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Production, Manufacturing and Logistics

Estimating customer service in a two-location

continuous review inventory model with

emergency transshipments

Kefeng Xu a, Philip T. Evers b,*, Michael C. Fu c

a

College of Business, University of Texas at San Antonio, San Antonio, TX 78249-0634, USAb Robert H. Smith School of Business, University of Maryland, College Park, MD 20742-1815, USAc Robert H. Smith School of Business and Institute for Systems Research, University of Maryland, College Park, MD 20742-1815, USA

Received 14 March 2000; accepted 20 December 2001

Abstract

In this paper, an approximate analytical two-location inventory transshipment model is developed that combines the

popular order-quantity, reorder-point Q;R continuous review ordering policy with a third parameter, the hold-backamount, which limits the level of outgoing transshipments. The degree to which transshipments improve both Type I

(no-stockout probability) and Type II (fill rate) customer service levels can be calculated using the model. Simulation

studies conducted to test the validity of the approximations in the analytical model indicate that it performs very wellover a wide range of inputs.

2002 Elsevier Science B.V. All rights reserved.

Keywords: Inventory; Emergency transshipments; Distribution; Logistics; Pooling; Continuous review policy

1. Introduction

Found in both military and commercial set-

tings, emergency lateral transshipments represent

one strategy for enhancing customer service whilereducing costs in a multi-location inventory sys-

tem. An organization may have many service lo-

cations spread throughout a wide geographical

region to satisfy numerous, dispersed customers;

however, one location may have unfilled demand

due to insufficient stock while another has excess

stock. Here, lateral transshipment (also referred to

as pooling, transfer, and redistribution) of product

could be used to satisfy the unanticipated demand

at one location with the surplus from another lo-cation.

Although there has been a significant amount of

work studying the effects of transshipments, most

of it provides no control over the degree of

transshipment, with the notable exceptions of the

periodic review models of Cohen et al. (1986) and

Tagaras and Cohen (1992). However, like most

transshipment models in the literature, both of

these assume a periodic review system with no

fixed ordering cost, whereas we consider a set

European Journal of Operational Research 145 (2003) 569584

www.elsevier.com/locate/dsw

* Corresponding author. Tel.: +1-301-405-7164; fax: +1-301-

405-0146.

E-mail address: [email protected] (P.T. Evers).

0377-2217/03/$ - see front matter 2002 Elsevier Science B.V. All rights reserved.

PII: S0 3 7 7 -2 2 1 7 (0 2 )0 0 1 5 8 -3
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of independently functioning locations operating

under a continuous review inventory policy that

includes a fixed setup cost for placing orders. Our

setting is motivated by the fact that many orga-nizations allow autonomy of operations at indi-

vidual locations, and these locations are likely to

use independent lot size-reorder-point ordering

policies. Modeling the degree to which this or-

dering policy can incorporate transshipments to

improve service is the major thrust of this paper.

In particular, we develop a two-location inventory

transshipment model that incorporates an inde-

pendent continuous review three-parameter in-

ventory control policy at each location: the usual

lot size-reorder-point Q;R inventory control po-licy combined with an additional transshipment

parameter H that controls the amount of on-hand inventory held back by a location making

an outgoing transshipment. An approximate ana-

lytical model is developed that can be easily solved

by numerical methods. The accuracy of the model

is tested against simulation for the case of Poisson

demands. For key physical characteristics of the

inventory system, such as the average transship-

ment and stockout amounts, analytical results for

this model are generally within 1015% of simu-

lated results for non-identical locations and within35% of simulated results for identical locations.

A brief review of the literature reveals that the

majority of reported studies examining multi-

location inventory control problems with trans-

shipments assume a periodic review policy where

at the end of each period inventories are evalua-

ted, replenishment orders generated if appropriate,

and emergency transshipments scheduled simulta-

neously at several locations if needed. These

studies usually assume no order setup cost such

that an order-up-to base-stock inventory policyis appropriate. In particular, the early pioneering

works of Gross (1963) and Krishnan and Rao

(1965) examine a periodic review policy in a single-

echelon, single-period setting. Gross (1963) con-

siders a multi-location system but assumes that

ordering and transshipment decisions are made

before the realization of demand. Das (1975) ex-

tends Gross (1963) by allowing transshipments in

the middle of each period. Assuming identical cost

parameters across locations, Krishnan and Rao

(1965) develop an N-location newsboy (i.e., order-

up-to) model that forms the basis of many later

studies. The two-location model with different cost

parameters was analyzed by Aggarwal (1967) andKochel (1975) for the single-period case. In par-

ticular, Kochel (1975) proves: (1) the expected one-

period cost function is convex with respect to the

inventory levels and thus a generalized order-up-to

policy is optimal in the multi-location model; and

(2) complete pooling is optimal under far less re-

strictive conditions on the cost parameters. These

results are further extended to the infinite horizon

case by Kochel (1982, 1988) for the discounted

criterion and the average criterion, respectively.

Notably, Tagaras (1989) explicitly generalizes the

model of Krishnan and Rao (1965) by allow-

ing different unit ordering, holding, penalty, and

transshipment costs across locations in a two-

location setting. Tagaras and Cohen (1992) further

extend this to the case of non-zero deterministic

replenishment lead time by examining, along with

complete pooling, a specific partial pooling policy

in which one location maintains a certain inven-

tory level (or inventory position) after transship-

ping to the other location. Kochel (1990)

formulates a model similar to Tagaras (1989) in a

multi-location framework, though he considers anet profit objective function and (stock) selling

decision at the beginning of a period. Using a

model similar to those of Krishnan and Rao (1965)

and Tagaras (1989), Chang and Lin (1991) exam-

ine the cost function conditions that make the

case with transshipments among locations more

desirable than the case without them. Kochel

(1998) provides a survey of multi-location inven-

tory models with lateral transshipments.

