Coordinated Replenishment Strategies in Inventory/Distribution Systems

1

Coordinated Replenishment Strategies in Inventory/Distribution Systems

Mustafa Cagri Gurbuz, Kamran Moinzadeh*, and Yong-Pin Zhou

University of Washington Business School

January 2005; revised January 2006; revised July 2006

To Appear in Management Science

Abstract

In this paper, we study the impact of coordinated replenishment and shipment in inventory/distribution

systems. We analyze a system with multiple retailers and one outside supplier. Random demand occurs at

each retailer, and the supplier replenishes all the retailers. In traditional inventory models, each retailer orders

directly from the supplier whenever the need arises. We present a new, centralized ordering policy that orders

for all retailers simultaneously. The new policy is equivalent to the introduction of a warehouse with no

inventory, which is in charge of the ordering, allocation, and distribution of inventory to the retailers. Under

such a policy, orders for some retailers may be postponed or expedited so that they can be batched with other

retailers’ orders, which results in savings in ordering and shipping costs. In addition to the policy we propose

for supplying inventory to the retailers, we also consider three other policies that are based on these well

known policies in the literature: (a) Can-Order policy, (b) Echelon Inventory policy, and (c) Fixed

Replenishment Interval policy. Furthermore, we create a framework for simultaneously making inventory and

transportation decisions by incorporating the transportation costs (or limited truck capacities). We numerically

compare the performance of our proposed policy with these policies to identify the settings where each policy

would perform well.

Corresponding author. Box 353200, University of Washington, Seattle, WA 98195. [email protected]

Yong-Pin Zhou ! 1/24/05 9:23 PMDeleted:

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1 Introduction

Effective management of distribution systems has been one of the most challenging issues facing both the

practitioners and the academicians for years. This has become even more critical in recent years because the

advancement in information technology has facilitated information sharing between the parties in the supply

chain, and more efficient transportation systems have emerged in many supply chains. Replenishment,

shipment consolidation/coordination policies, different inventory allocation methods, and effective utilization

of information are among the most important supply chain management issues today.

Aided by the sharing of information about demand, inventory, and supply between the parties in the

supply chain, companies are increasingly using coordinated replenishment/shipment to manage their

distribution systems, with the aim to reduce logistics (transportation and warehousing) costs which accounts

for over 10% of U.S. GDP (Delaney 1999). Coordinated replenishment is defined as the practice where a

central decision maker, based on the information provided by other parties in the system, decides how much

and when to order (Cheung and Lee 2002) for the whole system. Among several methods to handle the flow of

goods across the supply chain such as direct shipment, merge-in-transit, and cross-docking (see Brockmann

1999 and Kopczak et al. 2000), cross-docking is considered to be the most efficient one to facilitate

replenishment/shipment coordination.

In this paper, we consider the management of a single product in a centralized distribution system

consisting of one warehouse and N identical retailers who face stochastic demand. The retailers hold stock of

the item and unfulfilled demand is fully backordered. The warehouse functions as a stylized cross-docking

terminal with no inventory and coordinates shipments to retailers. As soon as shipments arrive at the

warehouse they are dispatched to the retailers (Gue 2001). The warehouse, as the central decision maker, has

full system information, and replenishes all the retailers’ inventory from an outside supplier with ample

supply. In addition to holding and backorder costs at the retailers, costs are incurred to transport goods both

from the supplier to the warehouse and from the warehouse to the retailers. We use a general transportation

cost function that allows the incorporation of limited transportation capacity.

The form of the optimal warehouse replenishment policy for managing distribution systems in

general, and the system described here in particular, is unknown. Most of the effort in the literature has been

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to find reasonable and practical heuristic policies built upon the three broad categories of echelon based,

installation based, and time based replenishment policies.

Although echelon-based inventory policies dominate the installation-based policies in serial systems

(Axsater and Rosling 1993), neither dominates the other in general distribution systems. To the best of our

knowledge, Axsater and Juntti (1996) is the only study that attempts to identify the settings where echelon

policies are superior to installation policies in distribution systems via simulation.

In this paper, we propose a new policy, called the hybrid policy, that utilizes both echelon and

installation inventory information to coordinate the replenishment of all the retailers. Under the hybrid policy,

the warehouse monitors the inventory position at all retailers, and places an order at the outside supplier to

raise every retailer’s inventory position to an order-up-to level, SH, whenever any retailer’s inventory position

hits sH, where sH<SH, or the total demand at all the retailers reach QH. We will optimize the triplet (sH, SH, QH)

to minimize the total system cost, consisting of linear holding and backorder costs at the retailers, ordering cost

at the warehouse, and transportation costs.

The hybrid policy improves upon the coordinated replenishment/shipment policies discussed in

Cheung and Lee (2002), Cachon (2001) and Cetinkaya and Lee (2000), where the replenishment policies are

based on the echelon stock (i.e. there is no s). By monitoring the echelon inventory only and overlooking the

installation inventory information, these policies can result in huge backorder costs at some retailers (e.g. in the

extreme all the demand could come from one retailer alone). The hybrid policy helps to reduce the backorder

costs at the retailers by introducing the installation reorder point. Moreover, the hybrid policy replenishes all

the retailers at the same time, reducing ordering and transportation costs at the warehouse. These benefits

come at a cost, however. Under the hybrid policy, some retailers might get replenished even when they have

observed very low demand. A better policy should also have a can-order level for each retailer (see Balintfy

1964, Federgruen et al. 1984, Ozkaya et al. 2003 for a discussion of can-order policies) so that replenishment

will occur only if the inventory position is below the can-order level at that particular retailer. This introduces

another layer of complexity and the resultant policy is intractable.

In this paper, we also compare the hybrid policy with the purely installation based and echelon based

policies, referred to as Policies IB and EB. Although Policies IB and EB are special cases of the hybrid policy

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and their performance is dominated by that of the hybrid policy, they provide good benchmarks as the optimal

form of the replenishment policy in such distribution systems is unknown and tight lower bounds for the

average total cost for such systems are hard to obtain. Thus, it’s instructive to identify the settings in which the

hybrid policy offers the maximum gain over these two effective and widely used policies.

We also compare a time based policy, which is referred to as Policy TB, with the hybrid in our paper.

Cachon (2001) considers full service periodic review (time based) and minimum quantity periodic review

(mixture of quantity and time based) replenishment policies. He finds out that retailers can gain additional

operational benefits by switching from periodic truck replenishment to continuous review truck replenishment

when store lead times are small.

Finally, we use the hybrid policy to represent coordinated replenishment policies, and compare it to a

traditional distribution model where retailers order independently using a continuous review (QNC, RNC) policy.

The goal is to see the impact of coordination through information sharing.

