MANAGEMENT SCIENCE
informs
Vol. 56, No. 3, March 2010, pp. 495–502
issn 0025-1909 eissn 1526-5501 10 5603 0495
®
doi 10.1287/mnsc.1090.1127
© 2010 INFORMS
Vertical Flexibility in Supply Chains
Wallace J. Hopp
Stephen M. Ross School of Business, University of Michigan, Ann Arbor, Michigan 48109, whopp@umich.edu
Seyed M. R. Iravani, Wendy Lu Xu
Department of Industrial Engineering and Management Sciences, Northwestern University,
Evanston, Illinois 60208 {s-iravani@northwestern.edu, wendy-xu@northwestern.edu}
J
ordan and Graves (Jordan, W. C., S. C. Graves. 1995. Principles on the benefits of manufacturing process
flexibility. Management Sci. 41(4) 577–594) initiated a stream of research on supply chain flexibility, which
was furthered by Graves and Tomlin (Graves, S. C., B. T. Tomlin. 2003. Process flexibility in supply chains.
Management Sci. 49(7) 907–919), that examined various structures for achieving horizontal flexibility within a single
level of a supply chain. In this paper, we extend the theory of supply chain flexibility by considering placement
of vertical flexibility across multiple stages in a supply chain. Specifically, we consider two types of flexibility—
logistics flexibility and process flexibility—and examine how demand, production, and supply variability at a single
stage impacts the best stage in the supply chain for each type of flexibility. Under the assumptions that margins
are the same regardless of flexibility location, capacity investment costs are the same within and across stages,
and flexibility is limited to a single stage of logistics (process) flexibility accompanied with necessary process
(logistics) flexibility, we show that both types of flexibility are most effective when positioned directly at the
source of variability. However, although expected profit increases as logistics flexibility is positioned closer to
the source of variability (i.e., downstream for demand variability and upstream for supply variability), locating
process flexibility anywhere except at the stage with variability leads to the same decrease in expected profit.
Key words: supply chain; flexibility; capacity investment
History: Received March 20, 2007; accepted April 25, 2009, by Paul H. Zipkin, operations and supply chain
management. Published online in Articles in Advance January 12, 2010.
1.
Introduction
Several authors have studied the problem of how
to use process flexibility. Jordan and Graves (1995)
showed that most of the benefits of full flexibility
(ability to produce and ship all products from all
plants) can be achieved by partial flexibility. Iravani
et al. (2005) introduced the concept of structural flexibility to capture the ability of a flexibility structure
to respond to demand or supply variability. Graves
and Tomlin (2003) presented a framework for analyzing the benefits of flexibility in a multistage supply
chain and developed a flexibility measure and guidelines for flexibility investment. Their paper addressed
the question of which flexibility structure is most efficient provided all stages of the supply chain make
use of the same flexibility structure. Other studies
include Fine and Freund (1990), Gupta et al. (1992),
and Van Mieghem (1998), among others. Taken as a
whole, this stream of research has provided a number
of useful insights that describe the impact of flexible
technology in a supply chain. However, all of these
studies have focused on process flexibility within a
single stage of the supply chain.
A multiechelon supply chain also presents the question of which stage to target for flexibility investment.
We term this the vertical flexibility problem because
of the analogy to vertical integration. Consequently,
The fundamental problem in any supply chain system
is efficiently matching supply with demand. Because
supply and demand are uncertain, we must make use
of various buffers, including safety stock, safety lead
time, and safety capacity, to facilitate this matching
problem. A well-known principle of factory physics
is that flexibility can reduce the amount of buffering needed to mitigate the effects of variability (Hopp
and Spearman 2008). Examples of flexible capacity
in a supply chain include (a) Dell sourcing multiple
mother boards from a single supplier, (b) HewlettPackard (HP) assembling voltage adaptors to printers
in its European distribution center before shipping
them to countries with different AC voltage standards, and (c) General Motors (GM) tooling stamping plants to produce body parts for more than one
model.
In each of these cases, by using capacity that can
be shifted from one product type to another, the firm
enhances its ability to adjust to fluctuations in either
the supply of materials or demand for products. However, as these examples highlight, the flexibility can
be positioned at different levels of the supply chain,
including suppliers (Dell), component plants (GM), or
distribution (HP).