Other authors have extended multi-location

inventory models with transshipments to incor-porate multi-echelon, or multi-period, aspects.

However, most of their extensions were made

possible by making further assumptions beyond

the single-echelon, single-period case, especially

regarding the unit cost of an activity across loca-

tions. Hoadley and Heyman (1977) consider a

general case of two-echelon, multi-location model

with a one-period order-up-to replenishment pol-

icy. Their model allows for balancing acts (pur-

chases, dispositions, returns, normal shipments,

570 K. Xu et al. / European Journal of Operational Research 145 (2003) 569584


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and transshipments) at the beginning of a period

and emergency acts (expedited shipments from an

upper echelon and emergency transshipments from

the same echelon) in the case of a stockout at theend of a period. Cohen et al. (1986) extend the

approach of Hoadley and Heyman (1977) even

further to the multi-echelon case, with particular

emphasis on low demand, high cost, high service

spare parts inventory. Their model differs from

most in that transshipments are not restricted only

to the lowest echelon. Jonsson and Silver (1987)

examine a two-echelon distribution system con-

sisting of a cross-docking central warehouse sup-

plying several branch warehouses and investigate

the desirability of complete redistribution of all

branch warehouse inventories one-period before

the end of the order cycle to minimize the (ap-

proximate) total expected units backordered.

Considering inventory transshipments in a

multi-period problem results in a class of dynamic

programming models, usually as an extension of

the single-location newsboy problem, or a broad

class of simulation models. Computational meth-

ods, rather than model implications and applica-

tions, are the focus of the former category, as

illustrated in Karmarkar (1987), Robinson (1990),

Klein (1990) and Archibald et al. (1997). Canta-galli (1987), Dorairaj (1989), Evers (1996, 1997),

and Tagaras (1999) are examples in the latter

category, simulating the effects of transshipment,

demand, and lead time parameters on system

performance.

The few studies examining continuous review

inventory systems, such as Lee (1987), Axsater

(1990), Dada (1992), and Sherbrooke (1992),

mainly focus on repairable items or spare parts

inventory under one-for-one ordering policy at

each location. In our setting, the fixed setup costfor ordering along with the allowance for emer-

gency transshipments makes plausible the efficacy

of a three-parameter inventory control policy, and

thus the basis for the form of the proposed in-

ventory control policy. In addition to the usual

continuous review lot-size and reorder-point pa-

rameters, a hold-back parameter is included to

establish the amount of on-hand inventory that a

location can reserve for itself prior to making an

outgoing transshipment.

The paper is organized as follows. In Section 2,

the approximate two-location continuous re-

view Q;R model with lateral transshipments

is developed. The accuracy of the model istested against simulation in Section 3. Section 4

contains a summary of, and conclusions from, the

work.

2. The analytical model

Consider a firm with two service locations much

closer in terms of transit times to each other than

to their respective supply sources. Managers at

each location closely watch the demand behaviorand inventory level and are thus able to determine

the remaining demand level near the end of an

order cycle. The locations receiving the transfer

request grant the transshipments based on their

own reserve levels.

Each location i (i 1; 2) serves the customersof its designated territory and controls its inven-

tory level independently and continuously with an

order-quantity, reorder-point Qi;Ri policy. Oncelocation i finds its inventory position Zi with

probability density function (p.d.f.) ui below

its reorder-point Ri, it places a replenishment order

of size Qi with a supplier having ample capacity to

fulfill all replenishment requests within a deter-

ministic replenishment lead time Li. Location is

inventory position is equal to its inventory level Yi

with p.d.f. gi and cumulative distribution func-tion (c.d.f.) Gi plus any replenishment ordersin transit.

2.1. Model assumptions

Our model adopts the usual assumptions of

continuous review models (cf., Lee and Nahmias,

1993). An infinite horizon problem is considered

with stationary, but stochastic, demand specified

by demand during lead time Di with p.d.f. fiand c.d.f. Fi. Demands are independent overtime and across locations, though demands at

the two locations need not be identically distri-

buted. With RiP 0, stockouts occur only during

the reorder lead time period. At most, only one

K. Xu et al. / European Journal of Operational Research 145 (2003) 569584 571


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replenishment order is outstanding at each loca-

tion. Thus, inventory position and inventory level

are different from each other only during the re-

order period. And before the replenishment orderarrives, there is a chance that demand Di willoutstrip planned available stock Ri even if somesafety stock is maintained.

Assume location i can determine its actual de-

mand during lead time near the end of its order

cycle since most of its demand during lead time

Di has already been realized and a very shorttime period remains to be forecasted (usually a

small fraction of the replenishment lead time). This

treatment is similar to those in most studies of

periodic review inventory policies, such as Krish-

nan and Rao (1965) or Tagaras (1989), which

assume transshipments occur after demand is re-

alized but before it must be satisfied. If necessary

then, management at location i could request an

emergency transshipment during its order cycle

from location j (j 6 i). Here the order cycle refersto the period between two successive order deliv-

eries to a location. Any demand that cannot be

satisfied either from on-hand inventory or by

emergency transshipment is backordered (the in-

ventory level becomes negative with a backorder)

and satisfied when the next replenishment arrives.Further assume that emergency transshipment

times are negligible relative to regular replenish-

ment lead times and that there can be at most one

transshipment request per cycle (typical assump-

tions in most transshipment studies including

Tagaras and Cohen, 1992). The transshipment

quantity that location i receives from j depends on

location js inventory level and control policy. For

example, when location j receives a transfer re-

quest from i:

(A) location j could make available all of its on-

hand inventory;

(B) location j could hold back enough of its on-

hand inventory to cover its reorder-point

and only make available the remainder of its

on-hand inventory; or

(C) location j could hold back some amount of

on-hand inventory beyond the zero inventory

indicated in (A), but unrelated to the reorder-

point.