For more recent studies that show the value of information sharing in distribution systems, the reader

is referred to Moinzadeh (2002), Cachon and Fisher (2000), Gavirneni (2001), Axsater and Zhang (1999),

Graves (1996), Aviv and Federgruen (1998), and Cheung (2004). Atkins and Iyogun (1988), Viswanathan

(1997), and Federgruen and Zheng (1992) also study joint replenishment strategies in distribution systems

where information sharing is available.

An important aspect of our model is the incorporation of transportation capacity considerations into

inventory replenishment policies. Even though there exists a body of literature on joint inventory and

transportation decisions, most of the models assume deterministic demand. Chan et al. (2002) consider

incremental and all unit discount transportation cost structures under a cross-docking setting, whilst Li et al.

(2004) consider dynamic lot sizing with batch ordering and truckload discounts. Federgruen and Zheng (1992)

consider a general ordering/transportation cost structure. For lot sizing and dispatching models with

deterministic demand and capacitated transportation resources, see Lipmann (1969), Lee (1989), Cetinkaya

and Lee (2002), Lee et al. (2003), Toptal et al. (2003), and Anily and Tzur (2005).

For cases with stochastic demand, the work is rather limited. Federgruen and Zipkin (1984b) integrate

routing and inventory decisions with capacitated vehicles, and Federgruen and Zipkin (1984a) consider joint

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replenishment with general ordering/transportation cost structure. Aviv and Federgruen (1998) consider fixed

cost of shipment with a capacity limit on the quantity to be shipped. Cachon (2001) assumes transportation

costs that depend on the number of capacitated trucks dispatched to replenish the retailers, and Cetinkaya and

Lee (2000) consider a fixed cost of dispatching plus variable unit shipment cost and employ a time based

policy for shipment release. In our model, there also exist truck capacity limits, but the shipment sizes are

allowed to exceed the truck capacity, in which case additional transportation cost is incurred (e.g. a common

carrier could be used to transport the goods). Under the hybrid policy, we can reduce the variation in the

shipment size and avoid incurring additional transportation cost by optimizing QH, which is an upper bound on

the total shipment size from the supplier to the warehouse.

When the warehouse receives the shipment from the supplier, the hybrid policy allocates the goods to

the retailers based on their inventory positions at the time the order is placed, not the more up-to-date inventory

positions at the time the shipment is received. Clearly, the system would benefit from the postponement of

allocation decision that will in turn lower inventory related costs at the retailers. The analysis of such problems

is quite complicated, and beyond the scope of this paper. Readers interested in postponement/allocation and

stock rebalancing policies along with replenishment coordination are referred to Maloney (2002), Cheung and

Lee (2002), Schwarz (1989), Eppen and Schrage (1981), Federgruen and Zipkin (1984a), Jackson (1988),

McGavin et al. (1993), and Gavirneni and Tayur (1999).

Ordering in batches is considered one of the five main causes of the bullwhip effect, usually defined

as the increase in demand variability as one moves up the supply chain (Chen et al. 2000). Lower demand

variance, and less bullwhip effect, will lead to lower costs for the supplier. The replenishment policy, which

determines the order size at the supplier as well as the length of the order cycle, clearly impacts the demand

variance at the supplier (thus the bullwhip effect). Cachon (1999) analyzes the supply chain demand variability

in a distribution system and finds that retailers switching from synchronized ordering to balanced ordering

policies reduce supply chain holding/backorder costs. For all the replenishment policies considered, we also

analyze the variance of the demand rate at the supplier.

Our contribution is three-folded: First, we propose a model for the analysis of coordinated

replenishment/ordering policies in distribution systems. We fully characterize the system dynamics under the

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proposed hybrid policy and compare its performance against three other well-known coordinated

replenishment policies via a numerical experiment. Furthermore, we identify the settings where the hybrid

policy offers the maximum benefit. While we mainly focus on average total cost rate in our comparisons, we

also measure the bullwhip effect associated with each of the policies considered. Second, our model allows

joint inventory and transportation decisions under stochastic demand. By comparing the systems with and

without transportation costs, we quantify the additional value of the hybrid policy in distribution systems with

limited transportation capacity. Third, we investigate the value of coordinated replenishment by comparing the

hybrid policy with a Non-Coordinated policy with no warehouse, where each retailer orders independently

from the outside supplier.

The rest of the paper is organized as follows: In Section 2, we introduce the model and propose the

policies. Section 3 analyzes the various policies, and is followed by the numerical analysis in Section 4.

Finally, we conclude with managerial insights and future research directions in Section 5.

2 The Model

We consider a distribution system consisting of N identical retailers and an outside supplier with ample supply.

Each retailer faces random demand that follows Poisson distribution, holds inventory, and backorders all

excess demand. There is a centralized decision maker (for ease of exposition, we call it the warehouse) that

has access to information about demand, inventory levels and relevant costs at the retailers and is responsible

for ordering, receiving, allocating, and dispatching shipments to retailers. This warehouse, which is a stylized

representation of a cross-dock operation, holds no inventory. The transit time from the outside supplier to the

warehouse, L0, and from the warehouse to each retailer, L, are both constant.

For all the policies considered in this paper, the objective is to minimize the expected cost rate of the

supply chain, which consists of ordering/shipment costs, and holding/shortage costs at the retailers. We assume

that holding and shortage costs are linear. In our model, the ordering cost is fixed and we assume a general

functional form for the shipment cost. Under these assumptions, we propose to use the following hybrid policy

to manage such a distribution system:

HYBRID POLICY: The warehouse orders to raise all the retailers’ inventory position to SH whenever any

retailer’s inventory position drops to sH or the total demand at all the retailers reaches QH.

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The hybrid policy is based on both the installation and the echelon inventory positions. The system

replenishment may be triggered in two ways: (1) installation trigger: whenever any retailer’s inventory position

drops to sH; or (2) echelon trigger: whenever the echelon inventory position drops to NSH-QH. The values of sH,

SH, and QH will be optimally determined. When a retailer’s inventory position first drops to sH, it sends a

signal that other retailers may soon need to be replenished. The hybrid policy then orders to bring every

retailer’s inventory position back to SH. Hence, the installation trigger in the hybrid policy is proactive in the

sense that most retailers will be replenished before their reorder point is reached. The echelon trigger in the

hybrid policy is designed to reduce the transportation penalty cost, because it reduces the variation in the total

quantity shipped to the warehouse from the outside supplier. We will verify this numerically in Section 4.

To understand the effectiveness of the hybrid policy, we will compare it to the following three well-

known coordination policies:

POLICY IB (Installation Based): The warehouse orders to raise all the retailers’ inventory position to SIB

whenever any retailer’s inventory position drops to sIB.

POLICY EB (Echelon Based): The warehouse orders to raise all the retailers’ inventory position to SEB

whenever the total demand at all the retailers reaches QEB.