495
Hopp, Iravani, and Xu: Vertical Flexibility in Supply Chains
496
Figure 1
Management Science 56(3), pp. 495–502, © 2010 INFORMS
(a) and (b): Examples of Horizontal Flexibility (How to Use Flexibility in a Given Stage); (c) and (d): Examples of Vertical Flexibility
(What Stage to Make Flexible)
(a) Full flexibility
(c) Single-stage full logistics flexibility
Stage 5
Plant 1
ABC
Plant 2
ABC
Plant 3
ABC
Demand
A
Demand
B
Demand
C
Stage 4
Plant 1
Plant 1
Plant 1
ABC
ABC
ABC
Plant 2
Plant 2
Plant 2
ABC
ABC
ABC
Plant 3
Plant 3
Plant 3
ABC
ABC
ABC
(b) Chained flexibility
AB
Demand
A
Stage 2
Stage 1
Stage 0
Plant 1
Plant 1
Demand
A
A
Plant 2
Plant 2
Demand
B
B
B
Plant 3
Plant 3
Demand
C
C
C
Stage 1
Stage 0
A
(d) Single-stage full process flexibility
Stage 5
Plant 1
Stage 3
Stage 4
Plant 1
Plant 1
A
A
Stage 3
Plant 1
ABC
Stage 2
Demand
Plant 1
Plant 1
A
A
Plant 2
Plant 2
Demand
B
B
B
A
Plant 2
Demand
Plant 2
Plant 2
BC
B
B
B
Plant 3
Demand
Plant 3
Plant 3
Plant 3
Plant 3
Plant 3
Demand
CA
C
C
C
ABC
C
C
C
Plant 2
ABC
Notes. Unshaded boxes denote specialized plants. Shaded boxes denote flexible plants.
we refer to flexibility within a stage as horizontal flexibility. Figure 1 contrasts sample horizontal flexibility structures (Figure 1(a) and (b)) with sample vertical flexibility structures (Figure 1(c) and (d)). These
structures contain two types of flexibility: logistics flexibility (the ability to ship products to different locations) and process flexibility (the ability to produce different types of products). Aprile et al. (2005) used
numerical studies to compare lost sales resulting from
different process and logistics flexibility configurations in a fixed-capacity, five-product, two-stage supply chain. They observed that, given some degree
of logistics flexibility, process flexibility in the supply stage enables the system to cope with demand
variability. They also noted that process flexibility in
the assembler stage is more beneficial when there is
capacity variability in both supplier and assembler
stages.
Our paper goes beyond the results of Aprile et al.
(2005) toward a theory of vertical flexibility by providing analytical results of where to locate flexibility within a supply chain. We do this by proving a
principle that describes how the optimal location for
full logistics and process flexibility in a multiechelon
multiproduct supply chain is affected by variability in supply, demand, and intermediate processing.
Our main insight is that if (a) margins are the same
regardless of flexibility location, (b) capacity investment costs are the same within and across stages,
(c) only one stage in the supply chain has variability,
and (d) flexibility decisions are limited to locating a
single stage of full logistics (process) flexibility accompanied with necessary process (logistics) flexibility,
then logistics (process) flexibility is most effective
when positioned directly at the source of variability.
However, while the effectiveness of logistics flexibility increases with proximity to the source of variability, the effectiveness of process flexibility is equally
suboptimal when located at any stage other than the
stage with variability.
2.
Model Formulation
We consider a multiechelon, multiproduct supply
chain, which produces I different products, indexed
by n = 1 2 I, and has K + 1 stages, indexed by
k = 0 1 K, and I plants per stage. Note that the
number of plants at each stage is assumed to be the
same as the number of products so that, for a supply chain with no flexibility, each product has a dedicated plant at each stage of the supply chain. Stage 0
denotes demand, so that node n of stage 0 represents
the retail outlet for product n, whereas stage K represents the initial (supply) stage. We assume that there
Hopp, Iravani, and Xu: Vertical Flexibility in Supply Chains
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Management Science 56(3), pp. 495–502, © 2010 INFORMS
are always sufficient raw materials at stage K and that
the cost of these materials is included in how much
the company earns for selling one unit of the product.
Shipping routes between plants in stage k and stage
k − 1, or between plants in stage 1 and demand
nodes at stage 0, are represented by an arc set Ak ,
k = 1 2 K, where plant i at stage k can supply
plant j at stage k − 1 (or demand node j if k = 1) iff
i j ∈ Ak . Plant i at stage k can produce product n
iff i n ∈ B k , k = 1 2 K. Set A = A1 AK and
set B = B 1 B K , respectively, represent the logistics and process flexibility configurations of the supply chain.
