Make-to-order, Make-to-stock, or Delay Product Differentiation? – A ...
Transcript of Make-to-order, Make-to-stock, or Delay Product Differentiation? – A ...
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: [email protected].
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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[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.
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[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.
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[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 .