POLICY TB (Time Based): The warehouse orders to raise all the retailers’ inventory position to STB every T

time units.

Policies IB and EB are special cases of the hybrid policy, where QH and SH–sH, respectively, are

sufficiently large. The trigger for ordering is installation based for Policy IB and echelon based for Policy EB.

Furthermore, Policy IB is also a special case of the can-order policies where the can-order levels are equal to

the order-up-to level for all the retailers as there are no fixed transportation costs for orders shipped to a

retailer. Policy TB is a time-based policy, analogous to the periodic-review order-up-to policy. Similar to the

hybrid policy, Policy IB is proactive in reducing retailer’s shortage as whenever one retailer triggers the order,

all other retailers are replenished, even though their inventory positions are above the reorder point. In contrast,

Policies EB and TB are echelon based and time based, and they opt towards reducing ordering costs.

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We define the relevant notation in Table 1. We use subscript “0” for the warehouse related and “i”

( Ni ,...,1= ) for retailer related parameters and variables. Also, we define Δ=S-s, )0,max(xx =+ ,

),min( baba =! , and, as in Hadley and Whitin (1963), !

);( j

ejp

jµµ

µ!

= and );();( !"

=

=rj

jprP µµ .

L0 Transit time from the supplier to the warehouse under coordinated replenishment Li Transit time from the warehouse to retailer i under coordinated replenishment li Transit time from the supplier to retailer i (under Non-Coordinated policy) LT Total leadtime for any retailer

iT!

Time of the occurrence of Δth demand for retailer i

QT

Time of the occurrence of Qth demand for all the retailers

τ Time between two consecutive replenishments (cycle time)

IPi(t) Retailer i’s inventory position at time t

ILi(t) Retailer i’s inventory level at time t

Z0 The total amount shipped to the retailers in an order cycle

Zi The amount shipped to retailer i in an order cycle

Di(t) Demand at retailer i for a period of time t N Total number of retailers in the system

λi Mean demand rate at retailer i (i = 1, …, N)

K0 Fixed order cost for orders placed by the warehouse to the supplier under coordinated replenishment ki Fixed order cost for orders placed by retailer i to the supplier under Non-Coordinated policy C0 Maximum capacity of the truck from the outside supplier to the warehouse

Ci Maximum capacity of the truck from the warehouse to retailer i

πi Unit backorder cost/time at retailer i

hi Unit inventory holding cost/time at retailer i f(.) Probability density function of τ F(.) Cumulative distribution function of τ g(n,C0) Penalty cost the warehouse pays to the carrier when the inbound quantity is “n” units

gi(n,Ci) Penalty cost retailer i pays to the carrier when the outbound quantity is “n” units

CR(.,.) Expected total cost rate

α0 Inbound unit penalty cost under coordinated replenishment αi Outbound unit penalty cost for retailer i

Table 1: Notation

Coordinated replenishment reduces ordering and transportation costs as the warehouse places a large

order for all the retailers at the same time and receives all the inbound shipments at once. Therefore, in

addition to examining the effectiveness of the hybrid policy within the class of coordinated replenishment

policies, we would also like to evaluate the performance of the hybrid policy when compared to the non-

University of Washington Buisness SchooComment: Page: 2

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coordinated replenishment policy. We will compare the hybrid policy with the following policy via a

numerical analysis in Section 4.

NON-COORDINATED POLICY: Each retailer uses a continuous review (QNC, RNC) policy independent of

one another and is responsible for its own replenishment process.

Our model considers both the inventory and the transportation costs. To discern the effect of

transportation costs, we will do the numerical analysis in Section 4 with and without the transportation costs.

We define the quantities shipped from the outside supplier to the warehouse as “inbound” and those from the

warehouse to the retailers (or from the supplier to the retailers under the Non-Coordinated policy) as

“outbound”. Note that the inbound quantity in every order cycle is constant under Policy EB, but random

under the other three coordinated replenishment policies. This will play a critical role in the numerical

comparisons with and without the transportation penalty costs in Section 4. Among the coordinated shipment

policies, the inclusion of transportation penalty costs will improve the relative performance of the hybrid

policy and Policy EB in most cases as QH and QEB can be optimized to minimize the transportation cost

penalties. Moreover, since the outbound quantities under Non-Coordinated policy are constant, the benefit of

using coordinated shipment policies (relatively to Non-Coordinated policy) should decrease when

transportation penalty costs are included. As a result, whether and how the warehouse incurs additional

transportation costs may have a significant impact on which policy should be employed.

3 Analysis

The Non-Coordinated policy is the traditional (Q,R) policy. Interested readers can find detailed analysis in

Section 4.7 of Hadley and Whitin (1963). In this section, we focus on the coordinated replenishment policies

for distribution systems where all the retailers are identical (and suppress the retailer subscript i sometimes).

The analysis applies to heterogeneous retailers as well, but since the additional notation makes the analysis and

computations cumbersome, it is not covered in this study.

3.1 Analysis of the Hybrid Policy

When an order is triggered, all the retailers’ inventory positions are raised to SH. The system as represented by

all the retailers’ inventory positions, therefore, is a regenerative process, and the order epochs are the

regenerative points. We denote the time between two consecutive order epochs by τ, and refer to it as the

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“cycle time”. The cycle time is the minimum of the time for any retailer to realize ΔH units of demand and the

time for all the retailers to realize a total of QH units of demand. Therefore,

),,,..,(min1

HHH Q

NTTT

!!=" (1)

where i

H

T!

~Erlang(ΔH,λ) for all i and HQ

T ~Erlang(QH,Nλ). While the i

H

T!

s are i.i.d., HQ

T is dependent on

the i

H

T!

s. Let Di denote the random demand at retailer i during a cycle (excluding the possible triggering

demand), then we have the following relation:

,);(......1);(1

1)(,1)(,.....,1)(Pr1)Pr(1)(

1

1

2

2

21

21 0 0 0 11...

1,...,, 1

1

1

!! !"! "

!

= = = =

#$+++

#%$ =

=

#=#=

&'

()*

+#$#%$#%$#=>#=

U

d

U

d

U

d

N

i

i

Qdddddd

N

i

i

H

N

i

iHNH

N

N

HN

HN

tdptdp

QtDtDtDttF

,,

-

(2)

where )1,1min(1 !!"= HH QU and )...1,1min( 11 !!!!!!"= iHHi ddQU for i=2, … ,N.

Proofs of the lemmas and propositions in this section are provided in the Appendix.

Lemma 1: The probability density function of τ, f(.), is given by:

! ! "!= =

#

==

#

#

=

1

1

1

1

2

20 0

1

10

);();(.......)(U

d

U

d

N

i

Ni

U

d

N

N

tUptdpNtf $$$ (3)

From (3) we note that the distribution of τ depends on QH and ΔH (the difference between SH and sH)

but not on SH and sH individually.