We assume both demands and production capacities can be random. However, to focus on the effect
of variability on the optimal location of flexibility,
we restrict our attention to cases where only a single
stage has variability. For a given flexibility configuration (i.e., fixed A and B), we formulate the problem of
maximizing expected profit as a two-step sequential
decision process:
1. Capacity Investment Decision: First, before
demand is observed, the firm chooses production
capacity levels for all plants, ki , by taking into
account demand distributions and unit capacity
investment costs cik , i = 1 2 I, k = 1 2 K.
For a plant with yield loss and machine failures
and other sources of variability, we consider the
production capacity to be a random variable, Qik ki ,
which follows distribution f Qik ki that depends on
the level of ki . We use qik to represent a realization
of capacity Qik ki and let = ki and c = cik be
corresponding matrices of capacity and capacity
investment cost, with ki and cik representing the
entries at the ith row and kth column.
2. Production Flow Decision: After all uncertain
demand or production capacities, or both, have been
observed, the firm chooses a matrix of production and
k
k
shipping flows, X = Xijn
, where Xijn
represents the
quantity of product n produced in plant i of stage k
for plant j of stage k − 1. Note that this flow matrix is
constrained by the flexibility configuration of the supk
ply chain. A flow Xijn
can be nonzero only if i j ∈ Ak ,
k
i n ∈ B , and j n ∈ B k−1 . In other words, a flow of
product n from plant i to plant j can only exist if there
is a shipping route from i to j and both plants are
able to process the product. Taking into account the
fact that distribution center i at stage 0 is for product type i, to make our model concise, we define a
set B 0 for the demand nodes, such that j n ∈ B 0 iff
j = n. To simplify the notation, we define set F k as
k
the set of all triples i j n for all feasible flows Xijn
k
k
k−1
that satisfy i j ∈ A , i n ∈ B , and j n ∈ B . Let
rn denote the unit selling price of product n minus
k
the cost of raw materials; pin
the unit production cost
k
the unit transof product n in plant i of stage k; tijn
portation cost of product n from plant i at stage k to
plant j of stage k − 1; and r, p, and t the corresponding vectors (matrices). To maximize profit, the firm
observes demand vector d = d1 d2 dI and production capacity matrix q = qik , and then chooses its
production flow matrix X as the optimal solution to
the following linear program, which we call problem
P2
A B d q, where · represents the maximum
profit:
ABdq
= max
X
ijn∈F 1
1
rn Xijn
−
K
k=1 ijn∈F k
k
k
k
pin
+tijn
Xijn
(1)
subject to
u u i n∈F k+1
k+1
Xuin
=
j i j n∈F k
k
Xijn
i n = 1 2 I k = 1 2 K − 1
jn i j n∈F k
i j i j n∈F 1
k
Xijn
≤ qik
i = 1 2 I
k = 1 2 K
1
Xijn
≤ dn
k
≥0
Xijn
(2)
n = 1 2 I
(3)
(4)
i j n = 1 2 I
k = 1 2 K
(5)
Constraint (2) is the balance equation that sets the
total production flow into a plant equal to the total
flow out of it for each product, with the implicit
assumptions that to meet one unit of demand for
product n, (a) one unit of capacity is needed at each
stage and (b) all products consume the same processing capacity at the plant. Constraint (3) guarantees
that the total quantity of production of a plant does
not exceed its capacity. Constraint (4) avoids producing more than needed. Constraint (5) ensures nonnegativity of production flow.
Solving P2
A B d q is premised on first making capacity investments. To do this, the firm considers a random demand vector D = D1 D2 DI and selects a production capacity matrix = ki that decides the distribution of corresponding random matrix Q
= Qik ki . Profit is therefore a
random variable, A B D Q
, that depends on
the demand and capacity distributions. For a given
demand d and capacity q, profit is A B d q,
which is found by solving P2
A B d q. Hence, we
can express the capacity investment decision faced by
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the firm as solving the following problem, which we
label P1
A B D:
V ∗ A B D
I
K cik ki (6)
= max E DQ
!
A B D Q
" −
k=1 i=1
where E D Q
!
A B D Q
" is the expected
profit. The expectation is over random demand D and
random capacity Q
, and V ∗ · is the maximum
value of expected profit minus capacity investment
cost. The matrix ∗ that achieves V ∗ · is called the
optimal capacity investment strategy.
