheUniversity of Washington, Seattle, June 29-30, 1998, 67-72.

[11] Gupta, D. and S. Benjaafar, “Make-to-order, Make-to-stock, or Delay ProductDifferentiation? - A Common Framework for Modeling and Analysis,” Working paper,Michael G. DeGroote School of Business, McMaster University, Ontario, Canada, 1999.

[12] Hopp, W. and M. L. Spearman, Factory Physics, Second Edition, Irwin/McGraw-Hill, NY,2000.

[13] Jackson, J. R., “Networks of Waiting Lines," Operations Research, 5 (1957), 518-521.

[14] Kleinrock, L., Queueing Systems, Volume I: Theory, John Wiley and Sons, 1975.

[15] Lee, H. L., “Effective Inventory and Service Management Through Product and ProcessRedesign,” Operations Research, 44 (1996), 151-159.

34

[16] Lee, H. L.and C. Billington, “Designing Products and Processes for Postponement,”Management of Design: Engineering and Management Perspectives, S. Dasu and C.Eastman (Eds.), Kluwer Academic Publishers, Boston, MA, 1994, 105-122.

[17] Lee, H. L., and C. S. Tang, “Modeling the Costs and Benefits of Delayed ProductDifferentiation,” Management Science, 43 (1997), 40-53.

[18] Luenberger, D. G., Linear and Nonlinear Programming, Addison-Wesley PublishingCompany, Reading, MA, 1984.

[19] Magretta, J., “The Power of Virtual Integration: An Interview with Dell Computer'sMichael Dell,” Harvard Business Review, March-April (1998), 73-84.

[20] Swaminathan, J. M. and S. R. Tayur, “Managing Design of Assembly Sequences forProduct Lines that Delay Product Differentiation,” IIE Transactions, 31 (1999), 1015-1027.

[21] Swaminathan, J. M. and S. R. Tayur, “Managing Broader Product Lines Through DelayedDifferentiation Using Vanilla Boxes,” Management Science, 44 (1998), S161-S172.

[22] Whitt, W., “The Queueing Network Analyzer,” Bell System Technical Journal, 62 (1983),2779-2815.

[23] Wright, P., 21st Century Manufacturing, Upper Saddle River, Prentice Hall, NJ, 2001.

35

Appendix

A - Performance Metrics for a Pure Make-to-Stock System

Since demand is Poisson and processing times are exponentially distributed the two stages of

the production process behave like M/M/1 queues in tandem. Therefore, )(rπ , the probability

that there are a total of r jobs in process is

−−−−

==−+

=

−=

+

=∑

otherwise. )/1(

])/(1)[1)(1(

)1)(1(

)()()(

21

21

2121

212

01211

1

ρρρρρρρ

ρρρρρ

πππ

rr

r

r

r

ifr

rrrr

(A.1)

Consider the size of the average inventory for type j finished goods. The inventory buffer is non-

empty only if the number of type j jobs in process is less than bf. Thus, by definition, the average

inventory of type j finished items can be written as:

∑ ∑=

∞

=−=

f

j jj

b

k krrkjffjf rpkbbI

0,, ),()()( π (A.2)

where rk jp , is the conditional probability that kj jobs are present at stages 1 and 2, given that

there are a total of r jobs. Furthermore, owing to the equal arrival rates assumption, we have

.11

)!(!

!,

jj

j

krk

jjrk

M

M

Mkrk

rp

−

−

−= (A.3)

Due to symmetry, the total average inventory, )( ff bI , is simply M times )( ff bI . Upon

substituting from A.1 and A.3 into A.2, and simplifying, we obtain the expression for )( ff bI

shown in 1.

Next, we derive an approximate expression for the average order delay experienced by an

arbitrary type j product demand (tagged customer) arriving to the system. It experiences delay

only if the total number of type j jobs in the two production stages exceeds bf. Let there be k ≥ bf

type j jobs in the system at the moment of arrival of the tagged customer. Notice that we have

dropped the subscript j on account of symmetry. Then, the tagged arrival will experience a delay

which equals (k–bf+1)+Y service completions, where Y is a random variable with support [0, r-

k]. It represents all type l, l≠j, jobs that will have to be processed, on account of first-in-first-out

36

service discipline, until the (k-bf+1)th type j job is processed. The latter will be used to satisfy the

tagged customer. Then, using combinatorial arguments, the probability that Y=y, denoted by py,

can be written as follows:

.!