Next, we analyze the probability distribution of the inbound quantity, Z0. Later, we will use this

distribution to find the expected cycle time, the outbound quantity and demand during a cycle, and the

distribution of the inventory position at the retailers. Although it is possible to obtain these measures directly,

our derivations via the distribution of Z0 significantly reduces the computational complexity.

An order could be triggered either by a retailer experiencing ΔH units of demand or by all the retailers

experiencing a total of QH. We examine two cases: (1) When QH is sufficiently large (specifically, when

QH>ΔH+(N-1)(ΔH-1)), the echelon trigger can be ignored, and the distribution of Z0 is straightforward to

calculate. (2) When QH is not that large (specifically, when QH≤ΔH+(N-1)(ΔH-1)), Z0 is much harder to

characterize. Nevertheless, with a coupling argument, we can use the distribution of Z0 in the first case to

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derive Z0 in the second case. To differentiate the two cases, we will temporarily use Φ, in place of Z0, to

denote the inbound quantity in the first case.

Now suppose QH>ΔH+(N-1)(ΔH-1). Since QH can be ignored, we have !=

="N

i

iZ

1

, where Zi=Di+1 if

retailer i triggers the order and Zi=Di otherwise (where Zi is the random outbound quantity to retailer i and Di is

the demand for retailer i in a cycle).

It then suffices to derive the distribution of Zi.

Lemma 2: When QH>ΔH+(N-1)(ΔH-1), for any retailer i,

[ ] ,

0

11);(1);1(

01

)(0

1

!!"

!!#

$

%&

'%((%''

=

=& )*

=

'

H

H

t

N

Hi

n

ndttPtnp

n

nDP +++

and [ ]

!!"

!!#

$

%>

%&&%''

=

=( )*

=

'

H

H

t

N

Hi

n

ndttPtnp

n

nZP

0

1);(1);1(

01

)(0

1+++ . (4)

Next, we examine the second case where QH≤ΔH+(N-1)(ΔH-1). The random variable Z0 is in the range

[min(QH, ΔH), QH]. We will use a coupling argument to derive Z0 from Φ. Let there be two systems facing the

same demands. In System 1, Q1H>ΔH+(N-1)(ΔH-1) so Q1H can be ignored. In System 2, Q2H≤ΔH+(N-1)(ΔH-1).

All other parameters are identical. For any demand realization, if it results in an inbound quantity Φ>Q2H in

System 1, then this demand realization would certainly initiate the echelon trigger in System 2 so Z0=Q2H.

Conversely, if a demand realization initiates an installation trigger in System 2, then it would initiate the same

trigger in System 1. Hence we have the following result:

Lemma 3: When QH ≤ ΔH+(N-1)(ΔH-1), the inbound quantity Z0 satisfies:

HH QQZ <!"<0

and HH QQZ !"#=0

. Therefore,

!"

!#$

==%

<&'=%== (

)H

Qz

HHH

Qnz

QnQn

nZ

H

for )Pr(

),min(for )Pr(

)Pr( 0. (5)

12

We use the following relation to compute the expected cycle time:

Lemma 4:

!

"N

ZEE

][][ 0= (6)

Having derived the distribution of Z0 and using Lemma 4, we then use the following recursive relation

(based on QH) to compute the probability distribution of Zi (we use Z to simplify notation):

Lemma 5:

( )!!"

!!#

$

=%=

%=%=%+%&'

&%=%+%&'

=%'

HHH

QHHHHHHH

HHHHHH

HH

QQnN

QnNQNQnZP

QnNQNQnZP

NQnZPH

),min(1

),min(),,(),,1|(

)1,min(,...,1,0),,(),,1|(

),,|( (

(

(7) where,

( )]1,,|[]1,,1|[1

11

1

1),,(

11

!"!!!"!+#$%

&'( !#

$%

&'(##$

%&&'

(!!

="!+!

NnQENnQENNn

QNQ HHHH

nQn

H

HH

H

))*+ .

The recursive approach above is extremely useful for computing the probability distribution of Z for

large QH, ΔH, and N, because the direct method requires the consideration of all possible combinations of

demands at all the retailers that would make the summation domain prohibitively large and the problem

computationally intensive for large N, QH, and ΔH.

The expression for E[τ] in (6) is sufficient to derive the ordering cost at the warehouse. Now, in order

to find the holding and backordering costs at the retailers, we need to find the probability distribution of the

inventory level at each retailer at any point in time. As orders do not cross, we can use the following identity:

IP(t-LT)–D(LT)=IL(t), where LT=L0+L is the lead time. Therefore, to derive the distribution of the IL, it

suffices to derive the probability distribution for the IP. We start by conditioning on the demand at any retailer

during the cycle time, τ. It is well known (Ross 1993, page 223) that for a Poisson process conditioned on n

arrivals during time (0,t), the arrival times X1, X2, …, Xn have the same distribution as the order statistics of n

independent random variables uniformly distributed on the interval (0,t). Therefore, we have the following

conditional probability distribution of the inventory position:

HHH SnSnSjn

nDjIP ,......,1,for 1

1))(|Pr( +!!=

+=== " . (8)

13

Here, D(τ) denotes the demand faced by any retailer during the cycle time excluding the possible

triggering demand (as soon as inventory position drops to sH, it is raised to SH, so the inventory position is

equal to sH with probability 0). The following lemma gives the probability distribution of D(τ) :

Lemma 6: For any retailer, when QH=1, !"#

$

==$

10

01)|(

n

nQnDP H

; and when 2!HQ ,

!"

!#

$

%&

''%(('&

=

=&

),min(0

)1,1min1)1|(

01

)|(

HH

HHHH

Qn

Q(nQnZP

n

QnDP . (9)

Using Equation (8) and Lemma 6, we obtain the probability distribution of the inventory position as

follows:

. ..., ,,1

))(())(())(|()Pr( 1

11

HH

U

jSn

U

jSn

SUSjn

nDPnDPnDjIPPjIP

HH

!=+

======= ""

!=!=

### (10)

Lemmas 1-6 provide us with the framework to calculate the total system cost per unit time.

Proposition 1: The cost rate of the system under the hybrid policy is as follows:

( )( )

.)Pr();()Pr();(][

1

][)]([][)(*][

1

),min(

0),min(

00

0

!!"