As shown in Jordan and Graves (1995) and Graves
and Tomlin (2003), there are many ways a single stage
of a supply chain can be made flexible. Because the
focus of this paper is on the position, rather than the
type, of flexibility, we will focus on full flexibility and
will assume a single stage of flexibility. Full logistics
flexibility is achieved at a stage kf if what is produced in each plant at stage kf can be shipped to all
plants at stage kf −1. We use A1full kf to represent full
logistics flexibility configuration, where A1full kf =
A1 Akf AK , with Ak = i i ∀ i = 1 2 I
for k = kf , and Akf = i j ∀ i = 1 2 I j =
1 2 I. It is worth emphasizing that to make use
of full logistics flexibility of stage kf , stage kf and all
stages upstream to kf must have process flexibility. As
illustrated in Figure 1(c), to make use of full logistics
flexibility of stage 3, plants 1, 2, and 3 at stage 3 and
all upstream stages must be able to process products
A, B, and C.
Full process flexibility is achieved at a single
stage kf if all product types can be processed in
each plant of stage kf . We use B1full kf to represent full logistics flexibility configuration, where
B1full kf = B 1 B kf B K , with B k = i i ∀ i =
1 2 I for k = kf , and B kf = i n ∀ i = 1 2 I
n = 1 2 I. To make use of process flexibility at
stage kf , stages kf and kf + 1 must have logistics flexibility so that plants at stage kf are supplied with
subassemblies of all products and are able to ship
all products to the subsequent stage. As illustrated in
Figure 1(d), to make use of full process flexibility of
stage 3, plants 1, 2, and 3 at stage 3 all need to ship
products to all plants at stage 2. Also all plants must
have supply from plants 1, 2, and 3 at stage 4. Therefore, full logistics flexibility is required at stages 3
and 4.
In the remainder of this paper, we focus on the
location of a single stage of full logistics (process)
flexibility. Also, when we say a stage is flexible, we
mean it has full logistics (process) flexibility and the
corresponding process (logistics) flexibility to make it
possible.
We assume that implementing full logistics flexibility at stage k incurs a fixed cost $k ≥ 0,
k = 1 2 K, to establish the shipping channels to
all plants of stage k − 1. Also, full process flexibility
at stage k incurs a fixed cost % k ≥ 0, (k = 1 2 K),
to equip the plant with the necessary tooling to process all types of products. We use $K+1 to denote
the fixed cost incurred for plants at stage K to establish inbound logistics flexibility (supply channels) to
obtain all types of raw material. Hence, to evaluate a
flexibility configuration A B, we need to compute
V ∗ A B D and subtract from it the fixed cost associated with the flexibility structure.
To develop our results on the optimal position of
full logistics flexibility in a supply chain, we first need
to characterize the solution to P2
A1full kf B d q.
We define
mL in kf = rn −
−
kf −1
k=1
kf −1
k=1
k
pnn
+
K
k=kf
k
pin
K
kf
k
k
+ tinn +
tnnn
tiin
k=kf +1
as the unit contribution margin for production flow
from plant i of stage K to demand node n, where
rn is how much the company earns for selling one
kf −1 k
k
unit of the product, k=1 pnn
+ Kk=kf pin
is the production cost associated with the production flow, and
kf −1 k
K
kf
k
k=kf +1 tiin is the transportation cost.
k=1 tnnn + tinn +
We can show (see Lemma 1 in Online Appendix I,
provided in the e-companion)1 that if a supply chain
has logistics flexibility only at a single stage, the production flow allocation problem in the entire supply
chain can be simplified to a single stage production
flow allocation problem, where production flow from
kf
plant i of stage K to demand node n is given by Yin
and is associated with a unit profit margin, mL in kf .
With respect to process flexibility, we define
mP in kf = rn −
−
K
k=1
k=kf
k
pnn
K
k=1
k=kf kf +1
kf
+ pin
k
tnnn
kf
+ tinn
kf +1
+ tnin
as the unit contribution margin for production flow of
product n that is produced in plant i of stage kf and in
plant n of all other stages. The company earns rn for
1
An electronic companion to this paper is available as part of the online version that can be found at http://mansci.journal.informs.org/.