)!(!

)!()!1()!(!

)!1()!(

−

−−−−

−−−++−=

r

krk

ykrbbky

ykbrybkp

ff

ffy (A.4)

Let )(rD denote the average inter-departure time from the production system when there are r

jobs in it. Also, let Qi denote the steady-state queue lengths at stage-i. Since arrivals are

Poisson, we have:

+−

>−=

.otherwise)(

,0)(

2

ttT

QiftTrD (A.5)

From earlier arguments, the conditional probability that Q2=0, given that Q1+Q2=r, denoted by

0,rπ , can be written as follows:

−

−

==+

=

+otherwise.

)/(1

)]/(1[)/(

1

1

121

2121

21

0,

r

rr

ifr

ρρρρρρ

ρρρ

π (A.6)

Let )(, fjf bF denote the average waiting time experienced by a type j tagged customer. Then,

∑ ∑ ∑ ∑∞

= =

−

=

+−

=

−≈

f f

f

br

r

bk

kr

y

ybk

yrkfjf rrDppbF0 0

,, ).()())(()( πl

l (A.7)

Note that equation A.7 is an approximation since it ignores dependence among states of the

system. Due to symmetry, an arbitrary type j customer experiences the same average delay as an

arbitrary arrival, irrespective of class, i.e., )()( , fjfff bFbF = . Therefore, upon substituting

from A.1, A.3, A.4, and A.5 into A.6, and simplifying, we obtain 2.

37

B - Input Process for Stage-2 in a System with Delayed Differentiation

Lemma B.1: The input process to production stage-2 can be characterized as follows.

++Λ++Λ

Λ−Λ−

+ΛΛ

=+ΛΛ

=

otherwise.sss

s

s

bifs

sAdb

d

211

21

*2

))()((

)(

,0

)(

1

µµ

µρ (B.1)

where A*(s) is the Laplace Stieltjes transform of the random variable A.

Proof: If bd=0, the stage-2 input process coincides with the departure process of stage-1. Here

Burke's theorem applies and therefore )/()(*2 ssA +ΛΛ= . For bd ≥ 1, let’s consider the jth

arrival to stage-2 under steady state operations. It coincides either with an external arrival (this

happens when the intermediate buffer is not altogether empty),or with the moment of a service

completion at stage-1. Let jτ denote the arrival epoch of the jth arrival, and A2,j the inter-arrival

time. Clearly,

,1 ,1,2 ≤∀−= − jA jjj ττ

where we define 00 =τ . Let )(1

jQ denote the steady state number of customers in stage-1 at the

moment of jth arrival to stage-2, and Q1 denote the steady state number in stage-1 at an arbitrary

observation epoch. Since stage-1 queue is an M/M/1 queue and each stage-2 arrival epoch

coincides either with a stage-1 arrival epoch or with a stage-1 departure epoch, it can be argued

that )(1

jQ is identical in distribution to Q1 for all j.

We define three random variables X, X1 and X2. All three are exponentially distributed with

parameters 1µ+Λ , Λ and 1µ respectively. Thus, X1 is the time between consecutive external

arrivals and X2 is the time between departures from stage-1 when )(1

jQ > 0. Similarly, X is the

time until next event which could either be an arrival or a service completion at stage-1.

Consider, ,0 , ≥∀ jjτ the moment at which jth customer has just arrived. The time until next

arrival to stage-2, A2,j+1, can be written as:

38

>

=+ΛΛ

+

+Λ+

<

=

>

>

+

.

,))((

))((

,

)(12

)(12

1

1

11

1

)(11

1,2

12

21

dj

dj

XX

XX

dj

j

bQifX

bQifX

XX

bQifX

A

µ

µµ

(B.2)

In B.2, the notation 211 XXX > denotes the duration of X1 given that X1 > X2. The following

contingencies give rise to relationship B.2:

1. If at dj

j bQ <)(1,τ , next arrival to stage-2 will coincide with the next external arrival. Since

the forward recurrence time of X1 is identical in distribution to X1, it follows that Aj+1 = X1

whenever Q1(j) < bd.