#$$%

&=+=+

'+++=

( ((=

)

=)=

+

N

i

Q

n

ii

Q

Qn

HHH

HH

nZCngnZCngE

IPELTDEILEhNE

KCR

*

++* (11)

Proposition 2: For any fixed value of ΔH and QH, CR is convex in SH. Furthermore, the optimal SH is the

largest integer satisfying the following inequality:

!+"#+#= +

$#=

H

HHHH

S

SQSj hLTjPjIP

)1,1,1max(

))*;(1)(Pr(%

%& . (12)

3.1.1 Special Case of the Hybrid Policy: Policy IB

Policy IB is a special case of the hybrid policy where QH>ΔH+(N-1)(ΔH-1). As a result, the probability

distributions of the demand during the cycle and the outbound quantity are given in Lemma 2, the distribution

of the inbound quantity is given as !=

="N

i

iZ

1

, and the probability distribution of the inventory position is

given in Equation (10). Moreover, we can simplify the probability distribution of τ from Equation (3) and the

cost rate from Proposition 1.

14

( ) ( )[ ]( )0,;1;1)(

1!"##"=

#ttPtpNtf

N

IBIB $$$ . (13)

Proposition 3: The cost rate of the system under Policy IB is as follows:

( )( )

.);()Pr();()Pr();(][

1

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

2

1

0

)1)(1(

0

0

!!"

#$$%

&'+=+=(+

)+++=

* **=

)'

=

)')+'

'=

+

N

i

IBi

n

i

N

n

CgnZCngnCngE

IPELTDEILEhNE

KCR

IBIBIB

IB+

,,+

(14)

3.1.2 Special Case of the Hybrid Policy: Policy EB

Since the time to observe QEB units of demand, TQ, is Erlang(QEB,Nλ), we have:

Lemma 7: Under Policy EB,

!"

"=

+""

##+"$%&

'() "$

%&

'()$$%

&''(

)"

==

1

1 ,1

111

)Pr(EBQ

jSn

EBEBEB

jSnjS

EB

SjQSNNjS

n

QjIP . (15)

Furthermore, the inbound quantity is a constant QEB. Note that the warehouse still pays the additional penalty if

QEB exceeds inbound truck capacity. For the outbound quantities, we have:

EB

nQn

EBQn

NNn

QZ

EB

!!"#$

%&' ("

#$

%&'""#

$%%&

'==

(

0,1

11

n)Pr( . (16)

Proposition 4: The cost rate of the system under Policy EB is as follows:

( )[ ]

.)Pr();();(][

1

][)]([][)(

1 0

0

0

!!"

#$$%

&=+!!

"

#$$%

&+

'+++=

((= =

+

N

i

Q

n

iEB

EB

EB

nZCngCQgE

IPELTDEILEhNQ

NKCR

)

**+

(17)

3.2 Analysis of Policy TB

Under Policy TB, the warehouse places an order with the outside supplier every T time units (τ=T) to bring the

inventory position of every retailer to STB. Therefore, Policy TB resembles the periodic review, order-up-to,

<R,T> policy in Hadley and Whitin (1963). Under Policy TB, both the inbound and the outbound quantities

(0Z and Z ) are Poisson, with rates of NλT and λT, respectively. The following proposition makes use of

function B(1,STB-1,T), which is derived in Hadley and Whitin (page 260). For completeness, we also describe

B(1,STB-1,T) in the Appendix.

Proposition 5: The cost rate of the system under Policy TB (c.f. Hadley & Whitin 1963, p. 260) is:

15

.);();();();(1

),1,1()(2

*);1(

0 1 0

0

0

!"

#$%

&++

'()

*+,

-++!"#

$%& --+=

. ../

= =

/

=n

N

i n

i

TBTB

TnpCngTNnpCngT

TSBhT

LTShNTNPT

KCR

00

10

00 (18)

4 Numerical analysis

We first investigate the performance of different coordinated replenishment policies via a numerical

experiment and identify the settings where the hybrid policy provides significant improvement over Policies

IB, EB, and TB (recall that hybrid policy dominates Policies IB and EB). In 4.1.1, we present our results for

the case with no transportation penalty costs. Then we include the transportation penalty costs in 4.1.2. In

order to better understand the value of coordinated replenishment and the settings where it is beneficial when

compared to traditional systems where retailers order independently, we conduct a numerical experiment in

section 4.2 and quantify the value of coordinated replenishment by comparing the hybrid policy with the Non-

Coordinated policy. Thus far, we have focused on the expected total cost rate in our comparisons between

policies. In Section 4.3, we turn our attention to another important measure, namely, the bullwhip factor and

discuss the bullwhip effect for all five policies considered. Finally, in Section 4.4, we present insights on

supply chain design issues when the hybrid policy is used.

4.1 Numerical Analysis within Coordinated Replenishment Framework

4.1.1 Numerical Analysis without Transportation Penalty Costs with Coordinated Replenishment

There are situations where the transportation capacity is rarely exceeded and the transportation penalty costs

can be ignored. In this section, we assume no limitations on truck space and examine the effectiveness of the

various coordination policies from a pure ordering/inventory perspective. The relevant costs are ordering,

inventory holding and backorders.

The following set of parameters is used in the experiment:

{ } { } { } { }. 20 ,10 ,5 , 100 ,50 ,2 , 99.0 ,95.0 ,9.0 ,1 , 2 ,1 ,5.0 ,1 ,10

0 !!!=!==+

NKhLhL

L

"

"#

For all the policies, the optimal value of S is calculated given other variables (QH and ΔH, ΔIB, QEB, and

T). Although we were unable to prove convexity of the expected total cost rate over Δ, Q, or T, we observed it

16

in our numerical experiment. Therefore, in the numerical tests, we assume reasonable maximum limits and

search over all possible values within this range (i.e., we do not stop the search when the cost rate increases).

SH,IB,EB,TB* ΔH,IB

* QH* QEB

* T* N K0 L0/L π/(π+h)

Table 2: Trends in average optimal values of decision variables

We consider 81 cases in total. In Table 2, we present the behavior of the optimal decisions for each

policy that reflects the tradeoff between inventory costs and the fixed ordering cost. In systems with few

retailers, large transit time, and/or fixed ordering cost, the warehouse tends to order less frequently (higher

E[τ]) and the retailers carry more inventory (higher S*) . Moreover, order frequency/inventory increases as

shortages become more expensive, as expected.

To compare the effectiveness of our proposed hybrid policy with the other three policies, we compute

the percent deviation of Policies IB, EB and TB from the hybrid policy defined as:

. , )(

)() (Policy for % IB,EB,TBi

hybridCR

hybridCRiPolicyCRideviation =

!=

N L0 π /(π+h) K0 Avg. % Deviation 5 10 20 0.5 1 2 0.9 0.95 0.99 2 50 100

Policy IB 0.14 0.21 0.34 0.22 0.28 0.20 0.28 0.27 0.15 0.21 0.14 0.35 Policy EB 2.73 1.69 0.90 2.18 1.77 1.37 0.96 1.50 2.85 1.04 2.17 2.11 Policy TB 7.38 4.01 2.04 5.35 4.48 3.59 3.32 4.16 5.95 3.31 5.13 4.99

Table 3: Average % deviations of Policies IB, EB, and TB from the hybrid policy

The percent deviations of Policies IB, EB and TB from the hybrid policy are summarized in Table 3.

