Coordinating an Innovation in Supply Chain Management

European Journal of Operational Research 123 (2000) 568±584

www.elsevier.com/locate/dsw

Theory and Methodology

Coordinating an innovation in supply chain management
Bowon Kim

*

Graduate School of Management, Korea Advanced Institute of Science and Technology (KAIST), 207-43 Cheongryangri Dongdaemoon,
Seoul 130-012, South Korea
Received 1 July 1998; accepted 1 January 1999

Abstract
The importance of a long-term relationship between a manufacturing ®rm and its supplier(s) has been emphasized in
the literature on supply chain management. Essential to such a relationship is the coordination among participants in a
supply chain. In order to sustain the relationship, the coordination should enhance the pro®tability of not only the
manufacturer, but also the supplier(s). In this paper, we consider a particular supply chain situation in which the
manufacturer coordinates, e.g., supports, its supplier's innovation that can eventually lead to supply cost reduction.
Developing a mathematical model, we show that although the coordination could improve the manufacturing ®rm's
own pro®tability, it might not be attractive to the supplier unless the supply cost reduction should ultimately increase
the market demand to a certain extent. Under particular circumstances, if the market demand stays constant, the
manufacturer's pro®t increase due to the coordination equals the amount of pro®t loss to the supplier. The analysis
presents exact mathematical criteria to determine whether the coordination strategy can be agreeable to both the
manufacturer and the supplier. Numerical examples are employed to show the applicability of the criteria. Ó 2000
Elsevier Science B.V. All rights reserved.
Keywords: Purchasing; Supply chain management; Optimal control theory; Supplier innovation

1. Introduction
Studies on supply chain management have
emphasized the importance of a long-term strategic relationship between a manufacturing ®rm and
its suppliers (Spekman, 1988; Doyle, 1989; Choi
and Hartley, 1996). The fundamental assumption
underlying this emphasis is that the long-term re-

* Tel.: +82 2 958 3610; fax: +82 2 958 3604.
E-mail address: [email protected] (B. Kim).

lationship makes both the manufacturer and the
suppliers better o€ than when there is no such relationship (Asanuma, 1989; Cusumano and Takeishi, 1991; Parlar and Weng, 1997). Therefore,
we can infer that in order to be sustainable, the
supplier±manufacturer relationship must result in
enhancing the pro®tability of the suppliers as well
as the manufacturer itself (Iyer and Bergen, 1997).
The essence of this relationship is concerned
with coordination between the two participants
(Reyniers, 1992; Whang, 1995; Sox et al., 1997).
Much work has been carried out for research on

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B. Kim / European Journal of Operational Research 123 (2000) 568±584

coordination mechanisms (Malone and Crowston,
1994). Anand and Mendelson (1997) showed that
the ®rm's coordination structure is jointly determined by its decision-rights structure and its information structure (Eisenhardt and Tabrizi,
1995). They modeled the e€ects of alternative coordination structures characterized by the two dimensions on the performance of a ®rm that faces
uncertainty in market demand. Similarly, Desiraju
and Moorthy (1997) examined how the coordination dynamics could be a€ected by the information
asymmetry between manufacturer and supplier
(Gurbaxani and Whang, 1991; Clemons and Row,
1992; Blair and Lewis, 1994).
E€ective supplier relationship can contribute to
improving operations performance in such attributes as quality, delivery and price or cost, that in
turn are regarded as important criteria for supplier
selection (Weber et al., 1991; Akinc, 1993; Lau and
Lau, 1994; Ingene and Parry, 1995). Reyniers and
Tapiero (1995) examined issues of delivery and
control of quality in supplier±manufacturer contracts, highlighting the importance of strategic and
contractual aspects in supply quality management
(Trevelen, 1987). Hartley et al. (1997), on the other
hand, investigated a related issue, i.e., on-time
performance or lead-time, from a perspective of
new product development, showing a signi®cant
relationship between supplier-related delays and
overall project delays. In a similar vein, Clark
(1989) underscored the importance of supplier involvement for e€ective product development (De
Meyer and Van Hooland, 1990; Cooper, 1994).
As alluded already, a sustaining supplier±
manufacturer relationship cannot be possible
without tangible bene®ts accrued to both partners
(Monczka et al., 1993). Such a phenomenon was
described as Pareto-improving by Iyer and Bergen
(1997). They postulated that with the arrangement
of quick response (QR) between a buyer (retailer
in their paper) and a supplier (manufacturer in
their paper), the buyer can always get bene®ts, but
it might not be attractive to the supplier. Based on
empirical studies, they further suggested that the
buyer's service level commitments could reduce the
burden as well as uncertainty the supplier should
bear. In addition, other industry practices to
overcome the unbalanced bene®t were presented

569

such as contractual commitment between the
manufacturer and the supplier, and cooperative
advertisements.
Malone and Crowston (1994) de®ned coordination as the process of managing dependencies
among activities, and suggested di€erent kinds of
dependencies associated with such aspects as
shared resources, task assignments, producer/consumer relationships, design for manufacturability
and so forth. In this paper, we consider the manufacturer's supporting supplier innovation as a
way to deal with interorganizational coordination,
e.g., managing the dependence between manufacturing ®rm and its supplier(s) (Whang, 1993):
supplier innovation could relate to improvements
in quality, yield, delivery time and also supply cost
(Monczka et al., 1993; Lau and Lau, 1994; Hartley
et al., 1997). That is, in order to make it attractive
for the supplier to participate in the supply chain
coordination, the manufacturer must convince the
supplier that such a relationship will bene®t the
supplier too. The manufacturer's support can be
regarded as an incentive for the supplier.
Hammond (1992) proposed that the implementation of speci®c coordinating mechanisms
between ®rms in a supply chain can facilitate individual decision-making processes so that performance much closer to the system-optimal level
can be attained (Weng, 1995). If we view the
manufacturing ®rm and its supplier as consisting
in a network, their network pro®t can be de®ned as
the pro®t achieved through the coordination between themselves, but before being allocated to
each of them individually (Jeuland and Shugan,
1983). Thus, we can suggest the ®rst condition for
a sustainable supplier±manufacturer coordination:
that is, the network pro®t attained through the
coordination must be larger than the sum of uncoordinated individual pro®ts. Since the network
pro®t is before the allocation to each individual
participant, an optimal strategy at the network
level might not necessarily be optimal to each individual ®rm in the supply chain. Thus, the second
condition for a sustainable supplier±manufacturer
coordination is that at least in the long run each
individual participant in the supply chain must
perceive its allocated pro®t is larger than that it
could attain with no such coordination. This

