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PRICINGANDREPLENISHMENTSTRATEGIESINADISTRIBUTION

SYSTEMWITHCOMPETINGRETAILERS

FERNANDOBERNSTEIN

TheFuquaSchoolofBusiness,DukeUniversity,Durham,NorthCarolina27708,fernando@

AWIFEDERGRUEN

GraduateSchoolofBusiness,ColumbiaUniversity,NewYork,NewYork10027,af7@

Weconsideratwo-echelondistributioandrateof

eachretailerdependsonalloftheretailers’prices,oralternatively,thepriceeachretailercanchargeforitsproductdependsonthesales

plierreplenisheshisinventorythroughorders(purchases,productionruns)fromanoutside

ere,ngcostsareincurredforallinventories,whileall

supplierordersandtransferstotheretailersincurfifirstcharacterizethesolutiontothecentralizedsystemin

whichallretailerprices,salesquantitiesandthecompletechain-widereplenishmentstrategyaredeterminedbyasingledecisionmaker,

e.g.,,thesupplierchoosesawholesalepricingscheme;theretailers

inguishsystematicallybetweenthecaseofBertrandandCournot

ormer,eachretailerindependentlychooseshisretailpriceaswellasareplenishmentstrategy;inthelatter,eachofthe

retailersselectsasalestarget,y,thesupplierrespondstotheretailers’choices

aseofCournot

competition,themechanismappliesadiscountfromabasicwholesaleprice,basedonthesumofthreediscountcomponents,whicharea

functionof(1)annualsalesvolume,(2)orderquantity,and(3)orderfrequency,respectively.

ReceivedDecember1999;revisionreceivedDecember2001;acceptedJune2002.

Subjectclassifications:Games/groupdecisions,noncooperative:priceandquantitycompetitionamongnonidenticalretailers.

Inventory/production,multi-item/echelon/stage:ory/production,policies,

marketing/pricing:coordinatingmechanismsviadiscountingschemes.

Areaofreview:Manufacturing,Service,andSupply-ChainOperations.

UCTION

Intheirattempttoimproveoroptimizeaggregateperfor-

mance,manysupplychainsincreasinglyinvestigateand

comparetheirperformanceundercentralizedanddecen-

entralizedsystem,each

chainmemberoptimizeshisownprofil-

lengethereforeconsistsofstructuringthecostsandrewards

ofallofthechainmemberssoastoaligntheirobjectives

withaggregatesupply-chain-wideprofiostand

rewardstructureisreferredtoasacoordinationmecha-

ecentralizedcostandrewardstructureresults

inchainwideprofitsthatareequaltothoseachievedunder

acentralizedsystem,thecoordinationmechanismiscalled

perfect.

Inthispaper,weaddressthesequestionsforthefollow-

ingprototypetwo-echelondistributionsystemwithcom-

ierdistributesasingleproductor

closelysubstitutableproductstomultipleretailers,whichin

tailer’ssalesoccurat

aconstantratewhichdependsonthepriceschargedbyhim

aswellasthosechargedbyallotherretailers,accordingto

agivenretailerspecifiatively,the

priceeachretailercanchargeforhisproductdependson

-

plierreplenishesherinventorythroughorders(purchases,

productionruns)fromanoutsidesourcewithamplesupply.

0030-364X/03/5103-0409$05.00

1526-5463electronicISSN

Fromthere,-

ryingcostsareincurredforallinventories,whileallsup-

plierordersandtransferstotheretailersincurfixedand

variablecosts,allwithfacility-specifi

consideroneadditionalcostcomponent:thesuppliermay

incuraspecificannualcostformanagingeachretailer’s

lthe“managementcosts”

associatedwitharetaileraccountbyaconcavefunctionof

theretailer’sannualsalesvolume,reflectingeconomiesof

stcomponenthasbeenconsideredbyChen

etal.(2001);seethereforadiscussionofhowsuchaccount

and

functionsandcostparametersarestationaryandcommon

knowledgeamongallchannelmembers.