We observe that Policy IB is almost as effective as the hybrid policy in all the cases − the average percent

deviation is no more than 0.35%, and the maximum percent deviation is 1.04%. However, having the

additional echelon trigger enables hybrid policy to slightly increase Δ* (we observe that Δ* for Policy IB is no

greater than that of the hybrid policy) and lower holding and ordering costs further, especially for systems with

large N and K0. The largest % deviations of Policy IB occur when N=20 and K0=100.

17

We also observe that the performances of the other two echelon based policies, Policies EB and TB

improve (even though they are still dominated by the hybrid policy) when the warehouse is responsible for

many retailers (large N) or when the warehouse is closer to the retailers relative to the supplier (large L0/L).

This is consistent with the results in Cheung and Lee (2002) and Axsater and Juntti (1996). From Table 3, we

note that the relative advantage of hybrid policy over Policies EB and TB is more significant for larger values

of π. This confirms our prior belief that installation-based trigger in the hybrid policy makes it more responsive

to balance of inventory and shortage costs. As K0 increases, the deviations for Policies EB and TB first

increase as a result of the increase in cycle time making them more vulnerable to shortages. However, when K0

is very large (K0=100), the impact of installation trigger in the hybrid policy diminishes, as the ordering cost

becomes the dominant factor in the expected total cost rate, making the hybrid also vulnerable to shortages.

Consequently, there is a slight reduction in the deviation figures for Policies EB and TB as K0 increases from

50 to 100.

In summary, we observe that the introduction of the installation trigger significantly improves the

performance of the hybrid policy over Policies EB and TB (echelon based), especially for large K0, π and small

L0/L, N. However, Policy IB seems to perform as well as the hybrid policy in most cases. Furthermore, Policy

IB dominates Policy EB in all but 10 cases while Policy TB is dominated by all other policies in all cases.

4.1.2 Numerical Analysis with Transportation Penalty Costs and Coordinated Replenishment

In this section, we include limitations on transportation capacities and examine its impact on the policies’

performance. Recall that in Section3, we allowed the transportation penalty cost, )(!g , to have a general form.

Among several alternatives to model transportation costs, the most commonly observed ones are: (i) a fixed

shipping cost plus a variable linear penalty cost for units beyond the transportation/truck capacity (Federgruen

and Zipkin 1984a and Cetinkaya and Lee 2000) or (ii) imposing an absolute limit on the quantity to be shipped

from the supplier (Aviv and Federgruen 1998) or (iii) having a stepwise transportation cost function where the

shipping cost is directly proportional to the number of trucks dispatched from the supplier (Cachon 2001).

Although our model can also handle options (ii) and (iii), we choose to work with option (i) in our numerical

experiment as it is used in practice and has also been employed in previous research. We believe it is a

18

reasonable one for it represents the cases where the warehouse uses its own fleet with a fixed transportation

capacity and uses a common carrier for additional units, who charges on a per-unit basis. The following are

the piecewise linear transportation cost functions for the inbound and outbound shipments, respectively:

000 )();( !+

"= CnCng and !+

"= )();( CnCngi . (19)

For each test case, the inbound truck capacity is expressed as a multiple of the optimal Q under Policy

EB calculated in 4.1.1: *

00* EBQC != . We call

0! , the inbound capacity factor and choose *

EBQ as the base

inbound quantity as it is equal to the constant inbound quantity used by Policy EB to balance the ordering and

inventory costs. To emphasize the relation of the total outbound capacity to the inbound capacity, we set

!*0CC = , where γ is called the outbound capacity factor. Below is a summary of all the parameter values:

{ } { } { }

{ } { } { }. , , 5.1 ,1 ,5.0 }, 2 ,1 ,5.0 { }, 2 ,5.0 { , 100 ,50 ,2

, 99.0 ,95.0 , 20 ,10 ,5 , 2 ,1 ,5.0 ,1 ,1 ,1

2.1100

0

0

NN

hL

L

K

NLh

!!!!!

!!!===+

""#

$

#

#

%%

In all, a total of 1944 cases are considered. The optimization procedure is almost identical to that in

4.1.1. The only difference is in the computation of the total cost (due to additional transportation cost).

The impact of N, K0, L0, and π on the average values of SH,IB,EB,TB*, ΔH,IB

*, QH,EB*, and T* is almost the

same as in 4.1.1. However, with the introduction of transportation penalty, at optima, all policies tend to order

more frequently (by decreasing ΔH,IB*, QH,EB

*, and T*) in order to reduce the likelihood of exceeding the truck

capacities. This occurs especially in cases with tight transportation capacity constraints and high unit penalty

cost. As a result of the increase in order frequency, average S* is also reduced to avoid high inventory holding

cost.

The inbound quantity, QEB, is constant under Policy EB. Therefore, the inclusion of limited inbound

truck capacities will relatively improve its performance. Among the rest of the policies, we expect Policy TB’s

inbound quantity to be less variable as it utilizes demand pooling and the time between two replenishments is

constant. Moreover, it can analytically be shown that the demand rate variance at the supplier is higher for the

hybrid policy and Policy IB (see Table 5). On the other hand, under the hybrid policy and Policy IB, the

inbound quantity has an upper bound of min(QH, ΔH+(N-1)(ΔH-1)) and ΔIB+(N-1)(ΔIB-1), whereas the inbound

19

quantity has no upper bound under Policy TB. Therefore, it is not clear how this will affect the other three

policies, where the inbound quantity is random.

Avg. % Deviations from hybrid

policy

Policy IB Policy EB Policy TB Min. 0 0.17 0.9 Avg. 0.23 1.77 4.48 without penalty

(Section 4.1.1) Max. 1.04 6.53 13.32 Min. 0 0 0.04 Avg. 1.21 1.45 4.74 with penalty

(Section 4.1.2) Max. 25.48 6.18 13 Table 4: The minimum, average, maximum deviations with/without penalty costs

We make the following observations as a result of our numerical experiment:

• Table 4 shows that the average % deviation is lower for Policy EB but higher for Policies IB and TB

(note that the change in Policy TB’s performance is not significant) when compared with those in

Section 4.1.1. We conjecture that this result is possibly due to larger variance in the inbound quantity

under Policies TB and IB compared to hybrid policy (as higher demand rate variance at the supplier

leads to higher cost), even though the demand rate variance at the supplier is lower for Policy TB

compared to that of the hybrid policy.