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Management Science 56(3), pp. 495–502, © 2010 INFORMS
selling the product. At the same time, this production
flow involves production cost
K
k=1
k=kf
kf
k
pnn
+ pin
Assumption 4. cik = cjk , i = 1 2 I, j = 1
2 I, i = j, k = 1 2 K
and transportation cost
K
k=1
k=kf kf +1
This assumption reflects the cost of flexibility, logistics or process flexibility, in the form of more sophisticated equipment, more highly trained staff, longer
routes, etc., which reduces the margins of products
produced in flexible plants or shipped along nonstandard distribution channels.
kf
kf +1
k
tnnn
+ tjnn + tnin We can show (see Lemma 2 in Online Appendix I)
that if a supply chain has process flexibility at only a
single stage, the production flow allocation problem
in the entire supply chain can be simplified to a single
stage production flow allocation problem, where flow
of product n that is produced in plant i of stage kf
kf
and in plant n of all other stages is given by Zin and
is associated with a unit profit margin, mP in kf .
Note that margin mL in kf or mP in kf depends on
the location (stage) of the logistics or process flexibility structure, respectively, and is therefore a function
of kf .
To generate our result, we will make use of the following assumptions throughout the paper.
Assumption 1. mL in k = mL in k
mP in k) for all i and n and all k = k.
(mP in k =
This assumption ensures that the unit contribution
margin is the same regardless of where logistics (process) flexibility is located, so that we can isolate the
role of variability from the role of cost.
k
Assumption 2. Unit variable production cost pin
and
k
unit transportation cost tijn are independent of whether the
capacity for product n at plant i of stage k is flexible or
dedicated.
The main components of the unit variable production cost are material and labor, which usually do
not depend on whether the capacity is flexible or
not. Consider, for instance, a flexible auto assembly
plant that produces models A and B, and a dedicated assembly plant that produces only model A. In
both plants, the material to produce model A is the
same, and the labor skill required (to install the doors,
for example) is the same. Furthermore, the cost of
shipping items from one plant to another is clearly
independent of whether those plants are flexible or
not. Thus, Assumption 2 represents many systems in
practice.
Assumption 3. mL ii kf > mL in kf , mL ii kf >
mL ni kf , and mP ii kf > mP in kf , mP ii kf >
mP ni kf ∀ n = i i n = 1 2 I 1 ≤ kf ≤ K
This assumption states that unit capacity investment costs are the same for all plants at the same
stage. This is a reasonable assumption in supply
chains where plants at one stage of the supply chain
perform similar processing functions and therefore
use similar equipment and facilities. For example, in
an electronics supply chain, semiconductor facilities
feed board assembly plants, which feed final assembly
plants. Because the complexity of the technologies at
each level is similar, the capacity costs of plants within
each level should also be similar. Of course, other supply chains may involve very different processes, with
very different capacity costs, at the same level. In such
cases, the differing capacity costs will obviously influence flexibility choices. However, because our focus is
on the influence of variability on the desired location
for flexibility, we consider only supply chains without
significant differences in capacity costs within stages.
The net effect of the aforementioned assumptions
is that variability will drive flexibility location decisions, not cost. Under these, we can show that for
the single-stage logistics (process) flexibility configuration, an optimal capacity configuration will have
the same capacity at all stages without variability or
(logistics or process) flexibility (see the proofs of Lemmas 3 and 4 in Online Appendix I).
3.
Optimal Location for Logistics
Flexibility
In systems where production facilities are already
flexible, increasing system flexibility can be achieved
by introducing logistics flexibility. This is often the
case in a pure distribution system consisting of warehouses, depots, and retail outlets, where all facilities
can process all products. But adding routes between
facilities may entail fixed and variable costs. Within
the Walmart distribution network, for example, personnel at any node (e.g., warehouse, depot, retail outlets) are able to process all types of products, and so
process flexibility can be realized without significant
costs, but logistics flexibility (e.g., supplying multiple
stores from depots) is not costless.
It can be shown that necessary conditions for
the optimal capacity investment strategies are that
if the single source of variability is downstream
(upstream) of the flexible stage, then plants downstream (upstream) of the flexible stage should have
Hopp, Iravani, and Xu: Vertical Flexibility in Supply Chains
500
larger total capacity than plants upstream (downstream) in order to hedge against the variability
(see Lemma 3 in Online Appendix I). This leads
directly to our main results for the case of a single
stage with logistics flexibility, which we present in
Theorem 1.