570

B. Kim / European Journal of Operational Research 123 (2000) 568±584

condition relates to the procedural justice in global
operations management (Kim and Mauborgne,
1991). For the purpose of our research, we do not
make a sharp distinction between the two conditions: in fact, it will turn out that the network
pro®t cannot be positive unless each participant should achieve higher pro®tability through
coordination.
In this paper, the coordination is re¯ected in the
manufacturing ®rm's support for the supplier's
innovation activities: we speci®cally focus on a
particular type of supply chain coordination, i.e.,
supplier±manufacturer coordination for the supplier innovation. The fundamental premise is that
the manufacturer's subsidy for its supplier's innovation can eventually lead to supply cost reduction, and thus the manufacturer's own product
price. The supplier's innovation is assumed to affect the supply quality and delivery time, both of
which are combined to determine the supply cost.
We will show that although the manufacturer's
subsidy for its supplier can be an incentive for the
supplier's innovation, a simple subsidy might not
be enough to induce the supplier's cooperation
under a certain market demand structure.
This paper is organized as follows. In the next
section, we establish a mathematical model to describe the supplier±manufacturer relationship
briefed above. Details of the model and its assumptions are also elaborated. In Section 3, we
show how the mathematical model is solved to
draw exact analytical criteria that can guarantee
the mutual bene®t for the two participants. It is
followed by Section 4 presenting numerical examples that can indicate the applicability of the
mathematical model. Finally, in the last section,

we draw some conclusions and inferences along
with suggestions for future research in this area.
2. Model formulation
In order to speci®cally focus on the intricate
relationship of manufacturing ®rm's supporting
the supplier innovation, we model a supply chain
consisting of only one manufacturer and its supplier (Kekre et al., 1995). Analytical and managerial implications from this sparse model can be
extended to more complicated situations. But, this
is hardly a unique assumption: Newman (1988)
studied single source quali®cation, implying that
single sourcing option, if utilized properly, can
provide long-term bene®t to a manufacturing ®rm
(Newman, 1989), while Trevelen and Schweikhart
(1988) probed sourcing strategies with a risk and
bene®t analysis framework (Ramasesh et al.,
1991). In an empirical study, Dyer and Ouchi
(1993) showed that one or two suppliers are usually enough to o€er bene®ts equivalent to those
provided by multiple suppliers to the manufacturing ®rm (Presutti, 1992).
Fig. 1 depicts the present model situation. After
receiving supplies, e.g., raw materials or intermediate inputs, from the supplier, the manufacturing
®rm produces its ®nal products and sells them to
the market. As the arrow from the Manufacturer
to the Supplier indicates, the primary decision
making is concerned with whether the Manufacturer should support its Supplier's innovation that
can lead to the reduction of the supply cost.
As shown in Fig. 1, there is another important
element that can a€ect the supplier±manufacturer

Fig. 1. A simple model structure.

B. Kim / European Journal of Operational Research 123 (2000) 568±584

571

Table 1
Comparison of the two models
Simple model (Section 2.1)

Extended model (Section 2.2)

Manufacturing capacity

Fixed capacity, D (much smaller than the
aggregate market demand)

Flexible capacity (insigni®cant ®xed cost, e.g.,
changeover cost)

Demand structure

Competitive market

Monopolistic/oligopolistic market (imperfect
market)
D as a function of p set by the manufacturer
Deterministic demand

Price, p given
Deterministic demand
Decision impact

Constant revenue due to price given
Cost reduced
Pro®t increase through cost reduction only
Supplier subsidy as pro®t loss
to the manufacturer

Revenue change via price change deterministic
demand
Cost reduced
Pro®t increase through cost reduction and revenue
increase
Supplier subsidy as pro®t loss to the manufacturer

Demand information

Insigni®cant delay between supplier and manufacturer

Analysis

Comparing pro®ts for with and without coordination (subsidy)

relationship, which is the Market Demand. In the
ensuing analysis, we consider two di€erent cases in
which the market demand structure behaves differently. In the ®rst case, we assume the market
demand for the ®nal product is constant. It is
equivalent to a premise that the manufacturing
®rm is in a competitive or perfect market with a
®xed production capacity: this situation is comparable with the competitive market condition
where a single ®rm cannot in¯uence the prevailing
market price, and therefore at least for the short
run the market demand for that particular ®rm is
constrained by the production capacity, assuming
the marginal revenue is larger than the marginal
production cost (Varian, 1992). From the manufacturer's viewpoint, there is little incentive to cut
its price since the price cut will surely be followed
by other ®rms given that the market is competitive.
Therefore, under such a circumstance, the market
demand for the ®rm's product will stabilize around
a ®xed quantity, e.g., the ®rm's production capacity. On the other hand, if the manufacturing
®rm can reduce the cost for the ®nal product, it
can increase its pro®t to the extent of cost saving.
We relax this assumption in the second case
where the market demand for the manufacturer's
product is a function of the product price set by
the manufacturing ®rm. That is, the lower the ®nal

product's market price, the larger the market demand for the product. This situation might be
comparable with an oligopolistic or monopolistic
market condition, where a ®rm can increase the
market demand by lowering the product price.
How sensitive is the market demand response to
the price change will be an important component
to be determined in the analysis. Table 1 compares
the two models analyzed in this paper.
In addition, for analytical simplicity without
loss of generality, we assume that there is little
information delay between the manufacturer and
its supplier (Whang, 1993). Thus, we do not consider any ineciency due to an information distortion in the supply chain (Desiraju and
Moorthy, 1997).
2.1. A simple model: The case of constant market
demand
The ®rst simple model assumes constant production capacity and a given market price for the
product. In this situation, the manufacturing ®rm
cannot unilaterally lower its price to attract more
demand. But, it can try to reduce the cost so as to
increase the pro®t. The production capacity of the
manufacturing ®rm, i.e., essentially the market

