European Journal of Operational Research 201 (2010) 112–122

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European Journal of Operational Research

journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

Evaluating the effects of distribution centres on the performance of vendor-managed inventory

systems
L. Yang, C.T. Ng *, T.C.E. Cheng
Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

article info abstract

Article history:
managed inventory (VMI) system comprising one manufacturer, one DC and n retailers. Adopting the order-up-to-level (OUL) replen-
Received 26 March 2008 Accepted 5 February
ishment policy, the system aims to maximize the overall system profit. We propose a model to evaluate the system performance by
2009 Available online 13 February 2009
considering the scale of the distribution network, influential cost factors, demand distribution, planning horizon, and facility locations. From
Keywords: the viewpoint of a supply chain, we examine the DC’s effects on the system in terms of net profit. Our findings reveal that the DC has effects
Distribution centre Vendor-managed inventory on demand variance and system profit, and there are some dominant factors that affect the overall system performance. The DC may lead to
system Order-up-to-level policy Location different system performance under a variety of cost factors, and in some situations, the DC may negatively affect system performance. We
problem
also suggest some innovative uses of the DC’s location to help enhance system performance.
This paper evaluates the effects of the © 2009 Elsevier B.V. All rights reserved.
distribution centre (DC) in a vendor- 1. Introduction

In the past decade, increasing attention has been paid to using the distribution centre (DC) to provide better services while reducing the cost of distribution
simultaneously. A good distribution network may lead to efficient logistics management. For instance, Benetton uses a DC in Ponzano, Italy, to serve over 6000 stores
in 83 countries around the world (Dapiran, 1992). It has been shown that the evaluation of the DC’s performance is becoming necessary and critical to long-term
business success (Kengpol, 2004).
In the literature on supply chain management, one of the well-known concepts is vendor-managed inventory (VMI) (see, e.g., Cheung and Lee, 2002; Disney et
al., 2003). Many successful businesses in reality have demonstrated the benefits of VMI, e.g., Wal-Mart and JC Penney (Cetinkaya and Lee, 2000; Dong and Xu,
2002). Under a VMI system, the performance of the supply chain can be improved in terms of four aspects, namely lower inventory holding cost, reduced stock-out,
improved service level, and reduced demand distortion (Aviv and Federguen, 1998; Angulo et al., 2004). The benefits from increased economies of scale in
purchasing and transportation operations can be achieved with the assistance of a DC under a VMI system. Coyle et al. (2003) estimated that transportation cost
represents approximately 40–50% of the total logistics cost and 4–10% of the product selling price for many companies. The above studies establish that both inven -
tory cost and transportation cost are important to the whole supply chain performance. It is noted that research on examining the DC’s role within agile supply chains
is very limited and partial in the literature. The related studies include Morgan and Dewhurst (2008), Baker (2008), and Quak and De Koster (2007).
Considering the multiple roles of the DC in a VMI system, Zhao and Cheng (2009) established a model to study the DC’s ability in modifying the manufacturer’s
ordering and delivery decisions in a VMI system. They provided a new definition of the DC’s modification ability in terms of system profit. In their model, they only
considered one retailer and inventory holding cost in the supply chain. In this paper we extend their research to include: (1) n retailers, (2) transportation cost, and (3)
time planning horizon. In addition, we provide an analysis of the DC’s location problem with respect to such parameters as number of retailers, demand distribution,
inventory holding cost, and transportation cost.
In this paper, based on the work of Zhao and Cheng (2009), we evaluate the effects of the DC’s modification ability on the performance of a supply chain
operating under a VMI system. We consider VMI in a consignment environment consisting of a manufacturer, a DC and n retailers. Under VMI, the manufacturer
takes care of product delivery, and stock and inventory control for the retailers. The manufacturer can increase its profit by reducing the total cost of product delivery
and storage with the application of DC. In a consignment environment,
 * Corresponding author. Tel.: +852 2766 7364; fax: +852 2330 2704. E-mail address: lgtctng@polyu.edu.hk (C.T. Ng).

0377-2217/$ -see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.02.006
L. Yang et al. / European Journal of Operational Research 201 (2010) 112–122

retailers only house and assist with the sale of the products under an agreement with the manufacturer on commission. We assume that the manufacturer and the
retailers have reached an agreement on VMI. We also assume that the manufacturer has full access to the sales profile leading to consolidated demand information.
Two scenarios are established to evaluate the DC’s modification ability, namely OneStage-Decision (OSD) and Two-Stage-Decision (TSD). Under OSD, there is no
inventory at the DC and the DC only makes order decisions from the manufacturer to the retailers. Under TSD, there is a certain inventory level at the DC and the DC
makes both ordering decisions from the manufacturer to DC, and delivery decisions from the DC to the retailers. Based on analytical comparisons of these two
scenarios, we offer some management suggestions, together with a new viewpoint on the DC’s location decision.
The rest of the paper is organized as follows. The literature review in Section 2 shows that our research fills a gap between theory and applications concerning the
role of the DC in a supply chain. Formulation of the general problem is presented in Section 3. In Section 4, two strategies of ordering and delivery decisions are
proposed and compared. In Section 5, the DC’s modification ability is defined and evaluated. Moreover, a new viewpoint of the DC’s location decision is proposed,
and its formulation is established. Conclusions and suggestions for further research are presented in Section 7.

2. Literature review

Within a supply chain, there are three important decisions, namely facility location decisions, inventory management decisions and distribution decisions (Shen
and Qi, 2007). In our study we combine these three issues in a VMI system with DC’s assistance. In the following we present a brief review of the literature on these
three problems.

2.1. Facility location decisions

Snyder (2006) reviewed studies in the literature on facility location problems. Nozick and Turnquist (2001) proposed a model to inves tigate the optimal DC
locations of inventories for individual products in a multi-product, two-echelon system. It was noted that the performance of a supply chain is affected by both the
location problem and inventory management. Shen et al. (2003) presented a joint location-inventory model to determine which retailers should serve as DCs and how
to allocate the other retailers to the DCs.