2. If at dj

j bQ =)(1,τ , next arrival to stage-2 occurs after 2 events take place. The first event

happens when either an arrival or a service completion occurs. Owing to the memoryless

property of the of the exponential distribution, the time until this event has distribution X.

The second event is either an arrival or a service completion depending on whether X1 > X2

or X2 > X1. This explains the two terms, 211 XXX > and

122 XXX > , which represent the

partial distributions of X1 and X2 respectively. 211 XXX > follows X if the first event is a

service completion which occurs with probability )/( 11 µµ +Λ . Similarly, 122 XXX >

follows X with probability )/( 1µ+ΛΛ .

3. If at bQ jj >)(

1,τ d, next arrival to stage-2 coincides with the next departure from stage-1.

Since the latter has a forward recurrence time of X2, it is clear that in this situation A2,j+1

equals X2.

Let )(211

aGXXX >

denote the cumulative distribution function of X1 given that X1 > X2. Using

notation gY to denote the probability density function of a random variable Y, we get

39

)(Prob

)and(Prob)(

21

211

211 XX

XXaXaG

XXX >>≤

=>

∫∫

∞=

∞=

=>

=>≤=

0 221

0 2211

)()|(Prob

)()|and(Prob

2

2

y

y

dyygyXXX

dyygyXXXaX

X

X

∫∫ ∫

∞

=

−Λ−

=

−

=

Λ−Λ=

0 1

0 1

1

1)(

y

yy

a

y

ya

yx

x

dyee

dyedxe

µ

µ

µ

µ

aaa eee Λ−−+Λ− −+Λ

−−= )1)((1 11

1

1)( µµµ

µ.

Upon differentiating the above, we get the following expression for the PDF of 211 XXX > :

aaX eeag

XXΛ−− Λ−+Λ=

>)1)(()( 1

1

1211

µµ

µ. (B.3)

Let )(1 rπ denote the probability that stage-1 has 0≥r customers at an arbitrary observation

epoch. Since )1()( 111 ρρπ −= rr , we can write

).()}(

)(){1()()1()(

212|2

21|111,2

11

1

1

1111

xgxg

xgxgxg

Xd

XXXX

XXXXd

Xd

jA

b

bb

++

+ΛΛ

++Λ

−+−=

>+

>++

ρµ

µµρρρ

(B.4)

Next, we take the Laplace Transforms on both sides of (B.4) and simplify to obtain the

following relationship:

).()

)()(1()1()(

11

1

1

1

1

11

111

1

11

*1,2

sss

ssssA

d

dd

b

bbj

+Λ+

++Λ−

++

++ΛΛ−

+ΛΛ

++Λ+Λ−+

+Λ−=+

µρµµ

µµ

µµµρρ

µµρ

B.5)

Since the distribution of A2,j+1 (the RHS of B.5) is independent of the index j, we drop this

subscript and simplify B.5 to obtain:

.))()((

)()(

211

211*

2sss

s

ssA

db

++Λ++ΛΛ−Λ−

+ΛΛ=

µµµρ

(B.6)

40

Notice that ∞→+ΛΛ→ dbas

ssA )(*

2 . This agrees with intuition: when buffer size is large,

the two stages operate like two independent M/M/1 queues. Hence proved.

Next we show that 12

2≈AC and that the stage-2 can be approximated by a M/M/1 queue.

From B.1, it is easy to obtain the following additional properties of A2:

2'*'

22*'

222

)0()(,1

)0(2 Λ

==Λ

=−= AAEAA if bd=0, and2

12

1

112 )1(

)1(22

ρµρρ

+

−−

Λ

db if 1≥db . Similarly,

12

2=AC if bd=0, and 2

112

1 )1/()1(21 ρρρ +−− +db if 1≥db . Thus, we have a M/M/1 queue at

stage-1 and a G/M/1 queue at stage-2. The steady state queue length distribution at stage-2,

)(2 rπ , can be obtained as )1()(2 xxr r −=π , for r = 0, 1, …, where )1,0(∈x is a solution to the

equation: )( 22*

2 xAx µµ −= (see, for example, Kleinrock [14], pp. 251).

