Research Article Store-Assistance Management for a Supply Chain with Consumer Return under ...

13 Pages·2017·2.14 MB·English

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Mathematical Problems in Engineering
Volume 2017, Article ID 4937217, 12 pages
https://doi.org/10.1155/2017/4937217
Research Article
Store-Assistance Management for a Supply Chain with
Consumer Return under Consignment Contract
ZhihuiWu,1,2DongyanChen,1,2andHuiYu1
1SchoolofManagement,HarbinUniversityofScienceandTechnology,Harbin150080,China
2SchoolofAppliedScience,HarbinUniversityofScienceandTechnology,Harbin150080,China
CorrespondenceshouldbeaddressedtoDongyanChen;dychen [e-mail]
Received3July2016;Revised10December2016;Accepted20December2016;Published19January2017
AcademicEditor:MohammadD.Aliyu
Copyright©2017ZhihuiWuetal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,
whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
Inthispaper,thestrategyproblemsofpricingandstore-assistanceserviceinvestmentareinvestigatedforasupplychainwith
consumerreturn,wheretheconsideredsupplychainunderconsignmentcontractconsistsofasinglemanufacturerandasingle
retailer.Firstly,weusedifferentialequationtomodeltheevolutionofstore-assistanceservicelevelanddepicttheeffectofstore-
assistanceservicelevelonthereturnrate.Byapplyingtwo-stagegame,bothoptimalpricingandstore-assistanceservicestrategies
areobtainedbasedonthepresentedoptimalcontrolmodel.Subsequently,thecommitteddynamicconsignmentpricecontractis
designedtocoordinateandimprovetheperformanceofsupplychain.Finally,anumericalexampleisprovidedtoillustratethe
impactsoftheeffectivenessofstore-assistanceservicelevelanddecayrateonthefeasibleregionofcorrespondingcontract.
1.Introduction withanintentiontoreducethereturnrate.Forexample,Best
Buycompany,astheworld’slargesthouseholdapplianceand
Theconsignmentcontractplaysacentralroleinimproving electronicsretailer,haslaunchedaservicecalledGeekSquad
the performance of supply chain [1, 2]. As a result, this which can reduce the return rate effectively and ensure a
contract has attracted considerable attention and is widely 20% rise of its earnings. Therefore, the aim of this paper is
applied in the large Internet market, such as Amazon, tofocusonthepricingandstore-assistanceservicestrategy
Alibaba,andTesco[3,4].Infact,theonlinemarketplacedoes problems for the operation management of supply chain
notprovidetheauthenticdescriptionofproductdirectlyand underconsignmentcontract.
experiences the products as brick and mortar stores. When Sinceconsumerreturnasanimportantfactoraffectsthe
actual products do not match with the expected value, the operationdecisionofsupplychain,itbecomesanimportant
consumer will return these products back to the merchant. topic in theoretical research of the effect of return policy
Fromthemerchant’sperspective,afull-refundreturnpolicy onsupplychaindecisionandthecoordinateissueofsupply
is commonly provided to the end consumer in order to chain [14–18]. To be specific, when the retail price is an
improve the market competitiveness of products. Note that endogenously given parameter, the effort has been made to
agenerousreturnpolicywouldincreasethemarketcompeti- investigate how to reduce the negative effect of consumer
tivenessbutitwillinevitablybringtheextrareturnedcosts.As return in [19]. Subsequently, by distinguishing the unsold
discussedin[5],itisnotsurprisingthatthereexistdifferences and returned products, a new buy-back contract has been
of the return rate in various industries. For instance, the designed in [20] to coordinate the supply chain under
fashion industry has accounted for approximately 35–40%. stochastic circumstance. In [21], by capturing the valuation
Accordingly, most industries promise a false failure return uncertaintyofconsumers,theimpactofthereturnpolicyhas
policytograbthemarketshare.Whenhandlingthefalsefail- beendiscussedontheperformanceofsupplychainandthree
urereturn,thestore-assistanceservicestrategywasprovided contractshavebeenproposedtocoordinatethesupplychain
by merchants to help consumers know the products better basedonthepresentedstrategies.Inaddition,theproblems
2 MathematicalProblemsinEngineering
Table1:Acomparisonofthepresentworkwithrelatedpreviousworks.