Policy (A) corresponds to the complete pooling

policy used by many authors in the periodic review

case (cf., Krishnan and Rao, 1965; Tagaras and

Cohen, 1992). Complete pooling is preferred inthose studies since replenishment orders are often

assumed to arrive at every location after the

maximally possible transfer amounts are used to

reduce system shortages to a minimum. However,

in a continuous review system, such a transship-

ment may result in another shortage situation

quickly arising at the supplying location.

Policy (B) tries to minimize each locations own

risk of stockout resulting from making an emer-

gency transshipment by assuring location ja certain

degree of protection from shortage after transfer-

ring out some of its inventory. Consequently,

managers at each location may favor policy (B) to

protect their own interests. Nevertheless, one

weakness of (B) is that it is conservative and po-

tentially myopic since it is possible that location j

could transfer stock to i now and then request an

emergency transfer later if necessary, hoping that

by then location i will be in a surplus situation.

While (A) and (B) represent two convenient

policy options, our model captures the range of

transshipment options implied by (C) through in-

corporation of an additional control parameter Hjthat represents the amount of on-hand inventory

held back before an outgoing transshipment is

authorized. Subject to stability considerations,

there are no theoretical limits for the choice ofHj,

though commonly HjP 0. For instance, Hj > Rjcould be used if management feels a locations

fixed order cost is too high and/or needs to reduce

the order frequency caused by transshipments

(another reason for Hj > Rj could be the existenceof a transshipment setup cost). The transshipment

quantity from location j to location i, given thatthe demand during lead time at i is x and the in-

ventory level at j is y, is defined by:

Xjix;y minfx Ri; y Hj

g: 1

Thus, a transshipment only takes place if the re-

alized demand during lead time exceeds the reor-

der-point at the receiving location and there is

excess inventory above the held-back point avail-

able at the sending location. The notation Xji XjiDi; Yj is used to define the corresponding



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random variable. Using the hold-back quantity

H to control transshipments by each location isappealing here because of its simplicity and ease of

implementation. This reserve parameter controlsthe window of opportunities to transship and can

be determined a priori.

As pointed out earlier, the only papers identi-

fied as providing some control over the hold-back

quantity are those periodic review models of

Cohen et al. (1986) and Tagaras and Cohen

(1992). Cohen et al. use a single sharing rate for

all locations in total to control the fraction of

excess supply at surplus locations used to satisfy

excess demand elsewhere. Our design is more

flexible as each location sets its own hold-back

amount. And, while similar in essence to ours, the

model of Tagaras and Cohen is more complicated

due to an additional control parameter: a thresh-

old inventory level that a location would like to

reach but not exceed after receipt of the trans-

shipment.

In order to obtain an analytically tractable

model for the performance of such a trans-

shipment system, we require the following as-

sumptions, which are exact for systems without

transshipments in general settings:

A1 ujz 1=Qj for Rj < z6Rj Qj

0 otherwise

2

for all j:

A2 Di and Yj are independent for all i 6 j:

A3 Yjt Lj Zjt Dj for all j: 3

(A1) means the steady-state inventory position

Zj, which we will assume to exist (true for appro-

priate values of Q;R;H), is uniformly distributedon Rj;Rj Qj and is independent of the lead time

demand Dj. This assumption is used by Zheng(1992) in a general treatment of the Q;R policy.According to Browne and Zipkin (1991), in the

case of continuous demand (A1) holds when the

cumulative demand can be modeled by a non-

decreasing stochastic process with stationary in-

crements and continuous sample paths. When the

demand is discrete and unitized and forms a sta-

tionary point process, these conditions are justified

if demand can be modeled as a Poisson process

(Hadley and Whitin, 1963). In this case, the re-

plenishment order is placed as soon as the inven-

tory position hits Rj exactly.

(A2) says that the demand during lead time at

one location is independent of the on-hand in-ventory level at the other. Since Di and Yi are

perfectly correlated during the reorder period, this

implies that Yi and Yj are independent at the end of

each locations order cycle.

(A3) is a well-known fundamental relationship

for an ordinary single-location facility relating the

inventory level at time t Lj, the inventory po-sition at time t, and the demand during a fixed lead

time (cf., Zipkin, 1986), stating that everything on

order at time t will have arrived in the system by

time t Lj, whereas new orders placed after t willnot have arrived by time t Lj. Thus, the on-hand inventory level Yjt Lj contains on-handinventory at time t and all orders in-transit on or

before time t but is reduced by the demand during

the lead time. In the stationary case t! 1, thetime dependence can be dropped, so Yj Zj Dj.Thus the c.d.f. for Yj is given by

Gjy PYj6y

ZRjQj

RjZ

1

zy

ujzfjx dxdz

ZRjQjRj

ujz1 Fjz y dz: 4

Once transshipments are introduced, these as-

sumptions do not hold precisely. For instance,

based on (A1) and (A3) it is assumed that the

inventory position at a location is affected by its

own demand process but not by the transshipment

process, even though transfers into and out of a

location actually do affect the on-hand inventorylevel and, consequently, the inventory position.