Since we need closed form expressions to be able to perform optimization, we set 122 ≈AC ,

i.e., we treat stage-2 queue as a M/M/1 queue. The reasonableness for this approximation can be

seen upon calculating 100]/)1[( 22

22 ×−=∆ AA CC , or the percent error upon assuming

12

2=AC . Notice that for 1≥db , the percent error ∆ is monotonically decreasing as ∞→b .

Therefore, we set bd = 1 and find that the maximum error is 7.7% and that it occurs when

686.01 =ρ . In fact for bd greater than 1, the error quickly goes to zero, as shown below.

For most practical situations, we expect b to be greater than 1 in order to keep response times

fast. Also, the intended use of our models lies in pre-design evaluation of buffer size and

location alternatives. Given these facts, we believe assuming 12

2≈AC is a reasonable

approximation. A close match was also observed between exact mean order fulfillment times

(obtained by solving the G/M/1 queue), and the corresponding approximate values obtained from

assuming 122 =AC . For example, if we set b=1, 686.0=Λ , and 11 =µ (notice that this is a case

in which our approximation is most inaccurate), we observe that our approximation

overestimates the exact value of the mean order fulfillment time at stage-2 by 2.85%.

41

bd=1 bd=5 bd=10 bd=20 bd=40 bd=80

ρ1=0.2 0.897 0.001 0.000 0.000 0.000 0.000

ρ1=0.3 2.288 0.018 0.000 0.000 0.000 0.000

ρ1=0.4 4.078 0.100 0.001 0.000 0.000 0.000

ρ1=0.5 5.882 0.348 0.011 0.000 0.000 0.000

ρ1=0.6 7.239 0.883 0.068 0.000 0.000 0.000

ρ1=0.7 7.667 1.740 0.288 0.008 0.000 0.000

ρ1=0.8 6.747 2.658 0.856 0.091 0.001 0.000

ρ1=0.9 4.209 2.722 1.590 0.454 0.066 0.001

C - Performance Metrics for a System with Delayed Differentiation

Recall that Qi denotes the steady state number of in-process jobs at stage-i at an arbitrary

moment of observation. It is easy to see that the number of semi-finished items in the

intermediate buffer = max(0, bd - Q1) and that Q1 is independent of stage-2. Therefore, the

average buffer inventory can be written as:

,)()()(0

1∑=

−=db

rddd rrbbI π (C.1)

where }{Prob)( rQr ii ==π , for 0≥r . We obtain equation (3) upon simplifying the above sum

after substituting )1()( 111 ρρπ −= rr .

To obtain )( dd bF , note that each arrival must wait for processing at stage-2, whereas it

waits at stage-1 only when the intermediate buffer is empty. Furthermore, since each external

arrival is a Poisson arrival, it sees time average behavior. Thus,

,)1(

)()1()(

2

211

ρρπρ−Λ

+Λ

+−= ∑∞

=brdd

rbrbF (C.2)

where the second term on the right hand side of C.2 is the expected order delay in a M/M/1

queue. Simplification of equation C.2 yields 4.

42

D - Performance Metrics for a System with Partial Delayed Differentiation

The probability, )( ii nπ , of having ni jobs of type i in process in stage-1 is given by [2]:

ii

i nn

n

ii rrM

Mn 111

11

11 )1())1((

)1( )( −=

+−

−= +ρρ

ρρπ , (D.1)

where

11

11

)1( ρρρ

+−=

Mr . (D.2)

Average inventory for each product i is given by:

∑∑==

−−=−=pdpd b

j

jpd

b

jipdpdipd rrjbjpjbbI

01

0, )1()()()()( , (D.3)

which, upon simplification, leads to:

( )pdbpdpdipd r

r

rbbI 1

1

1, 1

1)( −

−−= . (D.4)

Given the symmetry in the system, total average inventory is obtained by multiplying D4 by M

which yields 18.

Average order delay can be obtained by first noting that the average number of backorders

for each product is given by:

1

11

1,

1

)()1()()()(

r

r

rbnrnpbnbB

b

bn

npdi

bniipdipdipd

pdi

i

pdi

−=

−−=−=

+

∞

=

∞

=∑∑

(D.5)

Then average order delay, by virtue of Little’s law, is given by

( ) )1(1

)()(

2

2

1

11,

, ρρ

λ −Λ+

−Λ==

+

r

MrbBbF

b

i

pdipdpdipd . (D.6)

Given symmetry , we have Fpd(bpd) = Fpd,i(bpd) which leads to 19.