2.2. Inventory management decisions

With respect to inventory management decisions, we focus on VMI in supply chains in this paper. Dong and Xu (2002) proposed a VMI model to evaluate the
effects of VMI in supply chains. Angulo et al. (2004) evaluated the effects of information sharing on a VMI partnership. Cetinkaya and Lee (2000) presented an
analytical model for coordinating inventory and transportation decisions in VMI systems. Incorporating the dynamic dimension, Jaruphongsa et al. (2004) provided a
polynomial time algorithm to compute the optimal solutions for the replenishment plan and the dispatching plan. Bertazzi et al. (2005) compared the order-up-to level
policy and the fill-fill-dump policy of VMI. They showed that the fill-fill-dump policy leads to a lower average cost than the order-up-to level policy.

2.3. Distribution decisions

Three issues related to problems on distribution decisions include the vehicle routing problem, the DC consolidation problem, and the DC design problem. Anily
and Federgruen (1990) proposed a model to determine a feasible replenishment strategy to minimize the long-run average transportation and inventory costs. Based
on this model, some extensions have been made (see, e.g., Anily and Federgruen, 1990; Viswanathan and Mathur, 1997) along different directions. Some latest
research on the vehicle routing problem includes Hwang (2005), Aghezzaf et al. (2006), and Van Woensel et al. (2008). Regarding the problems of consolidation of
DCs, Chung et al. (2001) suggested that consolidation leads to lower total facility investment and inventory costs when demand follows certain distributions. Zhao
and Cheng (2009) demonstrated that there is a trade-off between the potential of the DC’s modification ability and the related costs. Research on the DC design
problem is limited in the literature. Bordley et al. (1999) pointed out that consolidating DCs can reduce lost sales. Kengpol (2004) developed a decision support
system (DSS) that can accommodate evaluation models and criteria to evaluate investments in a new DC. Baker (2008) investigated the design problem by means of
nine case studies on how individual business units designed and operated DCs to provide rapid response to their markets. To deal with the trade-offs between a
variety of factors, including location cost, transportation cost, holding cost, stock-out cost and capacity concerns, in supply chains, Romeijn et al. (2007) proposed a
generic modelling framework to address such issues.
From the above literature review, it is noted that the exact and precise role of the DC within agile supply chains has only been partially explored. Furthermore,
there is a blank area in evaluating the DC’s effect on the system performance of a supply chain. In this paper we seek to provide some strategic management
suggestions for, as well as a new viewpoint of, DCs by evaluating the DC’s effects on supply chains.

3. Problem formulation

In this study we use a two-echelon VMI system to examine two different decision-making strategies. The system consists of one manufacturer, one DC and n
retailers. Two alternative strategies are offered, namely one is delivering the product from the manufacturer to the retailers directly; the other is delivering the product
from the manufacturer to the DC first, and then delivering the product from the DC to the retailers. The order-up-to-level (OUL) replenishment policy is adopted in
the proposed model, which minimizes the total discounted holding and shortage cost over an infinite horizon (Heyman and Sobel, 1984). It is assumed that there is no
backlogging at the retailers. Denote Drt as the demand of retailer r in time period t(t = 1,2,3,...). We assume that Drt depends on Dr(t-1) as follows:
Drt ¼ dr þqrDrðt-1Þþert ðt ¼ 1; 2; 3; ... r ¼ 1; 2; 3; ...; nÞ; ð1Þ
 L. Yang et al. / European Journal of Operational Research 201 (2010) 112–122

where Dr0 =0;dr > 0 is a prior estimate of the average demand of retailer r in period t =1; -1 6 qr 6 1 is a constant coefficient that expresses the correlation of the
demands in two consecutive periods with respect to retailer r; and ert is the error term of the demand for retailer r in time period t, which is i.i.d. according to ert �
Nð0; r2 r Þ. In order to ensure the probability of a negative demand is negligible, we assume that ert is significantly smaller than dr. Such an assumption was used in
Lee et al. (2000). We further assume that the total cost is the sum of the inventory holding cost, product shortage cost, and transportation cost; while the revenue is the
retailers’ sales. The former two costs consist of the inventory cost. The net profit is the difference between the revenue and the cost. Two alternative strategies are
described as follows:

3.1. Strategy 1

In time period t, the DC makes ordering decisions for retailer r according to the OUL policy. The product is delivered from the manufacturer to the retailers
directly. This instance is characterized by the fact that the DC does not make changes to the product quantity during the product delivery process. Fig. 1a shows the
process of this strategy. It shows that the shortage and inventory costs in the considered VMI system occur at the retailers only. Since there is no backlogging at the
retailers, the cost of the product per unit at the retailers cR can be regarded as both the profit per unit and the shortage cost per unit from the viewpoint of the supply
chain (SC) system. hR is the holding cost per unit at the retailers. Furthermore, the product cost and holding cost at the retailers are assumed to be identical. For
simplicity, Strategy 1 is called the ‘‘One-Stage-Decision (OSD)” strategy. We use Lr to denote the lead time from the manufacturer to retailer r under the OSD
strategy.

3.2. Strategy 2

In time period t, the DC makes both ordering and delivery decisions simultaneously. Differing from Strategy 1, all the products are delivered from the
manufacturer to the DC first, and then delivered from the DC to the retailers. In other words, there is an inventory level at the DC. Due to limited capacity at the DC,
some products may be supplied from an ‘‘alternative” source. Strategy 2 is called the ‘‘Two-Stage-Decision (TSD)” strategy. We use lv and lr to denote the lead times
from the manufacturer to the DC and from the DC to the retailer r, respectively, under the TSD strategy. It is reasonable to assume that lv + lr > Lr. Moreover, we
assume that the DC needs to pay an additional product cost cv for a unit of the product acquired from the alternative source than from the manufacturer. Therefore, cv
can be regarded as the shortage cost per unit at the DC from the perspective of the SC system. is the unit inventory holding cost at the DC. In addition, we assume
that cR > cv. Fig. 1b illustrates Strategy 2. To simplify expression, by ‘‘given Drt” we mean ‘‘given Dri for all i =1, 2,...,t” in the following discussion.

4. OSD and TSD strategies

4.1. One-Stage-Decision (OSD) strategy

4.1.1. The order-up-to-level at retailer r Under the OSD strategy, the optimal OUL S
1
for retailer r in period t can be presented as follows (Heyman and Sobel,
1984):
rt

Fig. 1a. Illustration of the OSD strategy.