Likewise, (A2) could be violated if a large demand

Di at one location results in a large inbound

transshipment Xji which, in turn, decreases the

inventory level Yj at the other location. Violations

of these assumptions, however, pose few problems

as simulation experiments reported later indi-

cate that the quality of the resulting analytical

performance measures is quite good in many set-

tings.



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2.2. Model development

Beginning with the expected amount of trans-

shipment across locations during a typical ordercycle, given by EXji in (1), various modelperformance measures can be derived. Xji depends

on Di and Yj, which under (A2) can be treated as

independent. While the explicit derivation will

only be shown for the case where the demand

process and all decision parameters Q;R;H arecontinuous, the discrete case can be obtained in a

similar, straightforward manner. Differentiating

and using assumption (A1) for the p.d.f. ofZj, the

corresponding p.d.f. is

gjy G0jy ZRjQjRj

ujzfjz y dz

FjRj Qj y FjRj y=Qj

FjRj Qj y FjRj y=Qjif y6Rj;

FjRj Qj y=Qjif yPRj:

8>>>: 5

Note Fj0 0 under non-negative, continuousdemand as assumed in our case here.

Integrating over the joint density, the expected

amount of transshipment Xji during a typical cycleof location i is calculated as

EXji

Z1Ri

ZRjQjHj

Xjix;ygjyfix dydx

ZRiRjQjHjRi

ZRjQjHjxRi

x Ri

gjyfix dydx

ZRjQj

Hj Z1

RiyHj

y Hjgjyfix dxdy

ZRiRjQjHjRi

1 Fix

1 GjHj x Ri dx; 6

where (1) is used in the second equality to first di-

vide the region of integration into two parts. Then,

a change in order of integration plus some alge-

braic manipulations leads to the last line of (6)

for details, see Xu (1997). Note that GjRj Qj 1 (i.e., the inventory level does not exceed

Rj Qj. EXji depends on the relative magnitudesofHj and Rj. To demonstrate the flexibility of our

analytical model to capture a wide range of sce-

narios, we evaluate both Hj6Rj and HjPRj. IfHjPRj:

EXji 1

Qj

ZRjQjHj

FjRj Qj y

ZRiyHjRi

1 Fix dxdy: 7

Otherwise Hj6Rj:

EXji 1

Qj

ZRjQjHj

FjRj

" Qj y

ZRiyHjRi

1 Fix dxdy

ZRjHj

FjRj y

ZRiyHjRi

1 Fix dxdy

#: 8

In the case ofHj6Rj (and especially in the ex-

treme case ofQj Hj < Rj), it is conceivable that(8) is merely approximate since the inventory po-

sition after an outbound transshipment could drop

so low that a second replenishment order would be

needed. If so, then the conventional Q;R modelassumption of, at most, only one outstanding re-

plenishment is violated and model accuracy could

deteriorate. This issue is investigated in detail later.

The partial derivatives (shown in Appendix A)

have clear implications as to the effects of changing

decision parameters on expected transshipments. A

higher reorder-point at location i makes trans-

shipments less likely since a higher safety stock is

maintained by i. The expected average transship-

ment from jto iis not affected by location is order

size since i needs an emergency transshipmentduring its reorder period, nor is it related to loca-

tion is own held-back amount since the latter only

affects transfers to location j. On the other hand, a

higher reorder-point and order-quantity at loca-

tion jlead to higher on-hand inventory at location j

and thus increase the expected average transship-

ment quantity from j to i. Finally, a higher Hj re-

sults in location j curbing the outflow transfer to

location i. Later, these observations will be tested

using simulation.



7/16

The second important performance measure

considered is the expected amount of stockout

during a cycle. In the typical independent Qi;Ri

inventory system, the expected amount of stockoutper order cycle of i is given by:

niRi EDi Ri

Z10

x Rifix dx

Z1Ri

1 Fix dx: 9

Allowing for transfers between two locations at

the same echelon, the inventory availability at lo-

cation i during its reorder lead time is no longer

Ri, but Ri Xji. According to the transshipmentpolicy defined by (1), the transshipment quantity

Xji is always less than or equal to the requestedamount Di Ri. Therefore, the expected amountof stockout at location i per order cycle with

transshipments is:

EDi Ri Xji

Z10

ZRjQj1

x Ri Xjix;ygjyfix dydx

Z1Ri

ZRjQj1

x Ri Xjix;ygjyfix dydx

niRi EXji: 10

It is clear from (10) that, compared with the ex-

pected stockout in an independent system, the

expected stockout in the transshipment system is

reduced by the expected amount of transshipment

into the location per cycle.

The partial derivatives of EDi Ri Xji

with respect to the control parameters (shown in

Appendix A) indicate that by raising its own re-

order-point or the other locations reorder-point

level or replenishment order size, or by reducing

the others held-back amount, a location could

reduce its own expected stockout amount. For the

same reasons that EXji is not related to Qi andHi;EDi Ri Xji

is not related to Qi and Hi.One of most important reasons for holding in-

ventory is to ensure customer service. Here, the

levels for a given set of (not necessarily optimal)

operating parameter values Qi;Ri;Hi with andwithout emergency transshipments are computed.

Following Tagaras (1989), Type I and Type II

service measures are defined as follows:

PBPi no-stockout probability at location i beforepoolingFiRi,PAPi no-stockout probability at location i after

pooling,b

BPi demand fill rate at location i before pool-

ing 1 niRi=Qi,b

APi demand fill rate at location i after pooling.