Wefirstcharacterizethesolutiontothecentralizedsys-

teminwhichallretailerprices,salesquantities,andthe

completechainwidereplenishmentstrategyaredetermined

,ct

optimal(centralized)strategyisunknownand,inanycase,

ofsuchcomplexstructureastoprecludeitsimplementabil-

ity,evenifitcouldbecomputedinareasonableamount

ldsevenforthefarsimplercasewhereall

are,however,abletoderiveefficientlycomputablelower

andupperboundswhichareshowntobetight,when-

evertheretailers’grossprofitmargins[=(retailprice−

409

OperationsResearch©2003INFORMS

Vol.51,No.3,May–June2003,pp.409–426

410/BernsteinandFedergruen

wholesale

(say,

excessively

atleast

price)/wholesaleprice]arenotexcessivelylow

represents

large

20%),

(say,

and

less

theannual

than30%).

holding

The

costrateisnot

prices

icy

under

the

the

profit

optimal

ofa

power-of-two

strategywithstationary

lower

retailer

bound

ers.

to

when

(Under

service

apower-of-two

thecorrespondingsales

replenishment

ratesatallretail-

pol-

stant

theirinventoryisdown

policy,

tozero

all

and

facilities

eachuses

replenish

con-

power-of-two

replenishment

the

tionary

upper

multiple

intervals,

ofagiven

specified

base

as

period.)

afacility-specific

Similarly,

imum

retail

bound

prices

represents

employing

theprofit

alower

of

bound

astrategywithsta-

responding

setup

sales

andholding

at

costsincurredtoservice

forthe

the

min-

cor-

supplier

Weproceed

respond

chooses

with

the

awholesale

the

retailers.

decentralized

pricingscheme;

,the

variables.

to

of

We

this

distinguish

schemebyeachchoosingallof

the

his

retailers

policy

retailer

Bertrand

replenishment

independently

andCournot

systematicallybetweenthecase

selects

strategy;

chooses

ormer,each

inthe

his

latter,

retail

each

price

of

as

the

well

retailers

asa

ment

choices

strategy.

asalestarget,

Finally,

again

thesupplier

incombination

responds

withareplenish-

ishment

by

pricing

strategy.

implementing

Wefocus

her

initially

owncost-minimizing

totheretailers’

on

replen-

sale

Bertrand

price

schemes

foreach

where

unit

eachretailerpaysa

linear

constant

wholesale

whole-

librium

eter

may

andCournotcompetition

purchased.

that,

We

while

show,

a

under

Nashequi-

both

in

combinations,

failto

an

exist

equilibrium

undercompletely

(inpure

general

strategies)

param-

is,

retailer’s

fact,guaranteed

annual

priceelasticity

under

of

a

demand

condition

to

which

the

relatesthe

(This

isfied

condition

salesand

is,

his

again,

combinedinventoryand

ratio

setup

of

costs.

his

slightly

in

is

stronger

virtuallyallproduct

shown

categories.)

tobecomfortably

Arelated

sat-

and

librium

eed

condition

with

guarantees

acomparison

that

between

theequilibrium

theequi-

the

We

discount

case

next

under

of

derive

Bertrand

aperfect

andCournot

coordination

competition.

of

from

Cournot

abasic

competition,

wholesale

the

price,

mechanism

basedon

applies

the

a

(1)

three

frequency,

annual

discount

salesvolume,

components,

(2)order

whichareafunction

sum

of

optimal

in

centralized

respectively.

solution

Under

arises

this

quantity,and(3)order

as

discountscheme,the

which

theresulting

mal

allNashequilibria

ve

aNash

conditions

equilibrium

under

form

supply-chain-wideprofits,

inthe

thus

retailer

giving

game

rise

achieve

toastrong

opti-

competition,

ofperfect

arise

discounts

coordination.

basedon

In

the

the

annual

absence

sales

of

volume

retailer

as

in

demonstrated

onlyinthe

in

presence

Chenet

of

al.

account

(2001).