• Figures 1 and 2 suggest that when the inbound capacity becomes a critical factor (small γ0 and large

α0), Policy EB (IB) starts to perform relatively better (worse). Numerically, QEB*≤C0 for about 75% of

the time to completely avoid inbound penalty (QH*≤C0 63% of the time for the hybrid policy).

0

1

2

3

4

5

6

0 0.5 1 1.5 2

Inbound Capacity Factor

Av

e. %

Dev

iati

on

fro

m

hy

bri

d

Pol. IB

Pol. EB

Pol. TB

Figure 1

0

1

2

3

4

5

6

0 2 4 6

Inbound unit penalty

Av

e. %

Dev

iati

on

fro

m

hy

bri

d Pol. IB

Pol. EB

Pol. TB

Figure 2

20

• The hybrid policy still dominates the other three policies. Even though in some cases the benefit of

employing the hybrid policy is marginal, in other cases it is significant.

• Under the hybrid policy, the echelon trigger (QH*) is very effective in reducing the variance in the

inbound quantity and thus avoiding the penalty costs most of the time. Recall that Policy IB performs

almost as well as the hybrid policy without the transportation penalty costs. When transportation

penalty costs are included, however, the performance improvement of the hybrid policy over Policy

IB is more significant, especially in systems with many retailers, tight capacity, and low fixed

ordering costs (i.e. maximum deviations can be as high as 25.48%).

As mentioned before, the outbound quantity is random under all four policies, but it has an upper

bound of ΔH, ΔIB, and QEB (see Lemmas 5, 2, and 7) under the hybrid policy, Policies IB and EB, respectively

and there is no upper limit under Policy TB. Thus, one would expect the inclusion of outbound transportation

penalty cost to disfavor Policy TB. Our numerical results indicate that, to reduce outbound penalty cost, even

though the hybrid policy and Policy IB take advantage of the tight upper bound (ΔH* and ΔIB

*) when γ is low,

the change in performance is negligible.

Summarizing Table 4, we observe that when transportation penalty costs are included, the average

percent deviations in costs do not change drastically compared to those with no transportation penalty, but

there may be significant changes in the maximum deviation figures. Policy EB has the least variation from the

hybrid policy. Moreover, we observe that in the presence of the penalty costs, Policies EB and TB outperform

Policy IB in 37% and 8% of the cases, respectively, a large increase from 12% and 0% in Section 4.1.1. Since

all policies require the same information infrastructure, the hybrid policy is preferable to the other three

policies, unless the warehouse management is restricted to taking action only at certain points in time (which

favors Policy TB), or the contract with the carrier stipulates that inbound quantity must be a fixed number

(which favors Policy EB).

4.2 Numerical Analysis: Coordinated Policy (Hybrid Policy) vs. Non-Coordinated Policy

In this section, we investigate the value of having a warehouse coordinating replenishment rather than letting

retailers order independently. We are particularly interested in investigating the benefits of coordination

21

especially when system parameters such as order cost or lead times change (i.e., increase) as the system adopts

replenishment coordination. In doing so, we compare the performance of the hybrid policy to the Non-

Coordinated policy. We use the same set of parameters as in Section 4.1 for N, K0, L0, π, α. To see the effect of

the Non-Coordinated policy’s fixed ordering cost and lead time (denoted by k and l), we vary the values of k

and l (see Figures 4 and 5). Figures 3-6 show the results for some representative cases. In all these figures,

except Figure 6, and we assume ample transportation capacity.

In Figure 3, we assume the order costs are the same in both systems (k= K0), and vary K0 and N. Since

the main idea behind coordination is the possible savings in total fixed ordering costs due to coordination, the

hybrid policy enjoys the benefit of coordinated replenishment when the order costs K0 and N are larger as

replenishing all retailers simultaneously and thus, placing fewer orders per unit time results in savings. On the

other hand, we observe that the Non-Coordinated policy is more responsive against shortages and starts

performing relatively better as π increases (Figure 4). Moreover, our numerical results suggest that

coordination still pays off significantly even if handling at the warehouse (under coordinated replenishment)

introduces additional time or distance in shipping (i.e., l<L+L0).

In reality, there may be additional handling/sorting charges at the warehouse that may inflate the

ordering cost under coordinated replenishment. Thus, the fixed ordering cost in the Non-Coordinated policy, k,

might be less than that under the hybrid policy, K0. As can be seen from Figure 5, for a given K0, as k/K0

decreases, the performance of Non-Coordinated policy improves. In fact, there may exist a threshold level of

k/K0, below which the Non-Coordinated policy outperforms the coordinated policy and coordinated

% deviation of Non-Coordinated from hybrid

0

50

100

150

0 10 20 30N

% d

evia

tion K0=2

K0=50

K0=100

Target Service Level=95%, L 0 =1, l =2, k=K 0


40

50

60

70

80

90

0.85 0.9 0.95 1

Target service level

% d

ev

iati

on

l=max(L,L0)

l=(max(L,L0)

+L+L0)/2l=L+L0

N=10, k=K 0 = 50, L 0 = 1

Figure 3 Figure 4

22

replenishment may not be economical. That is, as a result of coordination, if the order cost increases

significantly, coordinated replenishment may not be appropriate anymore. Based on our numerical observation,

such threshold levels are usually small (for instance, it is about 0.2 in Figure 5), implying that in most practical

cases, the hybrid policy will be preferred to the Non-coordinated policy.

In Figure 6, we consider the two systems with truck capacity limitations. Note that the shipment size is

constant under the Non-Coordinated policy, where the inbound/outbound quantity is random under the hybrid

policy. Moreover, due to the economy of scale in ordering and shipping, the hybrid policy tends to order in

larger batches than the Non-Coordinated policy. These two factors worsen the hybrid policy’s relative

performance when the limitation on truck capacity is tight as shown in Figure 6.

Figure 5 Figure 6

In summary, our numerical experiment indicates that, coordinated replenishment results in significant

savings in supply chain costs in many practical settings. The magnitude of savings increase as there is little or

no increase in order cost and/or the total lead times when coordinated replenishment is adopted and for systems

with many retailers, large fixed ordering cost and large truck capacities.

4.3 Numerical Analysis of Demand Variability at the Supplier

It is known that one can reduce the system’s cost by mitigating the bullwhip effect, defined in this paper as the

ratio of the supplier demand rate variance to the variance of end customer demand. Since we assume ample

supply at the supplier (supplier’s cost is not modeled here), the bullwhip effect impacts the total cost rate in our

model only when the transportation capacity is limited. In this section, we examine the bullwhip effect under

all the policies considered in this paper. The results are given in Table 5. Detailed derivations can be found in

the Appendix.