When measuring service levels, consider the

original transshipment Xji defined in (1) as the

stochastic addition to the total inventory avail-

ability Ri during an order cycle at i. That is, withtransshipments a stockout occurs ifDi > Ri Xji:

PAP

i

PDi6Ri Xji

1

Z1Ri

ZHj1

gjyfix dydx

(

ZRjQjHj

Z1RiyHj

gjyfix dxdy

)

1

Z1Ri

fix dx

ZRiRjQjHjRi

1 GjHj x Rifix dx

PBPi

ZRiRjQjHjRi

1 GjHj x Rifix dx:

11

Thus PAPi > PBPi for Hj < Rj Qj, according to

(11). A convenient way of writing (11) is

PAPi PBPi

oEXji

oRi

whereoEXji

oRi

6 0 from 15 in Appendix A

:

12

Eq. (11) indicates that emergency transshipments

do indeed improve each locations no-stockout

probability, though only complete satisfaction of

the transfer request (i.e., full transshipment) can

prevent any stockout. Thus, the increase in the no-

stockout probability reflects the likelihood that the

location requested to make a transshipment has

enough extra stock to fulfill any emergency needs

at the requesting location under all scenarios of

demand during the replenishment lead time.



8/16

Defined as the ratio of satisfied to total cus-

tomer demand at a location (where a locations

total customer demand during its cycle is Qi), the

fill rate in the independent case is given by thefollowing:

bBPi 1 niRi=Qi

1

Z10

x Rifix dx=Qi

1 EDi Ri=Qi: 13

During an average cycle, a location i is ex-pected to receive a replenishment order of size Qifrom its supplier and an inbound transshipment

amount of EXji from the other location j to

fulfill its demand, but would also transfer out-bound amounts ofEXijTi=Tj to the other loca-tion. Here Ti and Tj are the average order cycle

lengths of locations i and j, respectively. Note that

outbound transshipment amounts should not be

counted as cycle demand at location i since they

are not direct customer demands from location i.

Therefore, the expected value of cycle demand can

be approximated as:

Ecycle customer demand at i

Qi EXji EXijTi=Tj:This expression implies that outbound transship-

ments shorten the cycle length in the same way

that inbound transshipments extend it. This is

confirmed by our simulation studies later. The

shorter the cycle length Tj is relative to Ti, the more

frequent the transshipment request is from j. Since

both EXji and EXij are small relative to Qi andtend to cancel each other out, partially if not

completely, we can approximate the cycle cus-

tomer demand as Qi. The fill rate with transship-

ments is approximately:b

APi 1 EDi Ri Xji

=Qi EXji EXijTi=Tj

% 1 niRi EXji=Qi

bBPi EXji=Qi: 14

The results in (12) and (14) resemble those of

Tagaras (1989), who derived the partial derivatives

of expected transshipments with respect to the

base stock in a two-location (periodic review)

newsboy model.

In general then, emergency transshipments

represent an effective approach for improvingservice levels as measured by no-stockout proba-

bilities and demand fill rates. This is achieved for

any level of the operating parameters Qi;Ri;Hi aslong as Hi is specified in a way that allows trans-

shipments. Under the no-transshipment system,

increasing service levels requires raising Ri, or

more precisely the safety stock, thus resulting in

higher total costs. However, as long as the savings

in shortage and holding costs outweigh the costs of

making emergency transshipments, an inventory

system with transshipments should result in lower

total costs than a system without transshipments

having the same operating parameters Qi and Ri.

3. Model validation

To examine the accuracy of the analytical

model, simulation is used. In contrast with the

analytical model (where only the distribution of

demand over lead time is needed), complete spec-

ification of the demand process is required for

simulation. In order to make the simulation spec-ification readily correspond to the aggregate in-

formation required for the analytical model, we

consider a Poisson demand process, which leads to

a Poisson distribution of demand during lead time

(average demand during lead time is denoted as

DDLT). Here, replenishment lead time is equal to

4 days. Three levels of demand inter-arrival time

(0.3333, 0.1667, and 0.1 days to represent low,

medium, and high demand, respectively) are sim-

ulated, resulting in three levels of DDLT: low of

12; medium of 24; and high of 40. Additional ex-periments having even lower or higher DDLT are

also conducted but not reported here, since com-

parable results are obtained. Shown in Table 1, 15

different levels of the inventory control parameters

Q;R;H are considered as well. A total of 270runs (3 demand levels, 3 control parameters, 15

control parameter levels, and identical and non-

identical locations), each of 1,000,000 days in

simulation run length, are tested. In the case of

identical locations, each location has the same



9/16

control parameters and average demand level. In

the non-identical case, the locations have the same

average demand level, but one different control

parameter (simulations not presented here verify

that transshipment behavior is similar even when

demands are different for two locations). A 1000-

day period is taken as one batch, leading to 1000

batches for each run-example. Batch means are

computed for the performance measures of inter-

est, resulting in a sufficiently high degree of pre-

cision in the estimates (standard error to mean

ratios of under 1% in almost all cases), so the

confidence intervals are neither reported nor dis-

played in the results here.

With respect to the validation, one parameter at

a single-location (either Qi; Ri, or Hi) is varied andplotted while all other parameters at both loca-

tions are fixed. The fixed Q;R values are obtained

through independent (no-transshipment) optimalconditions (cf., Nahmias, 1989) and by setting

H R. In order to consider a reasonable range ofthe parameters, the following costs are used: unit

holding cost $10 per unit per year, penaltycost $1 per unit, and order setup cost $5 perorder. Thus, without transshipments, the optimal

Q;R values are 36; 14, 51; 27, and 65; 46 forlow, medium, and high demand levels, respec-

tively.