Fig. 1b. Illustration of the TSD strategy. ?Material flow; information flow; M: the manufacturer; DC: the distribution centre; Rr: retailer r.
L. Yang et al. / European Journal of Operational Research 201 (2010) 112–122

qffiffiffiffiffiffi
S1 v1

rt ¼e1 rt þk1 rt ðt ¼1; 2; 3; ...; r ¼1; 2; 3; ...; nÞ; ð2Þ

( )( )()
PLr þ1 PLr þ1 -1 cR -1

where e1 ¼E DrðtþiÞjDrt ; v1 ¼Var DrðtþiÞjDrt ; k1 ¼Fc

Rþh , and Fð-Þis the inverse cumulative standard normal distribution
R

rt i¼1 rt i¼1 SS
function. From (1), we can deduce that

Lr þ1 Lr þ1 j Lr þ1 Lr þ1 j-1
1 -qrj j-1-i

X X XXX
DrðtþjÞ¼dr þ qDrt þ qerðtþ1þiÞ;
1 -qrr
r
j
( )

j¼1 j¼1 j¼1 j¼1 i¼0

qrrdr qrrqrr
Lr þ1
1 -qLr þ1 11 -qr

X 1 -qLr þ1 1 -qLr þ1
e ¼dr þDrt ¼ Lr þ1 -þDrt ;

j¼11 -q1 -q1 -q1 -q1 -q

rt

rrr rr
Lr þ1 1 r 2i

r2 X
¼ ð1 -qr Þ: ð3Þ ri¼1
rt
1 -q2

4.1.2. Product quantity delivered from the manufacturer to the retailers Based on the optimal OUL policy, the product quantity delivered from the manufacturer
to retailer r in period t can be presented as
1 1
yrt 1 ¼Drt þS -S rðt-1Þðt P 2Þ: ð4Þ
rt

From (1), we can deduce that

1 -qLr þ1 y1 ¼qLr þ1Drt þ r ðdr þert Þ: ð5Þ

rt r
1 -qr
Therefore, the total product delivery quantity to all the retailers in time period t in the SC system is
( )

XX 1 -qLr þ1 y1 ¼ y1 ¼ qLr þ1Drt þ r ðdr þert Þ : ð6Þ

1 -q
trt r rr r

4.1.3. The revenue of the SC system

To meet the demand in time period t + Lr + 1, the DC makes decisions for retailer r in time period t. DrðtþLr þ1Þis the demand in time period t + Lr + 1, i.e.,

2 L Lr þ1 L -
Lr 1
DrðtþLr þ1Þ¼dr 1 þqr þq þ---þqr r þq Drt þqr r erðtþ1Þþq erðtþ2Þþ---þerðtþL þ1Þ
r

rr r

Lr
¼dr r þqLr þ1 qLr -i

1 -qLr þ1 X
Drt þ erðtþ1þiÞ;
 1 -qrr
ri¼0
1 -qLr þ1 erðtþLr þ1Þ¼EDrðtþLr þ1ÞjDrt¼dr r þqLr þ1
Drt ;1 -qrr
!
2ðLr þ1Þ

21 -qr
rðtþLr þ1Þ¼Var DrðtþLr þ1ÞjDrt¼rr : ð7Þ r
1 -q2
Define p(-) as the revenue function of the system. Given Drt, the maximum expected revenue at retailer r due to the decisions made in time
(()) R1
period t is EpS jDrt ¼ cRx/r1ðxÞdx ¼cRerðtþLr þ1Þ, where /r1 is the probability density function of DrðtþLr þ1Þ for given Drt. The maximum
1

rt0
expected revenue of the whole SC system, for given Drt, due to the decisions made in time period t under the OSD strategy is
(()) P
1
EpS tjDrt ¼cR erðtþLr þ1Þ.
r

4.1.4. The inventory cost of the SC system

Ra Ra
Since demands are always non-negative, we assume that 0 f ðxÞdx ¼ -1 f ðxÞdx in the following discussion, where f(x) is the demand probability density function.
Based on pervious research (e.g., Lee et al., 2000; Zhao and Cheng, 2009), the minimum expected inventory cost, which consists of the shortage cost and holding
cost, at retailer r due to the decisions made in time period t for given Drt under the OSD strategy is
( ()) Z Srt 1 ()
Z 1( ) qffiffiffiffiffiffi qffiffiffiffiffiffi
EgS
1
¼hR S1 -x fr1ðxÞdx þcR x -S1 fr1ðxÞdx ¼ðcR þhRÞ½k1FSðk1ÞþfSðk1Þ� v1 -crk1 v1
rt jDrt rt rt rtrt
S1
0 rt

qffiffiffiffiffiffi
¼ðcR þhRÞfSðk1Þ v1 ; ð8Þ
rt

(1
Þ )1 r
2 t

ðx-e
where fr1ðxÞ¼pffiffiffiffiffiffiffiffiffi exp - . FS(-) and fS(-) are the cumulative standard normal distribution function and the standard normal prob
1
2pv12v

rt rt

ability density function, respectively. Define g(-) as the inventory cost function. Given Drt, the minimum expected inventory cost of the whole
(()) (())P
1 1
SC system due to the decisions made in time period t under the OSD strategy is EgS tjDrt ¼ rEg S jDrt .
rt
4.1.5. The transportation cost of the SC system
Define t1 r as the average unit transportation cost to retailer r under OSD and d(-) as the transportation cost function. Given Drt, the expected transportation cost at
retailer r in time period t under OSD is
 L. Yang et al. / European Journal of Operational Research 201 (2010) 112–122

( ())( )

1 -qLr þ1 EdS jDrt ¼Et1y1 jDrt ¼ qLr þ1Drt þ r ðdr þertÞ : ð9Þ
1

t1
rtrrtr r

1 -qr

Note that when Dri(i =1, 2, 3,..., t) are given, ert is also given. Therefore, the expected transportation cost of the SC system in time period t under the OSD strategy
with given Drt is
( ())X(())X( )

1 -qLr þ1
Lr þ1 r

1
EdS ¼ EdS1 ¼ qrt1 r: ð10Þ
e
tjDrt rtjDrt Drt þðdr þ rt Þ rr
1 -q
r

4.1.6. The profit of the SC system

Define w(-) as the profit function.The system profit equals the system revenue minus the system cost. Therefore, the SC system’s profit due to the decisions made
by the DC in time t under the OSD strategy is
( ())(())(())(())X Xqffiffiffiffiffiffi (())
EwS ¼EpS -EgS -EdS erðtþLr þ1Þ-ðcR þhRÞfSðk1Þ v1 -EdS : ð11Þ
1 1 1 1 1

4.2. Two-Stage-Decision (TSD) strategy

In this strategy, there is an inventory level at the DC. The DC orders products from the manufacturer to its inventory. With updated demand information, the DC
makes delivery decisions for the retailers, which are called the deliver-up-to-level policy. We assume that the shortage cost and holding cost at the DC are both
smaller than those at the retailers, respectively, due to economies of scale, i.e., cv < cR and hv < hR.