References Consumerreturnsreduction Consignmentcontract Static Dynamic Coordination
[6] Y N Y N N
[7] Y N Y N N
[8] Y N Y N N
[9] Y N N Y N
[10] Y N Y N N
[11] Y N Y N Y
[12] Y N Y N N
[13] N Y N N N
Ourpaper Y Y N Y Y
oforderandcoordinationhavebeenstudiedin[22]fortwo- introduced in [13] and the impact from customer return
stagesupplychainbytakingthepositiveimpactandnegative policyontoconsignmentcontracthasbeendiscussed,where
impactofconsumerreturnpolicyintoaccount. itcanbeseenthatcustomerreturnpolicydependsonsalvage
On the other hand, the store-assistance service can value of the product. Moreover, it has been revealed in [13]
effectively reduce the return rate of products but pay for that the vendor-managed consignment inventory is helpful
thecorrespondingservicecost.Hence,themerchantshould fortheperformanceofsupplychainwhetherthereturnpolicy
strikeabalancebetweencostandrevenue[12].Tomention isprovidedornot.Fortheissueofcontractcoordination,the
justafew,theequilibriumservicelevelandpricehavebeen methodhasbeenprovidedin[36]onhowtousethepayment
proposedin[8]fortwocompetitivefirms.Theinformation contracttocoordinatethesupplychainwhentheretailprice
revelation mechanism has been discussed in [10], where affects demands. Furthermore, the impact from shelf space
the store-assistance has been provided to reduce consumer onto demand has been studied in [37] and two-part tariff
return rate and increase demand. In [11], the return policy contracthasbeendesignedtocoordinatesupplychainsystem
ofthevendorandthestore-assistancepolicyofretailerhave underconsignmentcontractwithrevenuesharingcontract,
been discussed under vendor-managed inventory circum- whilethecustomerreturnscenariohasnotbeenaddressed.
stance. Recent studies on the consumer return are fruitful Motivatedbytheabovediscussions,theoperationdeci-
andthebulkofthesereportshavefocusedprimarilyonthe sionproblemisinvestigatedforasupplychainwithconsumer
staticmodel.Recently,theproblemsoftheinventorycontrol returnunderconsignmentcontract.Here,weusedifferential
andreturnmanagementunderthedynamicbackgroundhave equationtomodelevolutionofstore-assistanceservicelevel
become hot topics [9, 23–25]. For example, by taking the anddepicttheeffectfromservicelevelontothereturnrate.By
product design quality into account, the problems of the applyingtheoptimalcontroltheory,theoptimalpricingand
dynamicpricingdecision,thereturnstrategy,andthequality service investment strategies are derived under the central-
decisionhavebeenaddressedin[9].In[26,27],thedynamic ized and decentralized systems. Meanwhile, the contract is
preemptive policy has been studied for the supply chain designedtoimprovetheperformanceofdecentralizedsupply
system. However, it is worth mentioning that the decision chain. Finally, a numerical example is given to show the
analysis problem related to the dynamic consumer return changeofcontractfeasibledomainwitheffectivenessofstore-
hasnotbeenthoroughlyinvestigatedforsupplychainunder assistance service level and otherparameters and verify the
consignmentcontract. feasibilityofcoordinationcontract.Themaincontributionof
The issues of performance analysis and coordination this paper can be summarized as follows: (1) the operation
have been studied for supply chain under consignment problem of supply chain under consignment is investigated
contract in [28–31] and complex system under various from the perspective of dynamic; and (2) the coordination
performance indices [32–35]. For example, the effect from contractisdesignedforthesupplychainbytakingtheimpact
vendor-managedinventoryontotwo-stagesupplychainhas of consumer return into account, which can coordinate the
been discussed in [28]. As mentioned in [30], when the decentralized system in the dynamic environment. To be
consignment price and the retail price are exogenously specific, a systematic comparison between this paper and
given, there exist suitable conditions where the vendor- other related papers is given in Table 1 to show the novelty
managed consignment inventory is better than the retailer- andadvantageofthispaper.