43

E - Properties for a System with Partial Delayed Differentiation

Property 1: The ratio *d

*pd zz / → 1 as ρ1 → 1.

Proof: Recognizing that r1/(1 - r1) = ρ1/{M(1 - ρ1)}, the ratio *d

*pd zz / can be rewritten as:

[ ]

[ ]

[ ][ ]

[ ][ ]

−−

−−

+−

−−

+−

−−

=

−

+−

−

1ln

)11(1ˆln

)1(ln

)1(ln

111

111

11

11

11

1

111

**

1 ln

)1(ln)1(

)1(1

)1(ln

)1(ˆln)1(

/

11

1

11

ρρµα

ρρρ

ρµα

ρρρ

ρµαρ

ρρρρ

ρρρ

ρµαρ

!

!

M

M

M

M

zz dpd . (E.1)

Noting that

[ ]

+−

−

11

1

11

)1(ln

)1(ln

ρρρ

ρµα

M

! and

[ ][ ]

−ρ

ρµαln

)1(ln 11!

are very large number when ρ1 → 1, we can ignore integrality. Then, taking advantage of the

fact that )()ln()( xfaxf ea = , it is straightforward to show that

[ ] ( )

[ ][ ] ( )

.)1(ˆ1

ln

)1(ln)1(

)1(1

)1(ln

)1(ˆln)1(

lim/lim

1111

111

111

11

1

111

1

**

1 11 ρµαρρ

ρµαρ

ρµαρ

ρρρ

ρµαρ

ρρ −−−−−

−−−

+−

−−

=→→

!

!

M

M

zz dpd (E.2)

Let

44

[ ]

[ ][ ]

( ).)1(ˆ1)(

and ;ln

)1(ln)1()(

,

)1(ln

)1(ˆln)1()(

1110

1

1112

11

1

1111

ρµαρρ

ρρµαρρ

ρρρ

ρµαρρ

−−=

−−=

+−

−−=

f

f

M

Mf

!

Then, we can rewrite our limit as:

).)(

)()(

)()(

)()(

)(

)()((lim/lim

12

11

1012

12

11

011

1

**

1

1

11 ρρ

ρρρ

ρρρ

ρρ f

f

ff

f

f

ffzz dpd −

−=

→→ (E.3)

In order to show that that 1/lim **

11

=→

dzz pdρ, it is sufficient to show that

.1)(

)(lim

)()(

)(lim

)(

)()(lim

12

11

1012

12

11

011

1111

1

11

==−

=−

→→→ ρρ

ρρρ

ρρρ

ρρρ f

f

ff

f

f

ff

First, note that

1)(lim 011

=→

ρρ

f (E.4)

and

[ ][ ] [ ]ρρρ

ρµαρρρρ +−−

−−=→→ )1(lnln

)1(ˆln)1(lim)(lim

11

1111

1111 M

Mf ,

which, upon applying L’Hospital rule, reduces to

[ ] ∞=+−−−−−=

→→ ρρρρρµαρ

ρρ ])1(/[/1

)1(ˆlnlim)(lim

11

111

1111 MM

MMf . (E.5)

Similarly, we can show that.)(lim 2

11

∞=→

ρρ

f (E.6)

Consequently, we have

1})(

)(1{lim

)(

)()(lim

11

10

111

1011

1 11

=−=−→→ ρ

ρρ

ρρρρ f

f

f

ff (E.7)

and

1})(/)(1

1{lim

)()(

)(lim

121011012

12

1 11

=−

=− →→ ρρρρρ

ρρ ffff

f. (E.8)

45

Finally, we have

.)

))1(ln(

)ln(lim

)(

)(lim

11

1

1

2

1

1111

ρρρ

ρρρ

ρρ

+−

=→→

M

M

f

f

Since both the numerator and denominator are zero as ρ1 → 1., we apply L’Hospital’s rule to

obtain

.1])1(/[

/lim

)(

)(lim

1

1

2

1

1111

=+−

=→→ ρρρ

ρρρ

ρρ MM

M

f

f (E.9)

Hence, 1/lim **

11

=→

dzz pdρ

.