3.1. Results

The performance measures compared are the

expected amounts of transshipments and stockout

per order cycle. The complete set of results is

contained in Xu (1997), the vast majority of which

strongly support the model. Due to space limita-

tions, we present in figures here only a small subset

of cases, concentrating mainly on those where the

discrepancies between the analytical model and the

simulation results are most pronounced. A de-

scription of the particular experiment setting ac-

companies each figure.

According to the analytical model properties

(A.2) and (A.4), from the perspective of location

i, increasing order size Qi should not have any

effect on inbound transshipments EXji but shouldincrease outbound transshipments EXij. From

Fig. 1. Non-identical locations; low DDLT (Ri Rj Hi

Hj 14; Qj 36). (a) Comparison of transshipment quantitiesacross order quantities. (b) Comparison of stockout amounts

across order quantities.

Table 1

Inventory control parameters

DDLT Q R H

Low (12) 16; 18; 20;. . .

; 44 12; 14; 16;. . .

; 40 0; 2; 4;. . .

; 28Medium (24) 30; 32; 34; . . . ; 58 24; 26; 28; . . . ; 52 0; 3; 6; . . . ; 42

High (40) 50; 54; 58; . . . ; 106 36; 38; 40; . . . ; 64 0; 5; 10; . . . ; 70



10/16

Fig. 1(a), such predictions are generally confirmed

as the simulated average transshipment per order

cycle conforms to these general trends. In partic-

ular, the effects of Qi on Xij are relatively wellpredicted under all three demand scenarios.

However, according to the simulation results,

changing ones own order size does have some

minor impact on Xji, especially with low or me-

dium demand and a low Qi.

Similar in degree to that of expected trans-

shipments, the expected amount of stockout also

conforms to its simulation counterpart. Such con-

formance is better under lower Qi, as illustrated in

Fig. 1(b), though it is unclear why some deviation

(maximum 1020%) occurs in the mid-range ofQiunder the low to medium demand scenarios. Ac-

cording to the figure, the theoretical properties

(A.7) and (A.9), predicting that location is

stockout amount is not affected by its own order

size but decreases with the other locations rising

order size, are generally confirmed.

Unlike Fig. 1, which is based on low demand,

the analytically determined amounts of transship-

ments and stockouts strongly conform to the sim-

ulation averages in the case of high demand (see

Xu, 1997 for details). This suggests that as demand

discreteness and the coefficient of variation declinedue to higher demand, the analytical model better

predicts the simulation results.

Recall (A.1) and (A.3) raising ones own re-

order-point Ri reduces inbound transshipments

EXji, whereas raising the other locations reor-der-point Rj increases EXji. Such properties areclearly confirmed by the simulation results in Fig.

2(a). Moreover, the theoretical curves ofXji and Xijversus Ri match their simulation counterparts very

well, generally with less than 23% discrepancy.

An exception arises when Ri is less than or equal toits DDLTi (where roughly a 1020% discrepancy

between simulation and analytical results arise for

Xji), but since Ri < DDLTi usually implies stock-outs of 50% or more, it can be argued that such

a low reorder-point is not reasonable. Likewise,

the expected stockout amount at the receiving lo-

cation is closely matched by the simulation re-

sults (see Fig. 2(b)). For reasonable ranges of Ri(PDDLTi), the predicted stockout level at each

location is, in general, within less than 35% of the

simulated level. The figure provides evidence that

the model is well-behaved in relation to one loca-

tions reorder-point.

In Fig. 3, the effects of one locations trans-shipment control parameter, the held-back point

Hi, is shown. In all demand scenarios, the simu-

lated average outbound transshipment and stock-

out amounts as a function of Hi are extremely

accurately predicted by the analytical model (less

than 2% discrepancy). However, the model pre-

diction that inbound transshipments would not be

affected by the locations own held-back point is

not supported near the lower limit ofHi (especially

when close to 0) where a gap of roughly 3040% is

Fig. 2. Non-identical locations; medium DDLT (Qi Qj 51;Rj Hi Hj 27). (a) Comparison of transshipment quanti-ties across reorder points. (b) Comparison of stockout amounts

across reorder points.



11/16

present. We recognize that when Hi < Ri there is achance (though usually not large) that outbound

transshipments diminish the protection provided

by Ri during the replenishment lead time. This is

not accounted for in our model, making it only

approximately true when Hi approaches 0. Again,the relative extent of discrepancy is less severe in

the high demand scenarios.

All stockout quantities at both locations in re-

lation to ones held-back point are reasonably

predicted by the model in all demand scenarios.

Model property (A.7) predicts that the stockout

amount is unchanged with Hi but is only margin-

ally supported (though the actual magnitude of

change is not large). For instance, in the worst

scenario where demand is high, simulated stockout

amounts increase less than 20% when Hi decreases

from 40 ( DDLTi) to 0 (the lower limit).To summarize, under two non-identical loca-

tions most of the theoretical properties are con-firmed and the levels of physical quantities

approximately predicted, though the errors tend to

be larger when one of the inventory control pa-

rameters (Qi; Ri, or Hi) approaches its lower limit.A number of possible sources for the discrepancies

between the analytical and simulation models may

exist. These are briefly examined below.