4.2.1. The delivery-up-to-level at the retailers

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi
¼e21 v21 ¼e21 v21

The optimal delivery-up-to-level for retailer r in time period t under the TSD strategy is S
21
þk21 þk1 , where
rtrt rtrt rt

()() ()
P l þ 1 l þ 1 P c
e21 ¼E i ¼ 1 DrðtþiÞjDrt ; v ¼Var i ¼ 1 DrðtþiÞjDrt , and k21 ¼k1 ¼F cR þRhR.
r 21 r -1

rtrt S
jr

Plr þ1 Plr þ11-qPlr þ1 j Plr þ1Pj-1 j-1-i

The accumulative demand during the time period [t +1,t + lr +1] is j¼1 DrðtþjÞ¼dr j¼1 þ j¼1 qrDrt þ j¼1 i¼0qr erðtþ1þiÞ. Its
1-qr ()( )lr þ1 ()
lr þ1 rr

Plr þ1 qr ð1-qÞ qr ð1-qÞ Plr þ1

¼ dr qr qr qr

expectation and variance are respectively e21 ¼Ej¼1 DrðtþjÞjDrt lr þ1-þDrt, and v21 ¼Var j¼1 DrðtþjÞjDrt ¼
rt 1-1-1-rt

r2 Plr þ1

r 1-q2 i¼1
1-qr 2i .
r

4.2.2. Product quantities delivered from the DC to the retailers

Based on the optimal delivery-up-to-level policy, the product quantity delivered from the DC to retailer r in period t under the TSD strategy is
 21
y21 ¼Drt þS -S21 ; ð12Þ
rt rtrðt-1Þ

1 1
where S and S are the order-up-to-level for retailer r in time periods t and (t -1) under the OSD strategy, respectively. According to (1),
rt rðt-1Þ
we have

1 -qlr þ1 y21 ¼qlr þ1Drt þ r ðdr þert Þ: ð13Þ

rt r
1 -qr
( )
lr þ1

P 1-qr t rr 1-qrr

Therefore, the total product quantity of the SC system from the DC to the retailers in time period t is y21 ¼ qlr þ1Drt þd
þ
ert .

4.2.3. The order-up-to-level at the DC The optimal order-up-to-level of the DC in time period t is
qffiffiffiffiffiffiffiffi
S
22 22 v22

t ¼et þk22 t ; ð14Þ

( )(( ))( )(( ))

PPPPP P

Plv þ1 Plv þ1 Plv þ1 Plv þ1

where e22 t ¼E ri¼1 y21 rðtþiÞjDrt ¼ rE i¼1 y21 rðtþiÞjDrt ¼ re22 rt ; v22 t ¼Var ri¼1 y21 rðtþiÞjDrt ¼ r Var i¼1 y21 rðtþiÞjDrt ¼ r v22 rt , and
-1 cv

k22 ¼FS ðÞ.

cv þhv

The total product delivery quantity of the SC system from the DC to the retailers in time period t is

ðdr þert Þ Drt:1 -q

y22 ¼y21 þS22 -S22 ¼ 1 -qlr þlv þ2 þqlr þlv þ2 rt rtrtrðt-1Þ r r r

! !
lv þ1
ql þ2 ql þ2 22 21 dr rrrr
r r

X 1 -qlv þ11 -qlv þ1

ert ¼E yrðtþiÞjDrt¼ lv þ1 -þDrt; ð15Þ
1 -qr 1 -qr1 -qr

i¼1
!
lv þ1 lv þ1 22 21 r lr þ1þi

X r2 X 2

rt ¼Var yrðtþiÞjDrt¼ 21 -qr : i¼1 ri¼1

ð1 -qÞ

4.2.4. Product quantities delivered from the manufacturer to the DC

According to the optimal OUL policy, the product quantity delivered from the manufacturer to the DC under the TSD strategy in time period t is
L. Yang et al. / European Journal of Operational Research 201 (2010) 112–122

( )
X 1 -qlr þlv þ2
22 21
þS22 -S22 l þlv þ2 r
r

yt ¼yt t t-1 ¼ qr Drt þðdr þertÞ : ð16Þ

r 1 -qr

4.2.5. System revenue Given Drt, the maximum expected revenue at retailer r due to the delivery decisions made in time period t under the TSD strategy is
Z

( ())1
21
EpS jDrt ¼ cRx/r21ðxÞdx ¼cReðtþlr þ1Þ; ð17Þ
rt

where /r21 is the probability density function of Drðtþlr þ1Þ.

Given Drt, the maximum expected revenue of the whole SC system resulting from the delivery decisions made in time period t under the TSD strategy is
( ())(())X
2 21
EpS tjDrt ¼EpS tjDrt ¼cR erðtþlr þ1Þ: ð18Þ r

It can be deduced that the demand in time period t + lr +1 is

Drðtþlr þ1Þ¼dr 1 þqþq2 þ---þqlr þqlr þ1Drt þqlr erðtþ1Þþqlr -1erðtþ2Þþ---þerðtþlr þ1Þ
rrrrr r
lr r lr þ1 lr -i

1 -qlr þ1 X
¼dr þqDrt þ qerðtþ1þiÞ:
1 -qrr r

i¼0
1 -qlr þ1 ð19Þ erðtþlr þ1Þ¼EðDrðtþlr þ1ÞjDrtÞ¼dr r þqlr þ1Drt ;1 -qrr

!
2ðlr þ1Þ

2 1 -qr
rðtþlr þ1Þ¼VarðDrðtþl þ1ÞjDrtÞ¼r:
r

r
1 -q2 r
4.2.6. Inventory cost
Under the TSD strategy, the inventory cost of the SC system consists of the inventory costs at the DC and the retailers.