Theorem 1. For a multiechelon supply chain with full
logistics flexibility at only a single stage kf , process flexibility at all upstream stages of kf , and only a single source
of variability at stage kv , the following apply:
(1) If kv + 1 ≤ kf ≤ K − 1 (i.e., the source of variability is downstream from the stage with logistics flexibility),
then logistics flexibility at stage kf achieves higher expected
profit than does logistics flexibility at stage kf + 1.
(2) If 2 ≤ kf ≤ kv (i.e., the source of variability is
upstream from the stage with logistics flexibility), then
logistics flexibility at stage kf achieves higher expected
profit than does logistics flexibility at stage kf − 1.
Theorem 1 implies that if a supply chain has a single source of variability, and all stages have (full) process flexibility, it is most beneficial to place logistics
flexibility closest to the source of variability. The intuition behind this is as follows. Suppose we place the
logistics flexible stage adjacent to the source of variability on the downstream side (i.e., kv = kf ). This
allows us to use any available capacity of the plants at
the stage with variability kv (and its upstream plants,
because they feed the plants in the stage with variability kv ) to hedge against variability. Therefore, in
an optimal configuration, the capacity of plants at the
stage with variability kv (and upstream of stage kv )
will be higher than that of the dedicated plants downstream of the stage with variability kv . If we move
the flexible stage one level further from the source
of variability (i.e., kf = kv − 1), then we can still use
the plants at and upstream of the stage with variability kv to hedge against variability, but now we must
increase the capacity of the plants at an additional
stage (i.e., stage kf , because plants at stage kf should
have the capacity to process items that they receive
from plants at stage kv ). Hence, it costs more to get
the same amount of protection as the flexible stage is
moved away from the source of variability.
From a management perspective, these results suggest that demand variability (i.e., kv = 0) provides
motivation to make downstream stages of the supply chain flexible, whereas supply variability (i.e.,
kv = K) provides motivation to make the upstream
stages flexible. In systems where process flexibility is
inexpensive, this is a crisp insight. However, when
process flexibility is costly, downstream logistics flexibility becomes more expensive (because it requires
all upstream stages to have process flexibility). So,
when supply is variable, upstream logistics flexibility
is clearly preferable (because it is closer to source of
Management Science 56(3), pp. 495–502, © 2010 INFORMS
variability and requires fewer upstream stages to have
process flexibility). But when demand is variable, we
must balance the cost of the additional process flexibility with the benefits of positioning the logistics
flexibility further downstream. In Online Appendix II
(provided in the e-companion), we further investigate
this trade-off and show that the flexibility location
decision has a threshold structure.
4.
Optimal Location for Process
Flexibility
In addition to facilitating logistics flexibility, process
flexibility is effective in its own right. Indeed, in systems where full logistics flexibility is inexpensive (e.g.,
material is shipped between facilities via a third party
logistics firm and so additional routes can be added
without fixed cost), enhancing flexibility is solely a
matter of deciding where to add process flexibility. To
gain insight into this decision, we consider the problem of locating a single stage of full process flexibility.
It can be shown that the process flexible stage cannot have more (and may have less) capacity than the
other stages as a result of the ability to produce different products (see Lemma 4 in Online Appendix I
for details).
We can now state our main results for the optimal
location of process flexibility in Theorem 2. Note that
part (1) of the theorem holds when Assumptions 3
and 4 are relaxed, and with an additional assumption:
Assumption 5. cik = cik , i = 1 2 I k k = 1
2 K k = k This assumption states that unit capacity investment costs are the same for plants across stages. If
cik = cik , then the optimal location of flexibility could
be affected by both the location of variability and the
capacity investment cost structure. So we rule this out
to focus exclusively on variability location.
Theorem 2. For a multiechelon supply chain with
logistics flexibility at all stages, the following apply:
(1) If only stage kv = 0 (demand) has variability, then
the expected profit for a system with a single stage of process flexibility at stage k is equal for any k = 1 2 K.
(2) If only stage kv > 0 (plants) has variability, then
expected profit is maximized if the single stage with process flexibility is located at kv ; for all other positions of the
flexible stage, expected profit is equal.
Theorem 2 characterizes the optimal location for
process flexibility in a system where the fixed cost
of logistics flexibility is zero, and therefore it is costless to have logistics flexibility at all stages, including
stages adjacent to the stage with process flexibility.