3.2. Possible causes of discrepancies and examina-

tion of model assumptions

In the analytical model, the expected inbound

transshipment amount was shown in (A.2) not to

vary with Qi. However, when Qi is small relative to

DDLTi but the random realization of demand

during lead time at location i far exceeds DDLTi,

it is more likely that an inbound transshipment

request will be made in the next cycle since an

arriving replenishment Qi may not be enough to

bring the inventory level above Ri. The analytical

model assumes that location i will have inventory

position Ri to properly handle the random real-

ization of demand during lead time at the time ofreordering. Since the simulation is not limited by

this restrictive assumption, the actual inventory

position at the time of reordering could be lower

than Ri. In this case, the analytical model fails to

identify the higher need for inbound transship-

ments, which is captured by the simulation model.

Thus, a very low Qi relative to DDLTi represents

one instance where the analytical model cannot be

very accurately applied.

When Ri is less than DDLTi, there is a sig-

nificant chance of stockout and/or a significantamount of stockout, leading to relatively large

inbound transshipments and making the assump-

tion of uniform inventory position at the other

location tenuous. Thus, when Ri is low, the ana-

lytical model is less applicable because of its

assumptions. In this situation, the accuracy of

predictions for inbound transshipments decreases,

though estimates of other performance character-

istics from the analytical model tend to maintain

their validity.

Fig. 3. Non-identical locations; low DDLT (Qi Qj 36; Ri Rj Hj 14). (a) Comparison of transshipment quantitiesacross held-back points. (b) Comparison of stockout amounts

across held-back points.



12/16

Low levels of Hi (less than DDLTi and ap-

proaching 0) reflect another area where some of

the critical assumptions in the model only partially

hold. For instance, it is assumed in (A1) that theinventory position at a location is affected by its

own demand process but not by the transshipment

process. This is fairly acceptable if the extent of

transshipment is not large, such as when Hi is close

to or above DDLTi. However, an extremely low Hiinduces high outbound transshipments, resulting

in pronounced consumption of is on-hand inven-

tory, even during its own replenishment period.

Thus, even larger inbound transshipments are in-

evitably needed to avoid potential stockouts.

Since the analytical model assumes only one

outstanding order and, implicitly, the existence of

a steady-state, long-run stability could be a con-

cern when Hi Qi < Ri and the inventory of lo-cation i drops to Hi after a transshipment. If a

steady-state does not exist, then there will likely be

some discrepancies between the analytical results

and the simulation results. This is addressed by

prohibiting more than one outstanding replenish-

ment in the simulation model, too. In the experi-

ments examined here, stability was never an issue;

however, there is at least some chance that this

could be a problem when Hi Qi is pushed to thelower extreme.

Another source of discrepancy is the actual re-

alization ofGi (the distribution ofYi). To obtain(5), Eq. (3) was used, which is exactly true in the

independent (no-transshipment) case but only ap-

proximately so when transshipments are allowed

since transshipments into and out of a location

also affect on-hand inventory. Unfortunately, an

exact analysis is difficult to perform. Suffice it to

say, however, that a low Qi, Ri, or Hi intensifies the

disparity and that, if two identical locations en-gaging in transshipments are considered, the effects

of transshipments into and out of a location

should largely cancel each other out, resulting in

more precise model predictions relative to those in

the two-non-identical location case.

Other possible causes are violations of critical

assumptions of the analytical model. One as-

sumption is that the inventory position at a

location is uniformly distributed. Again, trans-

shipments into and out of a location distort this.

However, after experimenting with a reasonable

range of Q;R;H values, such distortion from theuniform distribution was found not to be excessive

for modest transshipment amounts and frequency,at least in comparison with the inventory distri-

bution of a typical, single-location Q;R model. Asecond assumption is that the random realization

of demand during lead time at location i and the

on-hand inventory level Yj at location j areindependent. If instead they are positively corre-

lated, then there will be more actual transship-

ments than when the two are independent.

However, experiments indicate that the correla-

tion is generally less than 0.1. A final assumption is

that a location has no more than one outstanding

order at any one time. This is violated only occa-

sionally in most experiments, less than 0.2% of

the cycles may need more than one order, though a

second order before the arrival of the outstanding

one is not allowed in our simulation. Thus it ap-

pears that breaches of the underlying assumptions

are not a significant factor (for details, see Xu,

1997).

3.3. The case of identical locations

While examining discrepancies, we hypothe-sized that the analytical model would match the

simulation results more closely if the locations

were identical. This notion is examined below with

a number of experiments under all demand sce-

narios. Since low demand usually resulted in more

inaccuracies in the case of non-identical locations,

only results from low demand scenarios are pre-

sented in the discussion of identical locations.

Since the locations are identical, only one lo-

cations results need to be graphed. Along the X-

axis of the identical location graphs, the sameparameters at both sites are changed simulta-

neously, in order to maintain the definition of

identical locations. Thus, application of equations

such as (A.1)(A.5) requires the use of the chain

rule. For instance, the effect of varying Qi and Qjsimultaneously in the same magnitude on is in-

bound transshipments would be equal to the sum

of their individual effects, represented by (A.2) and

(A.4), respectively. (The end result is that in-

creasing the order size of both locations will raise



13/16

each locations inbound transshipments while re-

ducing stockout amounts.)

Comparing Fig. 4(a) with Fig. 1(a) shows that

the analytical model predictions under changing

order sizes (Qi and Qj) are considerably more ac-

curate in the case of identical locations. The dis-

crepancy between the analytical and simulation

results is barely 23%, for both transshipments percycle (Fig. 4(a)) and stockouts per cycle (not

shown). Similarly, when the order sizes and held-

back points are fixed at both locations while the

reorder points at both locations are altered, as in

Fig. 4(b), the analytical model follows the simu-

lation fairly precisely, except at the point where

Ri DDLTi 12. The possible cause of suchdiscrepancy has been previously explained. Also in

the case of identical locations, the differences be-

tween the analytical and simulation results under

changing held-back points at both locations are

greatly reduced (compare Fig. 4(c) with Fig. 3(a)).