572

B. Kim / European Journal of Operational Research 123 (2000) 568±584

demand for the ®rm, is given as D at time t 2 ‰0; T Š,
and the market price for the product is given as p.
Manufacturer's total unit cost of the product
consists of two parts, internal production cost (c1 )
and supply cost (c2 ) charged by the supplier.
Therefore, the unit pro®t is p ÿ c1 ÿ c2 and the
total pro®t realized at t is D…p ÿ c1 ÿ c2 †. Since we
assume that p and D are exogenously determined,
the only way for the manufacturer to earn more
pro®t is to reduce either c1 or c2 . In this research,
we are interested in the collaborative relationship
between a manufacturing ®rm and its supplier, and
thus focus on supplier innovation to reduce c2 .
From the supplier's point of view, there would
be little incentive to lower c2 unless such innovation enables the supplier to earn more pro®t. If the
manufacturing ®rm wants to increase the unit
pro®t and the only remaining alternative is to reduce c2 , then it might o€er a certain amount of
support or subsidy for the supplier to innovate.
In the ensuing analysis, it is assumed that the
supplier takes on projects/experiments as innovation activities to reduce the supply cost. Thus, the
manufacturer's support for the supplier's innovation is to support or subsidize the innovation
projects. We also assume that the ®rst such project
was already implemented by the supplier before
the manufacturer starts subsidizing the innovation. Let us denote u…t† to represent the number of
projects the manufacturing ®rm subsidizes for the
supplier's innovation at t. We further impose a
constraint 0 6 u…t† 6 a which implies that the
manufacturing ®rm cannot support more than a
projects at any given time: the exact size of a depends on the ®rm's internal resource availability as
well as its capability. Since an accurate determination of a is not critical to our discussion in this
paper, a presumption is made that a is exogenously
assessed so as to be feasible to the problem.
For analytical simplicity without loss of generality, u…t† is considered as a continuous variable. In
addition, it is also presumed that the manufacturer's total cost to support one such project is a.
Let us denote x…t† as the cumulative number of
projects supported
by the manufacturer upto t,
Rt
i.e., x…t† ˆ 0 u…t† dt and dx…t†=dt ˆ x_ …t† ˆ u…t†.
Now, we can express c2 as a function of x…t†, satisfying dc2 =…dx† < 0. For analytical tractability, a

speci®c functional form is adopted, based on the
well-established learning curve model (Yelle,
1979), so that c2 ˆ cxm , where c is the base supply
cost, m ˆ ln /m = ln v, /m ˆ 1 ÿ N and 0 6 N 6 1
the learning rate, which is determined by the supplier's innovation capability. For instance, if N ˆ
0:05 and v ˆ 2, c2 is reduced by 5% each time x…t†
doubles while x…t† P 1. Although it can be conceptually easier to follow, adopting the learning
curve formula for c2 is not essential for the ensuing
analysis: c2 can be further simpli®ed by imposing a
more succinct constraint m < 0, not involving the
learning rate. Table 2 shows the summary of
variables and parameters for the ®rst simple
model.
Employing the variables and parameters de®ned so far, we can formulate the ®rst simple
model as follows, (P1).
ZT
Maximize

‰ D…p ÿ c1 ÿ c2 † ÿ auŠ dt;

Jˆ

…1†

0

subject to
x_ ˆ u
0 6 u 6 a;

…2†
x…0† ˆ x0 ˆ 1 specified:

…3†

As mentioned already, a is the manufacturer's total cost associated with supporting one innovation
project. We further assume that out of this a, cm is
Table 2
Summary of variables and parameters for the simple model
D: Production capacity (market demand) at t
p: Market price of the ®nished good
c1 : Internal unit production cost
c2 ˆ cxm : Supplier's unit supply cost to the manufacturer
c: Supply unit cost base
mˆ

ln /m
ln v

/m : Capturing supplier's innovation potential
v: Capturing `how fast the cost reduction occurs'
x…t†: Cumulative number of innovation projects supported by
the manufacturer
u…t†: Number of projects the manufacturer subsidizes at t
a: Maximum number of projects the manufacturer can a€ord at
any given time
a: Manufacturer's cost associated with an innovation project
‰0; T Š: Decision horizon

B. Kim / European Journal of Operational Research 123 (2000) 568±584

the net amount actually paid to the supplier:
0 6 cm < a is imposed. Thus, a ÿ cm is the amount
that is paid to the external entities outside the
supply chain.
x…0† ˆ 1 can imply that at the beginning the
supplier conducted one project (or experiment)
which enables it to provide the manufacturer with
supplies at a unit cost of c. It is premised that as
the number of projects/experiments the supplier
performs increases, the supply cost is reduced: the
speed of cost reduction depends on the parameters
constituting the cost function.
For the purpose of the analysis, a time value of
the pro®t is assumed insigni®cant, and thus the
discounting factor is not considered.
To solve (P1), we make use of the maximum
principle in the optimal control theory, the associated Hamiltonian being
H ˆ D…p ÿ c1 ÿ cxm † ÿ au ‡ ku:

…4†

In order to obtain an optimal solution, we take a
partial di€erentiation of Eq. (4) with respect to u.
Because H is a linear function of u we cannot use
the usual optimization criterion, i.e., oH =ou ˆ 0:
nor is it necessary to check the sucient condition
for optimality. Rather, we ®rst obtain oH =ou ˆ
ÿa ‡ k and ®nd out a binary solution,

if k P a;
 a
…5†
u
0 if k < a:
Applying the necessary conditions of the maximum principle,
oH
ˆ cmDxmÿ1 :
…6†
k_ ˆ ÿ
ox
Since m < 0; k_ < 0 always holds. We know additional characteristics of k: (i) it represents the
marginal value of x…t†, (ii) k…T † ˆ 0 because x…T † is
left free, and (iii) it is a continuous function.
Therefore, it must be that k…t† P 0 throughout
t 2 ‰0; T Š.
Along with k…t† P 0 and k_ < 0, Eq. (5) implies
there exists a time point t such that k…t † ˆ a,
0 6 t 6 T , and thus

0; 6 t 6 t ;

u …t† ˆ
0; t < t 6 T :

According to Eq. (2), we can further show

at ‡ 1;
0 6 t 6 t ;

x …t† ˆ

at ‡ 1; t < t 6 T :

573

…7†

We evaluate t by taking into account two conditions, k…t † ˆ a and k…T † ˆ 0. From Eqs. (6) and
(7) for t 6 t 6 T ;
mÿ1
k_ ˆ cmD…at ‡ 1†

and

k ˆ cmD…at ‡ 1†mÿ1 t ‡ k:
Since k…T † ˆ 0;
k ˆ ÿcmD…at ‡ 1†
k ˆ cmD…at ‡ 1†

mÿ1

mÿ1

T

and

…t ÿ T †:

Finally,
k…t † ˆ cmD…at ‡ 1†

mÿ1

…t ÿ T † ˆ a

and t must satisfy
…at ‡ 1†

mÿ1

…t ÿ T † ˆ

a
:
cmD

…8†

We need to do numerical analysis to evaluate t
satisfying Eq. (8). With t , the optimal costate,
control and state variables can be graphed as in
Fig. 2.
For evaluating whether the supplier subsidy
strategy makes the manufacturing ®rm better o€,
we need to compare the pro®t (J  ) based on the
strategy with that involving no such supplier subsidy. De®ne J0 as the total pro®t the manufacturing ®rm can earn without supporting its
supplier's innovation.
Theorem 1. In order for the supplier subsidy
strategy to be pro®table for the manufacturing ®rm,
it must be satis®ed that J  ÿJ0 > 0 and therefore,
n
o
D
c ÿ …1 ÿ amt †^
c ÿ aat
D…c ÿ c^†T ‡
a…m ‡ 1†
> 0;
where c^  cx…t †m ˆ c…at ‡ 1†m .
Proof. See Appendix A.

574

B. Kim / European Journal of Operational Research 123 (2000) 568±584

where b is de®ned as the net project bene®t the
supplier can obtain from the manufacturer's support
for a unit project/experiment.
Proof. See Appendix B.
What is more important is the following theorem based on the previous two theorems.
Theorem 3. Suppose that the manufacturer's net
outlay for the supplier subsidy per unit project/
experiment, a, is not less than the supplier's net
bene®t, b, and that the production capacity (thus,
the market demand) D is constant regardless of the
supplier innovation, the manufacturing ®rm's supplier subsidy strategy cannot be pro®table to both
the manufacturer and its supplier concurrently. If
a ˆ b, the manufacturer's pro®t increase due to the
supplier innovation is exactly the same as the
amount of loss experienced by the supplier due to
the same innovation.
Proof. From Theorem 1, the subsidy strategy is
pro®table to the manufacturer only if
n
o
D
D…c ÿ c^†T ‡
c ÿ …1 ÿ amt †^
c > aat :
a…m ‡ 1†
Fig. 2. A graphical interpretation of the optimal control solutions.

However, the condition in Theorem 1 is just
from the manufacturer's perspective. What makes
the manufacturing ®rm better o€ might not necessarily do the same thing for the supplier. For a
balanced analysis, we need to take the supplier's
view as well.
Denote J s as the supplier's total pro®t if the
supplier accepts the manufacturer's subsidy, and
J0s as that if the supplier does not accept the deal.
Theorem 2. The manufacturing ®rm's supplier
subsidy strategy is bene®cial to the supplier only if
J s ÿ J0s
ˆ bat ÿ D…c ÿ c^†T
n
o
D
c ÿ c^…1 ÿ amt † > 0;
ÿ
a…m ‡ 1†

Also from Theorem 2, the strategy can be acceptable to the supplier only if
n
o
D
c ÿ c^…1 ÿ amt †
bat > D…c ÿ c^†T ÿ
a…m ‡ 1†
> 0:
Therefore, the subsidy strategy would be adopted
by both manufacturer and supplier only if the two
inequality relationships are satis®ed, i.e.,
n
o
D
c ÿ c^…1 ÿ amt †
bat > D…c ÿ c^†T ÿ
a…m ‡ 1†
> aat :
That is, a necessary condition must be met that
bat > aat , i.e., b > a. However, according to our
earlier argument about the `realistic condition' in
Appendix B, b < a holds rather than b > a. As a
result, we can state that in general, the subsidy
strategy cannot be acceptable to both manufacturer and supplier simultaneously if the market

B. Kim / European Journal of Operational Research 123 (2000) 568±584

demand for the ®nal product does not change as
the product price varies.
Moreover, if a ˆ b, from Theorems 1 and 2, it
can be shown that J s ÿ J0s ˆ ÿ…J  ÿ J0 †, which
proves the theorem.
In this section, we have proved that a manufacturing ®rm's simple strategy to support its
supplier's innovation that can lead to the supply
cost reduction is not enough to guarantee increased pro®tability to both the manufacturer and
its supplier simultaneously. That is, in order for
the supply chain to have an increased pro®t as a
whole, it is not sucient to make an improvement
limited within the supply chain itself alone.
In the next section, we take a look at the possibility that the innovation subsidy strategy can
bene®t both the manufacturing ®rm and its supplier. In particular, we concentrate on the condition under which such strategy can make the two
participants better o€ by considering a market
demand structure sensitive to the price change of
the ®nal product.
2.2. An extended model: The case of price-dependent market demand