managementcosts,

required

thepresence

coordination

evenif

ofretailercompetition,

On

such

the

discounts

otherhand,

are

nale,

tionand

within

symmetric

the

mechanism

noaccount

context

thus

management

providesan

costs

economic

prevail.

ratio-

The

bargaining

ofamodel

power

with

for

complete

allretailers,

informa-

for

wholesale

sales

discount

volumes,

prices

one

tobe

ofthe

discountedonthebasisofannual

and

We

El-Ansary

,

most

Brown

prevalent

andMedoff

forms

1990,

ofprice

Stein

pricing

showthatfor

1992,

each

and

retailer

Munson

thiscoordinating

andRosenblatt

wholesale

1998).

incurred

scheme

nitude

forthis

is

retailer,

given

augmented

bytheper

by

unit

a

“indirect”costs

impact”,

of

presents

first

to

a

which

measure

increases

markup,themag-

theremainder

forthe

withtheso-called“competitive

of

degree

themarket.

ofcompetition

(Thismeasure

aretailer

was

compete

introduced

perfect

inprice

in

space

Bernstein

(i.e.,

et

face

al.2002.)Iftheretailers

more

coordinationcanbeachieved

Bertrand

withasimilar,

competition),

albeit

a

We

complex,discountscheme.

simple

decentralized

assessthe

system

value

by

of

analyzing

(perfect)coordination

settingsinwhich

within

retailers

linear

channel

andno

wholesale

othermeasures

pricing

are

schemeisofferedtothe

a

of

ket

thesystem,

members’

assuming

decisions.

either

Weanalyze

takento

the

coordinate

performance

the

or

powertospecifythelinear

that

wholesale

thesupplier

pricing

hasthe

scheme,

mar-

mize

that

chain

the

the

supply-chain-wide

constantwholesale

profits.

priceis

In

chosensoastoopti-

supplier

members

ing

as

areengagedinaStackelberg

thefirst

game

case,

with

the

the

Stackelberg

thenoncooperative

theleaderand

retailer

theretailers

gamedescribed

following

above.

byplay-

The

supply-chain-wide

solutionoftenresultsinmajorlossesinthe

on

Themarketingliterature

profits.

onchannel

a

model

simple

pricing

channel

decisions.

with

Jeuland

one

andShugan

coordination

(1983)consider

focuses

sions

authors

or

does

resulting

notconsiderany

supplier

inventory

and

replenishment

deci-

perfect

foundthat

setup

asimple

andinventory

quantitydiscount

carrying

results

costs.

in

The

a

model,

ishment

the

coordination

“quantity”

mechanism.

doesnotrefer

Because

tothe

theirsisastatic

native,

orderbuttotheannualsalesvolume.

size

As

ofa

an

replen-

alter-

setting,

Moorthy

two-part

perfectcoordination

(1987)showed

canbe

that

achieved

inthissingle-retailer

withasimple

cost

consider

plus

tariff,

afixed

i.e.,

franchise

bycharging

e

theretailerthemarginal

ing

authors

inprice

thespecial

space

caseoftwoidentical

and

retailers,

Staelin

compet-

(1983)

different

assumethat

under

thetwo

linear

retailers

procurement

aresupplied

costs.

by

These

two

grated

further

with

manufacturers

theirretailer

which

ornot.

are

See

either

Moorthy

vertically

(1987)

inte-

(1983),

Ingene

observations

authors

by

andParry(1995)

onthis

generalize

model.

for

JeulandandShugan

by

retailers.

anyconstant

show

allowing

thatperfect

for

coordination

twononidentical

cannot

retailers.

beachieved

The

by

Instead,

wholesale

theyderive

price

aperfect

which

coordination

isidenticalfor

scheme

both

the

discounting

posed

retailers’purchase

thewholesale

volumes.

price

While

asa

attractive,

linearfunctionof

than

isclearly

two

scheme

or

the

when

failswhenthenumberofretailersis

the

larger

pro-

casein

the

our

procurement

operational

costs

model

are

with

nonlinear,

inventory

as