Moreover, reasonable approximations (less than

15% discrepancy) of inbound transshipments per

cycle have been obtained for the lower limit ofHi(i.e., larger transshipments), which are hardly

achievable in the cases of non-identical locations.

The plots for two identical locations confirm the

hypothesis that identical locations lead to signifi-

cantly better model accuracy. In particular, for

some of the most problematic regions ofQi and Hi,

the maximum discrepancies between the analytical

and simulation results dropped from 20% (withrespect to low Qi) and 40% (with respect to low Hi)

in the case of non-identical locations to 3% and

15% in the case of identical locations, respectively.

4. Conclusions and further research

This research presents an approximate analyti-

cal model of a two-location inventory system with

emergency transshipments. It adds to the existing

literature in several ways. First, it includes acommonly used continuous review order-quantity,

reorder-point Q;R inventory policy at individuallocations, whereas past studies focused on a peri-

odic review inventory control policy or a one-

forone ordering policy in the case of continuous

review. Second, it incorporates a hold-back quan-

tity H, which allows each location to control its

extent of inventory sharing. A Q;R;H policy ispotentially attractive because it is a direct ex-

tension of the Q;R policy often employed at

Fig. 4. Identical locations; low DDLT. Comparison of trans-

shipment quantities across: (a) order sizes (Ri Rj Hi

Hj 14); (b) reorder points (Qi Qj 36; Hi Hj 14); (c)held-back points (Qi Qj 36, Ri Rj 14).



14/16

autonomous locations. Third, many interesting

properties of lateral transshipments can be inves-

tigated easily using the analytical model. For

example, transshipments are shown to alwaysimprove every locations customer service; fur-

thermore, given a cost structure on performance

measures, it would be easy to identify the appro-

priate inventory control parameters to adjust and

straightforward to predict their costs.

Since analytical tractability of the model re-

quired various approximations, simulation was

used to test the accuracy of the model. From these

results, the analytical model appears to work es-

pecially well (1) with medium to high demand

ranges, (2) with medium to high levels of each in-

ventory control parameter (relative to expected

demand during lead time), or (3) when all loca-

tions are identical.

Immediate future research in this area should

focus on utilizing the current model to explore

other issues not considered in this research. For

instance, how well does this model predict system

performance if some knowledge about the distri-

bution of demand during lead time exists but in-

formation regarding the details is unavailable?

More specifically, suppose that the only informa-

tion available for the demand process are the meanand standard deviation of demand during lead

time; will its approximation with, say, a normal

distribution be sufficient to analyze such a trans-

shipment system? The sensitivity of the perfor-

mance estimates to the actual form of the demand

process could then be evaluated. It would also be

desirable to collect the necessary demand and cost

information from a firm and conduct an empirical

analysis on the applicability of the model.

In this paper, a fixed hold-back amount Hj was

considered. Whether a constant Hj or a dynamicHj (one that varies during a cycle) should be em-

ployed raises some interesting issues. Ideally, a

dynamic Hj should provide more control over in-

ventory and allow managers to make better-

informed decisions regarding the appropriateness

of transshipments and their timing in cases of long

order cycle times. A flexible hold-back parameter,

however, is significantly more complex to imple-

ment and difficult to analyze mathematically (for

example, the timing within a cycle for the location

receiving the transshipment request is now sto-

chastic). Nevertheless, a dynamic Hj presents nu-

merous issues for future exploration.

Relaxation of some of the major assumptions ofthe analytical model, such as stochastic or rela-

tively long transshipment lead time, imperfect

information regarding demand during transship-

ment lead time, or more than one inbound trans-

shipment per order cycle, may make exact

mathematical analysis difficult. However, it would

be useful to investigate how well the analytical

model and its modification can approximate more

complex situations. In addition, the analytical

model could be extended to incorporate more than

two locations.

Appendix A

The partial derivatives ofEXji with respect tothe control parameters are as follows:

oEXji

oRi 1

ZRiRjQjHjRi

fix

1 GjHj x Ri dx

60; A:1

oEXji

oQi

oEXji

oHi 0; A:2

oEXji

oRj

ZRiRjQjHjRi

1 FixgjHj x Ri dx

1 FiRi1 GjHj oEXji

oRi

P0; A:3

oEXjioQj

Z

RiRjQjHj

Ri

fFjRj Qj Hj x

Ri 1=Qj GjHj x Ri=Qjg

1 Fix dx

ZRiRjQjHjRi

FjRj Qj Hj x Ri

1 Fix=Qj dx EXji=Qj

P 0; A:4



15/16

oEXji

oHj

ZRjQjHj

1 FiRi y Hjgjy dy

oEXji

oRj

6 0: A:5

The partial derivatives of EDi Ri Xji with

respect to the control parameters are as follows:

oEDi Ri Xji

oRi

1 FiRi

ZRiRjQjHjRi

fix

1 GjHj x Ri dx

6 0; A:6

oEDi Ri Xji

oQi

oEDi Ri Xji

oHi 0;

A:7

oEDi Ri Xji

oRj

1

ZRiRjQjHjRi

1 FixgjHj x Ridx

60; A:8

oEDi Ri Xji

oQj

1

ZRiRjQjHjRi

fFjRj Qj Hj x Ri

1=Qj GjHj x Ri=Qjg1 Fix dx60; A:9

oEDi Ri Xji

oHj

ZRjQjHj

1 FiRi y Hjgjy dy

P 0: A:10

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