-40

0

40

80

120

0 0.5 1 1.5k/K 0

K0=2

K0=50

K0=100

Target Service Level=95%, N= 10, L0= 1, l= 2


50

60

70

80

90

0 1.5 3 4.5 6C

unit penalty

cost=0.5

unit penalty

cost=2

Target Service Level=95%, N= 10, k=K0= 50, l= 1.5,

L0= 1, C

0= 10*C

23

In Table 5 we observe that there is no bullwhip effect under the Non-Coordinated policy, Policies EB

and TB, because their demand rate variance at the supplier is the same as the end customer demand rate

variance, Nλ. Note that this result is based on the assumption that the end customer demand is Poisson and

either the batch size or replenishment interval in these policies is constant. Different demand distributions at

the customer level could lead to different results in terms of the bullwhip effect.

Supplier’s demand rate Hybrid policy Policy IB Policy EB Policy TB Non-Coordinated policy

Mean Nλ Nλ Nλ Nλ Nλ

Variance !!"

#$$%

&+

)(

)(21

0

0

ZE

ZVarN'

!!"

#$$%

&+

)(

)(21

0

0

ZE

ZVarN' Nλ Nλ Nλ

Table 5: Supplier’s demand rate

We also observe that the bullwhip effect exists for the hybrid policy and Policy IB. Figures 7-9

illustrate the bullwhip factor, defined as the ratio of the demand rate variance at the supplier to the end

customer demand rate variance, for the hybrid policy and Policy IB with respect to N, K0, Ci. The following set

of values is used: {N=10, K0=50, π=19, L0=1, α0=α=2, C0=10* Ci}.

Figures 7-9 suggest that the hybrid policy can use the additional control variable, QH, to limit the total

demand from retailers, thus resulting in a smaller bullwhip factor than Policy IB. The variance under the

hybrid policy shows a concave behavior in N and K0, because an increase in either parameter leads to more

frequent echelon triggers, as the increase in QH* is not as significant as the increase in ΔH

*. On the other hand,

the demand rate variance under Policy IB increases at a faster rate when N, K0, or Ci increases, as higher ΔIB*

or N leads to a wider range for the random inbound quantity.

Figure 7 Figure 8 Figure 9

Demand rate variance at the

supplier

0

2

4

6

0 5 10 15 20 25N

Bu

llw

hip

fac

tor Hybrid

PolicyIB


supplier

0

2

4

6

0 50 100 150K0

Bu

llw

hip

fac

tor Hybrid

PolicyIB


supplier

0

2

4

6

0 1.5 3 4.5 6Ci

Bu

llw

hip

fac

tor

Hybrid

PolicyIB

24

It should be noted that, although hybrid policy and Policy IB result in the bullwhip effect and

increases the supplier’s cost, the overall supply chain’s cost may still be lower, due to the cost savings at the

retailers/warehouse.

4.4 Designing the Supply Chain Using Hybrid Policy

Our earlier numerical results suggest the hybrid policy to be employed as the coordination policy of choice in

such systems. In this section we study the design of the distribution systems that use the hybrid policy. Figure

10 shows that the average cost rate per retailer is decreasing in N. This implies that a regional warehouse,

rather than multiple local warehouses, should be used to serve a large number of retailers.

Furthermore, from Figure 11 we observe that it is economically beneficial to locate the warehouse

closer to the outside supplier. This would also lead to more bargaining power to reduce the fixed ordering cost

(K0) that will in turn decrease the cost rate at a faster rate (Figure 12). Moreover, it could be beneficial if the

management invests in increasing the capacity of its own transportation fleet, since increasing the truck

capacities (increasing inbound capacity factor from 0.5 to 1) while keeping the fixed ordering cost same would

reduce the average cost rate by almost 10%.

Figure 10 Figure 11 Figure 12

5 Conclusion

In this paper, we consider the practice of coordinated replenishment in a distribution system with multiple

retailers, a single outside supplier, and one warehouse that holds no inventory. To the best of our knowledge,

this is one of the few studies that use coordinated replenishment policies when both inventory and

transportation costs are considered in a supply chain under stochastic demand. We proposed a new policy,

called the hybrid policy, which is the mixture of a traditional echelon policy and a special type of can-order

policy. We also analyzed three other coordinated replenishment policies and compared their performance with

Avg. cost under hybrid

6

7

8

9

0 10 20 30

N


80

85

90

95

0 1 2 3

L 0 /L


60

75

90

105

0 50 100 150

K 0

25

the hybrid policy. For all policies considered, we derived expressions for the operating characteristics and the

expected total cost rate, and presented a procedure for obtaining the optimal policy parameters. To examine

the value of coordinated replenishment, we also compared the hybrid policy with the traditional distribution

system with no coordinated replenishment. Finally, we analyzed the resulting bullwhip factor in all policies

considered and observed that the two coordination policies that generally result in lower average total cost

rates result in the highest bullwhip. This raises an interesting question regarding the analysis of the entire

supply chain, which should include the supplier’s costs as well.

Our numerical results suggest that the hybrid policy provides significant improvement over the

echelon based replenishment policies, and the installation based policy (Policy IB) performs as well as the

hybrid when the transportation costs are not included. However, the marginal benefit of the hybrid policy over

Policy IB becomes significant when there are transportation penalty costs. We observed that most of the

parameters have significant effect on the relative performance of the hybrid policy over the other three

policies. However, the impact of the transit time from the supplier to the warehouse (L0) and the outbound

capacity factor (γ) on the deviations is minimal. Moreover, replenishment coordination seems to significantly

lower the system cost and the value of coordination increases in the cases of large fixed ordering/transportation

cost and truck capacities and many retailers.

There are a number of directions for future research. For instance, in this study we assumed that the

fixed shipping cost to a retailer is negligible. This is reasonable when the transportation fleet is owned by the

system such as in the case of Wal Mart. In cases when an independent carrier is used for delivery of shipments

to the locations in the supply chain, the assumption is violated as carriers usually charge on the basis of full

truck load (FTL). In such situations, a coordinated policy should incorporate a can order level not equal to the

base stock levels resulting in some retailers not to be replenished every time an order is placed to the supplier.

Another interesting extension to this research is the study of replenishment policies where inventory

can be reallocated among retailers when the inbound shipment is received at the warehouse (postponement of

allocation). Proposing/analyzing such policies that take advantage of real time information sharing would be a

significant contribution to the research in the supply chain management area.

26

Another extension to the model would be to consider the cases where the warehouse holds inventory

and the case with non-identical retailers. Finally, an interesting and practical, yet challenging extension would

be the study of coordination issues for systems with multiple products.

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Acknowledgements: The authors would like to thank the Department Editor, Paul Zipkin, the Associate editor, and the anonymous referees for their helpful suggestions. Funding for the second author was provided by Burlington Northern/Burlington Resources foundation.

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