4.2.6.1. Inventory cost at the retailers due to the DC’s delivery decisions in time period t. Under the TSD strategy, given Drt, the minimum expected inventory cost
with respect to retailer r due to the decisions made by the DC in time period t is

2 S21 x -S21 v21 v21

S
1

( ())Z rt ()Z 1( )qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi

21
EgS ¼hR -xxÞdx þcR xÞdx ¼ðcR þhRÞ½k21FSðk21ÞþfSðk21Þ�
rtjDrt rt fr21ðrt fr21ð rt -cRk21 rt

S
21
0 rt

qffiffiffiffiffiffiffiffi
¼ðcR þhRÞfSðk1Þ v21; ð20Þ
rt

( 21 2)
ðx-eÞ Plr þ1
rt
where fr21ðxÞ¼pffiffiffiffiffiffiffiffiffi1 ffi exp -2v21 is the probability density function of i¼1 DrðtþiÞ. The minimum expected system inventory cost occur
2pv21
rtrt

ring at the retailers, given Drt, due to the decisions made in time period t under the TSD strategy can be expressed as
( ( ))P (( ))Ppffiffiffiffiffiffiffiffi
v21
EgS
21
jDrt ¼ EgS21 jDrt ¼ðcR þhRÞfSðk1Þ rt .
trrt r

4.2.6.2. Inventory cost at the DC due to the DC’s ordering decisions in time period t. Given Drt, the inventory cost at the DC due to the decisions made in time period
t under the TSD strategy is

2
S
2

(())Z( )Z 1( ) qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi

22 22
S x -S v22 v22
tjDrt ¼hv tt -xf22ðxÞdx þcv tf22ðxÞdx ¼ðcv þhvÞ½k22FSðk22ÞþfSðk22Þ� t -cvk22 t
22
EgS

S
22
0 t

qffiffiffiffiffiffiffiffi
v22
¼ðcv þhvÞfSðk22Þ t ;
() ()
ðx-e22 Þ
2
Plv þ1 P
1t
y21

where f22ðxÞ¼pffiffiffiffiffiffiffiffiffiffi exp -is the probability density function of .

2pv 2 2v
2 22 i¼1 rrðtþiÞ t

4.2.7. Transportation cost Define t21 as the average unit transportation cost from the DC to retailer r, and t22

dr md as the average unit transportation cost from the manufacturer to the DC under the TSD strategy. Hence, given Drt, the expected transportation cost from the DC
to the retailers due to the DC’s delivery decisions made in time t under the TSD strategy is
( )

XX 1 -qlr þ1 Edy21 jDrt ¼ Et21 21jDrt ¼ t21 qlr þ1Drt þ r ðdr þertÞ :
yrt dr r rr 1 -qr
t dr

Given Drt, the expected transportation cost from the manufacturer to the DC due to the DC’s order decisions made in time period t under the TSD strategy is
 L. Yang et al. / European Journal of Operational Research 201 (2010) 112–122

( )
XX 1 -qlr þlv þ2
22
t22 lr þlv þ2 r

Edy22 ¼ Et22 ¼ md
qr :
t jDrt mdyrt jDrt Drt þðdr þ rt Þ rr e 1 -q
r

On the basis of the given Drt, the total SC system’s expected transportation cost due to the DC’s ordering and delivery decisions made in time period t is
( ())
2
EdS jDrt ¼Edy21 jDrt þEdy22 jDrt
ttt
()( )
X 1 -qlr þ1 1 -qlr þlv þ2

t21 lr þ1 r þt22 lr þlv þ2 r

¼ dr qr Drt þðdr þertÞ md qr Drt þðdr þert Þ : ð21Þ
1 -qr

4.2.8. The profit of the system Under the TSD strategy with given Drt, the profit of the SC system can be expressed as
( ())(())(())(())
2 2 2 2
EwS jDrt ¼EpS jDrt -EgS jDrt -EdS jDrt
tttt
qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi ( ())

X X
v21 v22
2
¼cR erðtþlr þ1Þ-ðcR þhRÞfSðk1Þ -ðcv þhvÞfSðk22Þ-EdS jDrt :
rt ttrr

5. Analyses and discussions

In order to simplify the analysis, we assume that all the parameters are identical for all the retailers. As such, to denote a parameter, we can drop the subscript of
the parameter, e.g., r = rr, d = dr, l = lr, L = Lr, Dt = Drt, and q = qr (r =1, 2,...,n).

5.1. Variance of the system based on the decisions

P
Define V
1
¼ v1 as the total demand variance of the system due to the DC’s decisions made in time tunder the OSD strategy.
t rrt

P P

v21 v21 v22

¼ is the demand variance due to the DC’s delivery decisions made in time t under the TSD strategy. v22 ¼ is the demand variance due to the DC’s ordering
decisions made in time t under the TSD strategy. V
2
¼v þv is the total demand variance of the sys
21 22

t rrt t rrt

tt t
tem due to the DC’s delivery and ordering decisions made in time t under the TSD strategy. From the standpoint of the DC, making decisions under the TSD strategy
is more complicated because the DC needs to undertake and coordinate both ordering and delivering decisions. In this section we characterize the DC’s decisions by
comparing the variances the DC faces under the OSD and TSD strategies, respectively.

¼v21 þv22 > V1

Proposition 1. V2 tt .
tt
P
P
Proof. Under the OSD strategy, we have V
1
¼ v1 ¼1n-r2 iL¼þ11ð1 -q2iÞ. Under the TSD strategy, we have V2 ¼v21 þv22, where
t rrt q2 tt t
P P
P P lþ1þi 2
v21 v21 nr lþ1 v22 ¼ nr lv þ1¼¼ i¼1ð1 -q Þ, and v ¼ð1 -q Þ . Without loss generality, it is reasonable to assume that l < L,t rrt 1-q t rrt ð1-qÞ i¼1
2i 22
2 2 2 2

lv < L and L < l + lv. Therefore, the difference in the ordering quantity variances received by the manufacturer under the TSD and OSD strategies is
" #
lþ1 Lþ1

XXX

1 21 1
2 1
V -V ¼nr2 ð1 -qiÞþ ð1 -q2iÞ-ð1 -q2iÞ
tt 2
1 -q21 -q2
ð1 -qÞi¼lþ2 i¼1 i¼1