575

while the product cost and its market price decreased together. In the following model, the price
is determined by the manufacturing ®rm as follows:
p ˆ r ‡ c1 ‡ cxm ;
r is the ®xed markup.
Then, D…t† ˆ d1 ÿ d2 …r ‡ c1 ‡ cxm †.
It is practical to assume D…t† ˆ d1 ÿ d2 …r ‡ c1 ‡
cxm † > 0 throughout the ensuing analysis: therefore, we focus on d2 that makes D…t† > 0 valid.
Now the manufacturing ®rm faces the following optimization problem, (P2):
ZT
Maximize

‰r…d1 ÿ d2 r ÿ d2 c1

Jˆ
0

ÿ d2 cxm † ÿ auŠ dt
subject to

x_ ˆ u;
0 6 u 6 a; x…0† ˆ x0 ˆ 1:

As in solving (P1), we obtain the new Hamiltonian,
H ˆ r…d1 ÿ d2 r ÿ d2 c1 ÿ d2 cxm † ÿ au ‡ ku;

As mentioned before, in this extended model,
we assume that the market demand for the manufacturing ®rm's product is a function of the
product price set by the manufacturer. More speci®cally, we adopt a simple linear demand curve
well established in the literature (Varian, 1992)

and optimal solutions for (P2) as follows.
Applying the same reasoning as in the simple
model case, we can derive t which should satisfy
mÿ1
…at ‡ 1† …t ÿ T † ˆ …a=cmrd2 † so that

D ˆ d1 ÿ d2 p;

x ˆ at ‡ 1

where d1 is the base (e.g., theoretical maximum)
demand for the product and d2 measures a demand
sensitivity in response to the price change, i.e.,
dD=dp ˆ ÿd2 , which represents the marginal
change in demand as the price changes by a unit.
Further we assume that the manufacturing ®rm
tries to keep its unit net pro®t constant when it
allows the product price to be reduced in order to
increase the market demand. This assumption is
not completely new: in an empirical study, Sterman et al. (1997) showed an example in which a
®rm's markup ratio remained remarkably constant

x ˆ at ‡ 1

while t 6 t ;
while t > t :



Denote J~ as the total pro®t the manufacturer can
earn by subsidizing the supplier innovation when
its market demand is a linear function of the

product price, and J~0 as the total pro®t the manufacturer can get without such subsidy.
Theorem 4. When the market demand is a linear
function of the product price, in particular
D…t† ˆ d1 ÿ d2 …r ‡ c1 ‡ cxm †, the supplier subsidy
strategy is bene®cial to the manufacturing ®rm if


and only if J~ ÿJ~0 > 0, thus

576

B. Kim / European Journal of Operational Research 123 (2000) 568±584

 
d2 r c 1 ‡

1
a…m ‡ 1†T





1 ÿ amt
T
ÿ c^ 1 ‡
a…m ‡ 1†T

> aat or d2 r…CA ÿ CB †T > aat ;

…9†

where


1
;
CA ˆ c 1 ‡
a…m ‡ 1†T


1 ÿ amt
CB ˆ c^ 1 ‡
a…m ‡ 1†T
m
m
and c^  cx…t † ˆ c…at ‡ 1† :

Proof. See Appendix C.
We need to examine the implication of Theorem 4 a little further. In order for the manufacturer
to subsidize the supplier innovation, the resulting
pro®t increase should be at least the same as the
amount of subsidy spent for the innovation projects, aat . From Eq. (9), the condition mentioned
above can be expressed as d2 r…CA ÿ CB †T ˆ aat .
That is, d2 r…CA ÿ CB †T can be regarded as the
pro®t increase due to the subsidy strategy for the
supplier innovation, which can be further decomposed by rearranging terms:
(a) if we suppose that CA ÿ CB is the average
cost saving (and therefore, price reduction) at
each t 2 ‰0; T Š because of the supplier innovation, from the manufacturer's point of view,
(b) d2 …CA ÿ CB † is the average demand increase
due to the supplier innovation per unit period,
and
(c) d2 r…CA ÿ CB † is the pro®t increase per unit
period, and ®nally we can conclude
(d) d2 r…CA ÿ CB †T is the manufacturer's total
revenue increase over t 2 ‰0; T Š resulting from
the supplier subsidy strategy.
As in the simple model analysis, the strategy
should bene®t the supplier if it can be practically
s
implemented. Denote J~ as the supplier's optimal
total pro®t when it accepts the manufacturer's
s
subsidy for the innovation, and J~0 as that without
involving any such subsidy.
Theorem 5. In order for the supplier subsidy
strategy under the variant market demand structure

to be acceptable to the supplier, it must be satis®ed
that
n
o
bat > d1 c^t ‡ SF T ÿ SE
n
o
…10†
ÿ d2 c^t SB ‡ SA SF T ÿ SC SE ÿ SD ;
where
SA ˆ r ‡ c1 ‡ c ‡ c^ ÿ cs ;
SB ˆ r ‡ c1 ‡ c^ ÿ cs ;
SC ˆ r ‡ c 1 ÿ c s ;
h
i
1
c^2 …at ‡ 1† ÿ c2 ;
a…2m ‡ 1†
h
i
1
c^…at ‡ 1† ÿ c ; SF ˆ c ÿ c^
SE ˆ
a…m ‡ 1†
SD ˆ

Proof. See Appendix D.
It is not easy to understand the practical implications of SA to SE . But, we can try to make an
indirect inference of the meaning of Eq. (10) by
rearranging the terms so that
n
o
c^t ‡ SF T ÿ SE
(
)
c^t SB ‡ SA SF T ÿ SC SE ÿ SD
;
d1 ÿ d2
c^t ‡ SF T ÿ SE
where
(

c^t SB ‡ SA SF T ÿ SC SE ÿ SD
d1 ÿ d2
c^t ‡ SF T ÿ SE

)

can be regarded as the average demand at t 2 ‰0; T Š
while fc^t ‡ SF T ÿ SE g as the supplier's revenue
loss per unit demand for t 2 ‰0; T Š. Thus,
n
o
c^t ‡ SF T ÿ SE
(
)
c^t SB ‡ SA SF T ÿ SC SE ÿ SD
d1 ÿ d2
c^t ‡ SF T ÿ SE
represents the total revenue loss to the supplier for
t 2 ‰0; T Š by accepting the manufacturer's subsidy
for supply innovation. As a result, Theorem 5