The intuition behind the result of Theorem 2 is as
follows. If the only source of variability is demand,
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Management Science 56(3), pp. 495–502, © 2010 INFORMS
then the amount of demand for a product, say product n, that can be satisfied is restricted by (1) the
capacity of plants that produce product n at all dedicated (i.e., nonflexible) stages and (2) the total capacity of plants at the flexible stage. No matter which
stage has process flexibility, these two restrictions are
the same. Therefore, investing in process flexibility at
any stage is equivalent. In contrast, if the only source
of variability is stage kv ≥ 1 (i.e., the capacity of stage
kv is random), then making the plants at stage kv flexible allows excess capacity of one plant at stage kv
to make up for a capacity shortage at another plant
at that stage. If, instead, any stage other than kv has
process flexibility, then such pooling is not possible
because production of each product is constrained by
the capacities at stage kv . Hence, investment in process flexibility at stage kv is optimal when it is the
only source of variability.
From a management perspective, the aforementioned results suggest that, when demand is the major
source of variability (i.e., kv = 0), the impact of variability is not the key consideration in decisions about
locating process flexibility. Because process flexibility is equally effective at almost all stages, it makes
sense to install such flexibility wherever it is least
expensive. In contrast, when supply is variable (i.e.,
kv = K), then there is incentive to make the suppliers
themselves flexible. In supply chain terms, this suggests that multisourcing from flexible suppliers may
be a helpful strategy for mitigating problems of yield
loss. However, it may not be a particularly attractive
option for dealing with uncertain demand, because it
may be cheaper to install (equally effective) flexibility
at a downstream stage.
5.
Conclusions
In this paper, we have focused specifically on the
impact of variability on the optimal placement of
logistics and process flexibility in a multiproduct,
multiechelon supply chain. Although we have only
discussed full flexibility, our insights about flexibility
position generally carry over to other configurations
(e.g., the “chaining” structure suggested by Jordan
and Graves 1995), provided that comparisons are
made between the same configuration at different
stages. To isolate the effect of variability, we have considered systems in which the capacity investment cost
is the same within and across levels of the supply
chain. For such systems, we have shown analytically that if there is only a single source of variability (in supply, demand, or any intermediate stage of
the supply chain), then positioning logistics flexibility as close as possible to the source of variability
or process flexibility at the source of variability is
optimal when the two types of flexibility are considered separately (i.e., either process or logistics flexibility is costless and the problem is only to locate
a single stage of the other type of flexibility). When
both types of flexibility are costly, the optimal configuration is more complicated, but still exhibits a
threshold structure that is informed by the behavior of the cases where process and logistics flexibility
are considered separately (see Online Appendix II for
details).
In practical terms, our results imply that systems
with a high degree of supply variability should make
use of upstream logistics flexibility provided process
flexibility is inexpensive. For example, supply chains
that rely on recycled materials may be subject to
uncontrollable fluctuations in their inputs and hence
would benefit from enhancing flexibility in this first
stage of the network (e.g., by using multiple recycling
plants to supply each downstream production plant).
In contrast, supply chains subject to volatile customer
demand may be better served by downstream logistics flexibility. For example, the automotive supply
chains that motivated the original Jordan and Graves
(1995) work must cope with fluctuations in individual
model sales that occur after plant capacity decisions
have been made. By making the final assembly plants
capable of supplying demands of different models,
their capacity can be used more efficiently to satisfy
demand.
Of course, variability is only one factor affecting optimal flexibility configurations. Another obvious factor is cost. For systems where flexibility is
very expensive at upstream stages (e.g., electronics supply chains in which the first stage is a very
costly and inflexible wafer fab), it may make sense to
use flexibility predominantly in downstream stages,
regardless of the source of variability. In contrast,
in systems where flexibility is very expensive at
downstream stages (e.g., some pharmaceutical supply
chains, in which cost, specialization, and regulations
may restrict the extent to which multiple products
can be produced in the same finishing plant), it may
make sense to use more flexibility in upstream stages
(e.g., commodity chemical plants). Further research is
needed to incorporate cost, variability, and the various process constraints of specific environments.
6.
Electronic Companion
An electronic companion to this paper is available as
part of the online version that can be found at http://
mansci.journal.informs.org/.
Acknowledgments
The authors thank two anonymous referees and an associate editor for thoughtful feedback, which greatly helped
to sharpen the ideas and presentation of this paper. The
502
authors are grateful to the department editor, Paul Zipkin,
for his encouragement and support. Finally, the authors
acknowledge the National Science Foundation for partial
support of this research under Grants DMI-0423048 and
DMI-0457412.
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