"
Lþ1 # 2 i-22i

X l0þ2 X
¼nrð1 þq þqi þ---þq1Þ-ð1 þq2 þq4 þ---þq2Þ > 0; i¼lþ2 i¼lþ2

¼v21 þv22

tt t > V
2 1
where l0= l + lv. Therefore, we have V t .h

5.2. Modification ability of the DC

5.2.1. Definition of the DC’s modification ability

From the viewpoint of the SC system, the definition of the DC’s modification ability is reflected in the system profit under the OSD and TSD strategies. If the
expected system profit under the TSD strategy is higher than that under the OSD strategy over the whole planning horizon, then the DC’s modification ability is
regarded as positive; otherwise it is regarded as negative. In the following discussion, we assume that the planning horizon is infinite. We have the following
definition.
((())(()))
Pþ1Pþ1
2 1
Definition 1. Over the whole planning horizon, if t¼1EðDEðwÞjDrt Þ¼ t¼1 EwSt jDrt -EwSt jDrt > 0, then the DC’s modifica
((())(()))
Pþ1Pþ1
2 1
tion ability for the SC system is positive; if t¼1 EðDEðwÞjDrt Þ¼ EwS jDrt -EwS jDrt < 0, then the DC’s modification
t¼1 tt
ability for the SC system is negative.
d
¼EðDt Þ

From (1), it can be deduced that . Note that the profit of the SC system consists of three components, namely system revenue,
1-q 1-qt

system inventory cost, and system transportation cost. The modification ability of the DC can be evaluated by
((())(()))
Pþ1Pþ1

t¼1 EðDEðwÞjDrtÞ¼ EwS2 jDrt -EwS1 jDrt .

t¼1 tt

(())(())(())(())X Xqffiffiffiffiffiffi (())

EwS jDrt ¼EpS jDrt -EgS jDrt -EdS jDrt ¼cR erðtþLr þ1Þ-ðcR þhRÞfSðk1Þ v1 -EdS jDrt ;
1 1 1 1 1

ttttrttrr
L. Yang et al. / European Journal of Operational Research 201 (2010) 112–122

(())(())(())(())
2 2 2 2
EwS tjDrt ¼EpS jDrt -EgS tjDrt -EdS jDrt
tt
qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi ( ())XX
v21 v22
2
¼cR erðtþlr þ1Þ-ðcR þhRÞfSðk1Þ rt -ðcv þhvÞfSðk22Þ t -EdSt jDrt ; rr

((())(()))((())(()))((())(()))
2 1 2 1 2 1
DEðwjDrt Þ¼ EpS jDrt -EpS jDrt -EgS jDrt -EgS jDrt -EdS jDrt -EdS jDrt
tttttt

¼DEðpjDrt Þ-DEðgjDrt Þ-DEðdjDrtÞ:

P -q

L
þ1lþ1

q
Each component can be expressed as follows:DEðpjDrt Þ¼cR ðerðtþlr þ1Þ-erðtþLr þ1ÞÞ¼ncRðq lþ1
-q ÞDt þncRd , and so the expected
Lþ1

r 1-q
difference in system revenue under two strategies in one period can be expresses as
Z þ1 ()lþ1 Lþ1Þ 1 -1 tþlþ1 tþLþ1Þ d

EðDEðpÞÞ¼ DEðpÞf ðxÞdx ¼ncREðDtÞðq-q¼-ncRðq-q;

01 -qt 1 -q
where f(x) is the probability density function of Drt. Therefore, over the whole planning horizon, the expected difference in system revenue under two strategies in
one period can be expresses as

dq T and
T lþ1 Lþ1
Þ Þ;
X
EðDEðpÞÞ¼-ncRðq-q2 ð1 -qt¼1 ð1 -qÞ
( ())( ())(qffiffiffiffiffiffiffiffi qffiffiffiffiffiffi)qffiffiffiffiffiffiffiffi
X
v21 v22
DEðgjDrt Þ¼EgS -EgS
2 1
¼ðcR þhRÞfSðk1Þ-v1

0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uu pffiffiffi ulv þ1

uX lþ1 Lþ1 X
uX u

nr nr

t ð1 -q2iÞ tAt ð1 -qlþ1þiÞ2

¼ðcR þhRÞfSðk1Þpffiffiffiffiffiffiffiffiffiffiffiffiffiffi@ -ð1 -q2iÞ þðcv þhvÞfSðk22Þ :

1 -q21 -q
i¼1 i¼1 i¼1

It is noted that DE(gjDrt) is independent of Drt and t. Therefore, the expected difference in system inventory cost under the two strategies in one period is E(DE(g)) =
E(DE(gjDrt)). Over the whole planning horizon, we have
0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u u pffiffiffi ulv þ1
T lþ1 Lþ1

X uX uX uX
Tnr Tnr 2
 EðDEðgÞÞ¼ðcR þhRÞfSðk1Þpffiffiffiffiffiffiffiffiffiffiffiffiffiffi@t ð1 -q2iÞ-t ð1 -q2iÞAþðcv þhvÞfSðk22Þ t ð1 -qlþ1þiÞ
1 -q21 -q
t¼1 i¼1 i¼1 i¼1

and

(())(())X ( )
d þert 21 d þert
t21 lþ1 þt22 l þ2 Lþ1 0
t
-t1 dr þt22 -t1 r 1 -q 1 -q

2 1
DEðdjDrt Þ¼Ed St jDrt -EdS tjDrt ¼ dr qmdqr qDt -þmd r :

The expected difference in system transportation cost under the two strategies in one period can be expressed as

[ 1 nd
t21 -t1 tt21 lþ1 þt22 l þ2 -t1 Lþ1
0

EðDEðdÞÞ¼ dr þt22 -qq;

mdr dr
qmdqr
1 -q
and over the whole planning horizon, we have

qð1 -qT Þ 21 lþ1 22 l nd

T
dr þt22 t þt 0 þ2 Lþ1

X
EðDEðdÞÞ¼ Tt21 md -t1 r -dr qmdq-t1 r q:

1 -q 1 -q
t¼1

Therefore, the difference in system profits under the TSD and OSD strategies is

T TTT

X XXX
EðDEðwÞÞ¼ EðDEðpÞÞ-EðDEðgÞÞ-EðDEðdÞÞ
t¼1 t¼1 t¼1 t¼1

qð1 -qT Þ
lþ1 Lþ1
Þ
¼-ncRdðq-q2ð1 -qÞ
2 0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3
ulþ1 uLþ1 pffiffiffi ulv þ1

uXuX uX 2

nr nr
5
-T4ðcR þhRÞfSðk1Þpffiffiffiffiffiffiffiffiffiffiffiffiffiffi@t ð1 -q2iÞ-t ð1 -q2iÞAþðcv þhvÞfSðk22Þ t ð1 -qlþ1þiÞ
1 -q21 -q
i¼1 i¼1 i¼1

qð1 -qT Þ q q
-Tt21 dr þt22 -t1 þt21 lþ1 þt22 l0þ2 -t1qLþ1 nd: ð22Þmdr dr md r 2
ð1 -qÞ
Define
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

vffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffi
ffiffiffi
 01
ulþ1 uLþ1 pffiffiffi ulv þ1

uXuX uX
nr nr
2
z ¼ðcR þhRÞfSðk1Þpffiffiffiffiffiffiffiffiffiffiffiffiffiffi@t ð1 -q2iÞ-t ð1 -q2iÞAþðcv þhvÞfSðk22Þ t ð1 -qlþ1þiÞ þt21 þt22 -t1 : ð23Þ
dr mdr
1 -q21 -q
i¼1 i¼1 i¼1
 L. Yang et al. / European Journal of Operational Research 201 (2010) 112–122

Therefore, (22) can be presented as

X
qð1 -q Þ qð1 -q Þ
EðDEðwÞÞ¼ -zT -ncRdðqlþ1 -qLþ1Þ T þt21 lþ1 þt22 l þ2 -t1qLþ1
0
T nd:
2 dr
qmdqr 2
t¼1 ð1 -qÞð1 -qÞ

It is noted that asT ? 1, two situations may arise, namely z – 0 and z =0.
PT
Case 1:If z – 0, then as T ? 1, zT will be the dominant term to determine t¼1EðDEðwÞÞ. Therefore, in the long run, the sign of z deter
^
mines the difference in system profits under the two strategies as positive or negative. From (23), we note that z consists of the expected difference in system
inventory cost and difference in average unit transportation cost from the manufacturer to retailers under the OSD and TSD strategies. It indicates that in the long run,
the differences in system inventory cost and average unit product transportation cost under the OSD and TSD strategies play a significant role in determining the
DC’s modification ability.Define A =(cR + hR)fS(k1) and
()() qffiffiffiffiffiffiffi
-1 cR -1 cv B 1þq

B =(cv + hv)fS(k22), where k1 ¼ Fand k22 ¼ F. Moreover, we set d ¼ and a ¼ d , where d 2(0,1). Furthermore, if we
S cRþhRScv þhv A 1-q
t1

assume that hv ¼
blv
and cv ¼ blv cR, where b is a constant parameter, then we have k1= k22.Set Dt ¼ t21 þt22 -; a ¼
l0 hR l0 dr mdr Plþ1 Plv þ1 lþ1þiÞ2 Plþ12 PLþ1 Pl þ22 nr
0
q

i
ð1 -q2iÞ; b ¼ð1 -; c ¼ i¼1ð1 -qiÞ; e ¼ i¼1 ð1 -q2iÞ and c0 ¼ b þc ¼ i¼1 ð1 -qiÞ. Then, we have z ¼ A pffiffiffiffiffiffiffiffi hpffiffiffi pffiffiffi pffiffiffii ¼1
i¼1

hpffiffiffi pffiffiffi pffiffiffii 1-q2

a nra
aþp ffiffi b -eþDt. We have the following conclusion:(i) If z ¼ A pffiffiffiffiffiffiffiffi aþpffiffi b-eþDt < 0, then the DC’s modification ability
n
hpffiffiffi pffiffiffi
n
pffiffiffii 1-q2

nra

is positive. (ii) If z ¼ A pffiffiffiffiffiffiffiffi aþpffiffi b-eþDt > 0, then the DC’s modification ability is negative.
n
1-q2
Case 2:If z = 0, then
T qð1 - 21 lþ1 22 l0þ2 1 Lþ1 qð1 -
X qT Þ qT Þ EðDEðwÞÞ¼ -ncRdðqlþ1 -qLþ1Þ þt þt -t q nd:

2 dr
qmdqr 2
t¼1 ð1 -qÞ ð1 -qÞ
z. Therefore, we have the fol
q PT
t21

dr q þt t q
lþ1 22 1 Lþ1

l þ2
mdq -r ð1-qÞ nd.As T ? 1, we have t¼1EðDEðwÞÞ!
0 2

z ¼-ncRdðqlþ1 lowing conclusion:As T ? 1, (i) if

^
qLþ1Þ
Define
q ð1-qÞ2
þ
-
^
z < 0, then the DC’s modification ability is positive. (ii) Ifz > 0, then the DC’s modification ability is negative.

5.2.2. Modification ability of the DC without consideration of transportation cost In the following discussion we only focus on the situation where z – 0. If the
difference in transportation cost under the OSD and TSD
strategies is negligible, i.e., Dt "0, then we focus on the system revenue and inventory cost of the supply chain. Consequently, pffiffiffi pffiffiffi pffiffiffi

hi
nra

z ¼ A pffiffiffiffiffiffiffiffi aþpffiffi b-e.

n
1-q2

Proposition 2. Without consideration of transportation cost, when q 2(0,1), the effect of the DC’s modification ability is related to the number of
^
retailers.
a2b a2 b

If n > pffiffi pffiffi 2 , then the DC’s modification ability is positive; if n < pffiffi pffiffi 2, then the DC’s modification ability is negative.
ð e- aÞð e-aÞ

Proposition 2 shows that even without consideration of transportation cost, the DC’s modification ability may be positive or negative. The choice of strategies
needs to consider the number of retailers. Only when the number of the retailers is more than a certain level will the DC’s modification ability become positive, which
means that the DC can make use of its economies of scale to improve total profit.
Proposition 3. Without consideration of transportation cost, when q = 0, the effect of the DC’s modification ability is related to the number of retailers.
h i

pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi
Proof. When q = 0, we have a = l +1, b = lv +1, e = L + 1, and z ¼ Anr l þ1 þpd ffiffi lv þ1 -L þ1.
n

ð
d2 lv þ1Þ ð
d2 lv þ1Þ

If n > pffiffiffiffiffiffiffi pffiffiffiffiffiffi 2, then the modification ability of the DC is positive; if n < pffiffiffiffiffiffiffi pffiffiffiffiffiffi 2, then the modification
ability of the DC is
ð Lþ1-lþ1Þ ð Lþ1-lþ1Þ
negative. h
Proposition 3 applies a particular situation where q = 0. It shows that there is a threshold level for the number of retailers to examine the DC’s modification
ability. Moreover, given n, we can find the optimal solution of l, which reflects the location of the DC. This result is presented in Proposition 4.