B. Kim / European Journal of Operational Research 123 (2000) 568±584

states that the supplier subsidy strategy is acceptable to the supplier if and only if the total net
subsidy from the manufacturer should be larger
than the total revenue loss to the supplier.
Theorem 6. Given the conditions assumed in this
section, the manufacturing ®rm's strategy to support
its supplier innovation can be acceptable to both the
manufacturer and the supplier if and only if Eqs. (9)
and (10) are satis®ed simultaneously. That is,


 

1
1 ÿ amt
ÿ c^ 1 ‡
T
d2 r c 1 ‡
a…m ‡ 1†T
a…m ‡ 1†T
> aat ;
n
o
bat > d1 c^t ‡ SF T ÿ SE
n
o
ÿ d2 c^t SB ‡ SA SF T ÿ SC SE ÿ SD :
As in the constant demand situation, if a > b is
held, the condition becomes
 



1
1 ÿ amt
T
ÿ c^ 1 ‡
d2 r c 1 ‡
a…m ‡ 1†T
a…m ‡ 1†T
n
o
> aat > bat > d1 c^t ‡ SF T ÿ SE
n
o
ÿ d2 c^t SB ‡ SA SF T ÿ SC SE ÿ SD :
…11†
Unlike Theorems 3 and 6 indicates that it could be
possible for both the manufacturing ®rm and its
supplier to become better o€ from manufacturer's
supporting the supplier innovation. Under what
circumstances Eq. (11) holds depends on many
factors such as characteristics of the market demand as expressed in d1 and d2 , decision time horizon captured by T, and the supplier innovation
capability embedded in m, i.e., its learning rate.
However, the most critical di€erence between the
model assumptions is concerned with the nature of
the market demand structure. Therefore, in the
numerical examples, it is reasonable for us to ex-

577

plore an impact of market demand structure on
the pro®tability of supplier innovation support.
3. Numerical examples
To draw more managerial insights from Theorem 6, we examine numerical examples. In this
section, we report key numerical results using parameter values in Table 3.
Since we are interested in how the market sensitivity a€ects the supplier innovation support
(coordination) strategy, we focus on the numerical
examples in which Eq. (11) is satis®ed by varying
d2 .
Numerical examples are depicted in Figs. 3±7:
since we are concerned with D > 0, we focus on d2
such that 0 < d2 < 50, which covers practically
meaningful ranges. Fig. 3 compares the manufacturing ®rm's total pro®t associated with the strategy to coordinate, e.g., support, its supplier
innovation with that not adopting the strategy as
the market demand sensitivity, d2 , varies, depicting
Eqs. (C.1) and (C.2) in Appendix C. Fig. 4 shows
the same comparison for the supplier, graphing
Eqs. (D.1) and (D.2) in Appendix D. Finally,
Fig. 5 integrates the two ®gures to describe the
relationship in Eq. (11).
From Fig. 3, we can see that overall the manufacturing ®rm gets better o€ by supporting the
supplier innovation regardless of the demand


sensitivity, i.e., J~ P J~0 . Although the di€erence
between two strategies, one with supplier subsidy


(J~ ) and the other without it (J~0 ), is negligible
when d2 is very small, in particular d2 6 5:5, after


the initial period, J~ becomes larger than J~0 .
For the supplier, we can draw a similar cons
s
clusion from Fig. 4: J~0 is larger than J~ until d2
s
s
becomes about 42.0, after which it holds J~ > J~0 .
Thus, we can infer that there is a nonlinear
s
s
relationship between J~ and J~0 for the supplier
case.

Table 3
Parameter values for numerical analysis
T

a

c

c1

cs

d1

/m

v

r

a

b

100

10

4

4

1

500

0.95

2

2

5

5

578

B. Kim / European Journal of Operational Research 123 (2000) 568±584

Fig. 3. Manufacturing ®rm's pro®t comparison.

Fig. 4. Supplier's pro®t comparison.

Conclusions derived from Figs. 3 and 4 can be
combined in Fig. 5 where at about d2 ˆ 42:0 the
coordination strategy becomes acceptable to the
supplier as well. For both the manufacturer and
the supplier, the range of 42:0 6 d2 < 50:0 makes it
mutually bene®cial to coordinate the supplier innovation that can eventually reduce the supply
cost and attract more market demand. In e€ect,
42:0 6 d2 < 50:0 can be called `zone of coordination' (Fig. 6).
From Fig. 5, we can make two practical observations. First, it is very large d2 s that can make

the subsidy strategy acceptable to the supplier.
This implies that the initial demand for the product is very small since D ˆ d1 ÿ d2 p. Second, a
large d2 also means that the demand increase as the
product price decreases is very large: a small reduction in the supply cost should have a signi®cant
impact on the demand increase.
We can ask what kind of industry can most
bene®t from this supplier±manufacturer coordination. Based on the two observations above, we
can infer that it might be an industry in between
introduction and growing stages in a product life

B. Kim / European Journal of Operational Research 123 (2000) 568±584

579

Fig. 5. Determining the joint positive pro®t range.