Property 2: *d

*pd zz = when )ˆ1/(ˆmin1 Λ+Λ=≤ ααρρ .

Proof: The result follows from the fact that when )ˆ1/(ˆ1 Λ+Λ≤ ααρ , we have )/(1ˆ 1 Λ−≥ µα . In

this case, a pure make-to-order system is optimal. Therefore, we have .0== *d

*pd zz

Property 3: *d

*pd zz / → M(1 – r1)/(1 – ρ1) as ρ1 → ρmin

+.

Proof: By the limit ρ1 → ρmin+, we mean that ρ1 = ρmin + ε, with ε very small. Thus, even though

ε is approaching zero, we have ρ1 > ρmin. Therefore, a pure make-to-order system is not feasible

(i.e., this is the smallest ρ1 for which a pure-make-to-order is not possible) and we have bpd* =

bd* = 1. If we substitute bpd* = 1 in zpd*, we obtain zpd* = M(1 – r1) and if we do the same for

zd*, we obtain zd* = 1 – ρ1. Taking the ratio leads to the desired result.

Note that this result is due to the integrality of b* so that if we slightly increased utilization

above ρmin, we move from a pure make-to-order to a system with an intermediate buffer. The

system with M products is forced to keep a buffer of one unit for each product while the system

with pooling can get away with just one buffer of unit size. Again, this is due to the fact that b is

an integer and not a continuous variable.

Property 4: If ]1)1(ˆ

[)1( 1

1

1

1 −−Λ−

=≥ρα

ρρ

ρmaxMM , then *

pdz = M(1 – r1).

Proof: Since the optimal buffer size is given by:

46

[ ]

+−

Λ−=

11

1

1*

)1(ln

)(ˆln

ρρρ

µα

M

bpd ,

bpd* = 1 if[ ]

,1

)1(ln

)(ˆln

11

1

1 ≤

+−

Λ−

ρρρ

µα

M

(E.10)

or equivalently

].1)1(ˆ

[)1( 1

1

1

1 −−Λ−

≥ρα

ρρ

ρM (E.11)

Substituting bpd* = 1 in *pdz leads to *

pdz = M(1 – r1).

Property 5: *d

*pd zz / → Mln[ρ1]/ln[r1] as α̂ → 0.

Proof: Noting that

[ ]

+−

−

11

1

11

)1(ln

)1(ln

ρρρ

ρµα

M

! and

[ ][ ]

−ρ

ρµαln

)1(ln 11!

are very large number when ,0→α! we can ignore integrality. Then, the limit can be written as

[ ]( )

[ ][ ] ( ) ))1(1(

1ln

)1(ln

))1(1(1

)1(ln

)1(ˆln

lim/lim

111

1

1

11

111

1

11

1

11

0

**

0 ρµαρ

ρρ

ρµα

ρµαρ

ρ

ρρρ

ρµα

αα −−−

−−

−−−

−

+−

−

=→→ !

!

!

!!

M

M

zz dpd . (E.12)

Since the limit of both numerator and denominator is infinity, we apply L’Hospital’s rule toobtain

[ ]

[ ].

)1(ln

ln

ln

1

)1(ln

lim/lim

11

1

1

1

11

1

0

**

0

+−

=

+−

=→→

ρρρ

ρ

ρ

ρρρ

αα

M

MM

M

zz dpd !! (E.13)

47

Property 6: *d

*pd zz / = M(1 – r1)/(1 – ρ1) when

Λ−≤≤

−Λ 11

21

ˆ)1(

1

µα

ρ

ρ.

Proof: The ratio *d

*pd zz / = M(1 – r1)/(1 – ρ1) when 1== *

d*pd bb . Since *

d*pd bb ≥ we have

1== *d

*pd bb when

[ ]1

]ln[

)(ˆln

1

1 ≤Λ−

ρµα

or equivalently when

)1(ˆ

1

21

ρρα−Λ

≥ .

The desired result follows by noting that (a) *pdb and *

db are non-increasing in α̂ and (b)

*pdb = *

db for )/(1ˆ 1 Λ−> µα .