hi
¼ l0 l0 22

R
Define l -e; 2 þe be the set of all possible l, where e is a parameter to determine the range of lR and 0 < e 6 l -1. 0

'
Proposition 4. Without consideration of transportation cost, when q = 0, for a given n, the optimal solution of l is l ¼ -e.
l0

2
h i

pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi
d 1d1

Proof. When q = 0, we have a = l +1, b = lv +1, e = L + 1, and z ¼ Anr l þ1 þpffiffi lv þ1 -L þ1 , where dz ¼ Anr pffiffiffiffiffiffi -pffiffi pffiffiffiffiffiffiffi , and

ndl 2 lþ1 n2 lv þ1 d z 1 d 1
2

¼ Anr -pffiffiffiffiffiffiffiffiffiffi -pffiffi pffiffiffiffiffiffiffiffiffiffiffiffi < 0. Therefore, z is concave in l.

3 n

4 ðlþ1Þ4 ðlv þ1Þ

2 3
dl

When the DC’s modification ability is positive, we need to maximize the system profit, which can be reflected by the value of z. A smaller value of z implies a
higher system profit. With respect to system profit, the objective is to maximize the system profit, i.e., Min
h i

pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi

dd l0
 '
z ¼ Anr l þ1 þpffiffi lv þ1 -L þ1 . Note that when pffiffi< 1, the optimal solution is at its lower boundary l ¼ -e. h
nn 2

Proposition 4 reveals that when the demand is constant over time, the optimal location of the DC is determined, which should be as close as possible to the
retailers. Under this situation, the DC links the manufacturer and retailers in the supply chain. In other words, the DC plays the role of a supplier for all the retailers.
L. Yang et al. / European Journal of Operational Research 201 (2010) 112–122

5.2.3. Modification ability of the DC with consideration of transportation cost

In order to examine the effect of transportation cost on the DC’s modification ability, we first consider transportation cost only. Second, we analyze the effect of
transportation cost on the whole system. We have the following definition.
( )()()

()PPP
þ1þ1þ1
2
Definition 2. If t¼1 EðDEðdÞÞ¼ t¼1 EdS -t¼1EdS1 < 0, then the DC’s modification ability in terms of transportation cost for
tt

( )()()

()PPP
þ1þ1þ1
2
the SC system is positive; if t¼1 EðDEðdÞÞ¼ t¼1EdS -t¼1 EdS1 > 0, then the DC’s modification ability in terms of
tt
transportation cost for the SC system is negative.
From

()X Tt21 t1 qð1 -q Þ t21 lþ1 þt22 l þ2 Lþ1 nd

T
0
T

t1
EðDEðdÞÞ¼ dr þt22 ---q;

mdr dr
qmdqr

1 -q 1 -q
t¼1

we see that when T ? 1, the second term in the above equation is finite; while the first term is infinite. Therefore, the determining term is Dt ¼ t21 t1

dr þt -md ¼ klv t0, t ¼ lt0, and t ¼ Lt0, where t0is the unit transportation cost per . Without loss of generality, we assume that t
22 21 1 22

mdr drr
time unit. k 2(0,1] is a constant parameter, which reflects economies of scale in transportation. A smaller k implies a larger transportation cost discount due to
economies of scale. k = 1 implies there are no economies of scale. Therefore, Dt = t0[kl0 -L +(1 -k)l], which is increasing in k and l.
Proposition 5

(I)If k > lL -l
0 - l, the DC’s modification ability with respect to transportation cost is negative.
(II)If k < lL --ll, the DC’s modification ability with respect to transportation cost is positive.
0

Proposition 5 reveals that when the discount is large enough, the DC’s modification ability can reduce the total transportation cost due to its economies of scale.
However, if the discount is small, even when there are economies of scale, the transportation cost will increase with the use of DC. Under this situation, the use of
DC reduces the system profit. Furthermore, noting the system objective of maximizing profit, we see that the objective function with respect to transportation cost is
Min Dt ¼ t0½kl0 -L þð1 -kÞl :
Since Dt is increasing in l, we have the following proposition:

Proposition 6. The optimal solution of l with respect to transportation cost is l * = min{ljl 2lR}. Particularly, if q = 0, for a given n, the optimal solution of l with
respect to system expected profit is l = min{ljl 2l }.
* R

It is reasonable that if we want to use economies of scale to reduce transportation cost, the location of the DC should be close to the retailers. Particularly, when
demand is constant, the use of DC can reduce the both inventory cost and transportation cost, due to its economies of scale. Therefore, the optimal DC’s location
should be as close to the retailers as possible.

6. Conclusions

This study evaluated the effects of the DC’s modification ability on a VMI system. We assumed that there are two strategies adopted by the DC, namely the OSD
and TSD strategies. Different strategies may lead to different system profits. Through analysis, we found that the determining factors that affect the DC’s modification
ability are inventory cost and transportation cost, rather than revenue. Focusing only on inventory cost, we found that there exists a range of the number of retailers
leading to positive DC’s modification ability. We found that the DC’s modification ability with respect to transportation cost relates to lead time and transportation
cost discount. The optimal solution of lead time reflects the location of the DC; while the transportation cost discount manifests in the effect of economies of scale in
transportation. In considering inventory and transportation costs simultaneously, we need to balance these two costs.. To conclude, making a decision to build a DC
needs to consider different conditions carefully. The DC’s modification ability may not always be beneficial to the system. The decision maker needs to consider
various factors, such as cost, demand distribution, number of retailers, lead time, and the location of the DC.
We assumed the demand of each retailer is independent and identical. However, in reality, the demand is variable and the demands of different retailers may affect
one another. This may be considered in further studies. Another direction for future research is to consider the situation with multiple suppliers, who may have
competition and cooperation in the system. Sharing the DC may lead to different benefit allocations to the suppliers. We also assumed that the system adopts the
order-up-to-level policy. Future studies can consider other ordering policies.

Acknowledgements

We are grateful to two anonymous referees for their helpful comments on an earlier version of our paper. This research was supported in part by The Hong Kong
Polytechnic University under a research studentship to Yang.

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