Fig. 6. Range for mutually bene®cial coordination.

cycle that can bene®t most: in this kind of industry, we can expect a relatively small initial market
demand, but a huge potential for demand growth
as innovation occurs.
We have tried other parameter values and were
able to derive qualitatively similar conclusions.
However, the principal implication of this numerical exercise is to show that it is possible for
both manufacturer and supplier to get better o€ by
coordinating the supplier innovation if the market
demand for the ®nal product is sensitive to the
product price set by the manufacturing ®rm to a
certain extent. Despite being relied on numerical
examples, this is a quite strong conclusion since
Theorem 3 mathematically proved that this kind

of mutual bene®t cannot be expected if the market
demand is constant regardless of the supplier innovation. An important managerial implication is
that the supply chain coordination cannot be
pro®table unless the eciency improvement inside
the supply chain can induce more revenues from
outside the chain, e.g., the market.
Since our research focuses on the e€ect of
market demand sensitivity on the supplier±manufacturer coordination, we have so far examined the
manufacturer's and its supplier's pro®t changes as
d2 varies. Utilizing the similar approaches along
with the analytical solution in Section 2, we can
analyze other dynamics involving di€erent parameters. In Fig. 7, we examine how the supplier's

580

B. Kim / European Journal of Operational Research 123 (2000) 568±584

Fig. 7. Changes in t as /m varies, given d2 ˆ 43.

innovation capability (captured in its learning
rate) a€ects t (the optimal time length of innovation support) for three di€erent Ts: although
d2 ˆ 43 is used for this particular numerical example, one can see a similar pattern for any reasonable value of d2 .
From Fig. 7, we can infer the following points.
When the supplier innovation capability is relatively low, e.g., for about /m > 0:96, t is relatively
short: the reason could be that the innovation
capability is too low to warrant a reasonable return for the innovation subsidy by the manufacturer. Likewise, if the capability is very high, e.g.,
for about /m < 0:65, t is relatively short: in this
case, the capability is so high that even a small
amount of subsidy can contribute a lot, and
therefore it is not necessary to support many innovation projects. Finally, when the supplier's innovation capability is moderate, e.g., about
0:80 < /m < 0:93, the need for innovation support
is the highest, implying that the coordination of
innovation is most e€ective in that range. Another
point we can make is that as the decision horizon,
T , increases, so does t : the longer the decision
horizon, the more endurable the e€ect of innovation support.
We can use the analysis model to investigate
other important dynamics associated with di€erent
parameters as well as variables.

4. Conclusion and managerial implications
Our primary research inquiry in this paper has
been to identify conditions associated with market
demand that can make the supplier±manufacturer
coordination for the supplier innovation mutually
pro®table to both participants. We showed that if
the market is not sensitive to the product price,
and thus eventually the supplier innovation itself,
then the supplier±manufacturer coordination
cannot satisfy both the manufacturing ®rm and the
supplier simultaneously. The extended analysis
along with numerical examples indicated that it
could be possible for such supplier innovation
subsidy to be acceptable to both participants at the
same time if the market demand increases in response to the favorable innovation coordinated by
the manufacturer and its supplier. An important
implication is that unless the revenue increases
outside of the supply chain itself, any eciency
gains within the supply chain cannot be sustainable since they are traded o€ between the chain
participants, i.e., it is a zero-sum game without a
substantive revenue increase from the outside. An
analogy might be that as long as the absolute size
of a pie remains ®xed for 2 persons, no method,
however brilliant that might be, can succeed in
allocating larger pieces to the two people than
those they got before.

B. Kim / European Journal of Operational Research 123 (2000) 568±584

Although the mathematical analysis as well as
numerical examples is logically drawn, we admit
that the models are based on rather constrained
assumptions. For instance, we limited our investigation within a simple supply chain consisting of
only one manufacturer and its single supplier. It
was also assumed that the market demand is either
not responsive to the product price change or only
linearly responding to it. Moreover, we opted to
consider neither any stochastic features nor information delay between the supply chain participants for the purpose of current research: we
believe future studies can generalize the research
conclusions by allowing relaxation of these stringent assumptions. In the numerical analysis, we
could have examined e€ects of changes in parameter values other than d2 on the pro®t di€erence
between the two alternative strategies. We have
mainly focused on the demand sensitivity parameter because that is key to our research question.
Although some of the limits mentioned above can
be easily recti®ed or might turn out to be insigni®cant, most of them can shed some light on this
line of research in exploring the supplier±manufacturer coordination in a supply chain.
Appendix A
With Eq. (7), we can calculate the optimal
pro®t the manufacturing ®rm can earn by employing the supplier support strategy. That is,
J ˆ

Zt

581

J  ˆ D…p ÿ c1 †T ÿ c^D…T ÿ t † ÿ aat
n
o
D
c ÿ c^…at ‡ 1† ;
‡
a…m ‡ 1†
where c^  cx…t †m ˆ c…at ‡ 1†m .
In order to evaluate whether the supplier subsidy strategy makes the manufacturing ®rm better
o€, we need to compare the pro®t (J  ) based on the
strategy with that involving no such supplier subsidy. De®ne J0 as the total pro®t the manufacturing ®rm can earn without supporting its
supplier's innovation.
We can calculate
J0 ˆ D…p ÿ c1 ÿ c†T :

…A:2†

J  can be rearranged so that
J  ˆ D…p ÿ c1 ÿ c†T ‡ D…c ÿ c^†T ‡ c^Dt ÿ aat
n
o
D
c ÿ c^…at ‡ 1† :
‡
…A:3†
a…m ‡ 1†
Therefore, in order for the subsidy strategy to be
attractive to the manufacturing ®rm, it must be
that J  > J0 or J  ÿ J0 > 0. By combining
Eqs. (A.2) and (A.3), we obtain the necessary
condition for the pro®table supplier subsidy
strategy as
J  ÿ J0 ˆ D…c ÿ c^†T ‡ c^Dt ÿ aat
n
o
D
c ÿ c^…at ‡ 1†
‡
a…m ‡ 1†
n
o
D
c ÿ …1 ÿ amt †^
c
ˆ D…c ÿ c^†T ‡
a…m ‡ 1†

m

‰ D…p ÿ c1 ÿ c…at ‡ 1† † ÿ aaŠ dt

ÿ aat > 0:

0
m

‡ ‰ D…p ÿ c1 ÿ c…at ‡ 1† †Š…T ÿ t †:

…A:1†

The second term in Eq. (A.1) represents the pro®t
for t 2 ‰t ; T Š: since x ˆ at ‡ 1 and u ˆ 0 for t >
t ;
ZT

m

‰ D…p ÿ c1 ÿ c…at ‡ 1† †Š dt

t

ˆ ‰ D…p ÿ c1 ÿ c…at ‡ 1†m †Š…T ÿ t †:
Therefore, the total manufacturer's pro®t becomes

Appendix B
We can calculate
J s ˆ

Zt

‰ D…c…at ‡ 1†m ÿ cs † ‡ baŠ dt

0

ZT
D…^
c ÿ cs † dt

‡
t

…B:1†

582

B. Kim / European Journal of Operational Research 123 (2000) 568±584

where cs is the supplier's unit internal manufacturing cost: it can be either raw material cost or
other base cost that cannot be reduced through the
innovation.
Let's de®ne b as the net project bene®t the
supplier can obtain from the manufacturer's support for a unit project/experiment. If the cost the
supplier must spend in order to implement the
project/experiment is cs P 0 (it is the external expenses, i.e., direct cost, the supplier must pay to
conduct an innovation project), b ˆ cm ÿ cs . From
the manufacturer's perspective, the total cost per
project is a, while the supplier's net bene®t from
the support is only b, which remains within the
supplier after the project has been done: b becomes
an embedded asset to the supplier. In general, we
can impose a realistic constraint that cs < cm < a
and b ˆ cm ÿ cs < a ÿ cs < a.
Eq. (B.1) takes the particular form since the
supplier's net pro®t from supplying products (intermediate goods) to the manufacturer is D…c…at ‡
1†m ÿ cs † at t 6 t , but D…c…at ‡ 1†m ÿ cs † at t such
that t < t 6 T .
Eq. (B.1) can be rewritten as

Appendix C
Each of the pro®ts can be determined as follows:
Zt



J~ ˆ

m
fr…d1 ÿ d2 r ÿ d2 c1 ÿ d2 c…at ‡ 1† † ÿ aag dt

0

‡

ZT n
t

ˆ r…d1 ÿ d2 r ÿ d2 c1 ÿ d2 c†T ‡ d2 r…c ÿ c^†T
o
d2 r n
c ÿ c^…1 ÿ amt † ÿ aat : …C:1†
‡
a…m ‡ 1†

J~0 ˆ r…d1 ÿ d2 …r ‡ c1 ‡ c††T

ˆ r…d1 ÿ d2 r ÿ d2 c1 ÿ d2 c†T :



J~ ÿ J~0 ˆ d2 r…c ÿ c^†T

‡

D
…^
c…at ‡ 1† ÿ c†
a…m ‡ 1†
D
…^
c…at ‡ 1† ÿ c†:
a…m ‡ 1†

On the other hand, if the deal is rejected by the
supplier, the total pro®t would be
J0s ˆ D…c ÿ cs †T :

…B:2†

Therefore, if the supplier subsidy strategy is bene®cial to the supplier, it must be satis®ed that
J s ÿ J0s > 0. From Eqs. (B.1) and (B.2), we have
the following.
J s ÿ J0s ˆ bat ÿ D…c ÿ c^†T
n
o
D
c ÿ c^…1 ÿ amt † :
ÿ
a…m ‡ 1†

o
d2 r n
c ÿ c^…1 ÿ amt † ÿ aat > 0
a…m ‡ 1†

implies
 

1
d2 r c 1 ‡
a…m ‡ 1†T


1 ÿ amt
T > aat :
ÿ c^ 1 ‡
a…m ‡ 1†T

c ÿ c†T ‡ …ba ÿ c^D†t
ˆ D…c ÿ cs †T ‡ D…^
‡

…C:2†

In order for the subsidy strategy to be bene®cial to
the manufacturing ®rm, it must be satis®ed that


J~ > J~0 . From Eqs. (C.1) and (C.2),

c ÿ cs †…T ÿ t † ‡ …ba ÿ cs D†t
J s ˆ D…^
‡

o
r…d1 ÿ d2 r ÿ d2 c1 ÿ d2 c^† dt

Appendix D
Applying the previous reasoning, we calculate
s
J~ ˆ

Zt

m

f‰d1 ÿ d2 …r ‡ c1 ‡ c…at ‡ 1† †Š…c…at ‡ 1†

m

0

ÿ cs † ‡ bag dt

‡

ZT nh

d1

t

i
o
c ÿ cs † dt;
ÿ d2 …r ‡ c1 ‡ c^† …^

…D:1†

B. Kim / European Journal of Operational Research 123 (2000) 568±584
s
J~0 ˆ ‰d1 ÿ d2 …r ‡ c1 ‡ c†Š…c ÿ cs †T ;

…D:2†

where cs is the ®xed, i.e., constant, portion of the
supplier's unit internal manufacturing cost.
By simplifying and rearranging Eqs. (D.1) and
(D.2), we determine
n
o
s
s
J~ ÿ J~0 ˆ bat ÿ d1 c^t ‡ SF T ÿ SE
n
o
‡ d2 c^t SB ‡ SA SF T ÿ SC SE ÿ SD ;
s
s
and therefore J~ ÿ J~0 > 0 implies

n
o
bat > d1 c^t ‡ SF T ÿ SE
n
o
ÿ d2 c^t SB ‡ SA SF T ÿ SC SE ÿ SD ;
where
SA ˆ r ‡ c1 ‡ c ‡ c^ ÿ cs ;
SB ˆ r ‡ c1 ‡ c^ ÿ cs ;

SC ˆ r ‡ c 1 ÿ c s ;

h
i
1
c^2 …at ‡ 1† ÿ c2 ;
a…2m ‡ 1†
h
i
1
c^…at ‡ 1† ÿ c ; SF ˆ c ÿ c^:
SE ˆ
a…m ‡ 1†

SD ˆ

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