managedconsignmentinventory.In[31],theoptimalpricing
and service level decisions have been investigated in two 2.TheProblemDescriptionand
supplychainsundervendor-managedinventoryandretailer- theBasicModel
managed inventory, and the corresponding performance of
thechannelhasbeenanalyzed.Itisworthwhiletonotethat We consider a supply chain formed of a manufacturer M
the performance of supply chain is more efficient in the andaretailerR,inwhichtheretailerhasastrongreputation
first scenario as mentioned in [31]. Subsequently, based on and the upstream manufacturer sells his products to the
the results in [31], the customer returns policy has been end consumer through the retailer’s consignment platform.
MathematicalProblemsinEngineering 3
To improve the market competitiveness, the manufacturer Inaddition,themarketdemand𝐷(𝑡) > 0isassumedto
providesafalsefailurefull-refundpolicytohisconsumerand dependonretailprice𝑝(𝑡)inanadditiveway;thatis,
requiressigningtheconsignmentpricecontractwithretailer.
Byconsideringtheimpactofconsumerreturn,retailershould 𝐷(𝑡)=𝛼−𝛽𝑝(𝑡), (3)
setafixedunitconsignmentprice𝑤tomanufacturerbased
on the actual sales, where we can determine the actual where 𝛼 is the market capacity and 𝛽 depicts the effect of
sales by subtracting the return amount from the product theretailpriceonmarketdemand.Specifically,theconsumer
sales. Clearly, the consumer return may affect the profits who returns the product will stop buying these products; it
of manufacturer and retailer. To get more benefits, both canbeclearlyseenin(2)and(3)thattheactualsalesofthe
themanufacturerandtheretailerinvestthestore-assistance product are given by 𝐷(𝑡)(1 − 𝑄(𝑡)). As in [12], the store-
service level to reduce the return rate. Generally speaking, assistanceservicecostsofthemanufacturerandtheretailer
fromtheperspectiveofphysicalstore,theaimofthestore- may be described by the means of convex and increasing
assistanceserviceinvestmentsistoensurethattheconsumer functionsoftheirownservicelevel;thatis,
finds the right products, which is mainly reflected by pro- 1
viding bigger space to display the full assortments/styles 𝐶(𝑢(𝑡))= 𝑢2(𝑡),
2
of products, installing special equipment to enhance the (4)
1
trial experience, and hiring highly qualified salespeople or 𝐶(V(𝑡))= V2(𝑡).
training the existing ones to improve their sale experiences 2
[8]. On the other hand, from the perspective of online
Throughout this paper, we assume that the production
store, the aim of the store-assistance service investments is
cost 𝑐 = 0. Given an infinite time horizon and a com-
to help the consumer make more rational decision, which
mon discount rate 𝜌, the objective functions of the retailer,
is mainly reflected by providing the online reference ser-
the manufacturer, and the supply chain are, respectively,
vices and offline after-sale services and adding the service
expressedas
with the ability of data mining. Here, the service with the
ability of data mining can provide a function that helps
+∞
the consumers check their own purchase records and finds 𝐽 =∫ 𝑒−𝜌𝑡[𝑤(𝛼−𝛽𝑝(𝑡))(1−𝑄(𝑡))
𝑅
thepurchaserecordsfromotherconsumers.Wedenotethe 0
(5)
store-assistance service investments of the retailer and the 1
manufacturerovertime𝑡as𝑢(𝑡)andV(𝑡),respectively.Then, − 2𝑢2(𝑡)]𝑑𝑡,
theevolutionofservicelevelcanbegivenby
+∞
𝐽 =∫ 𝑒−𝜌𝑡[(𝑝(𝑡)−𝑤)(𝛼−𝛽𝑝(𝑡))(1−𝑄(𝑡))
𝑆̇(𝑡)=(𝑢(𝑡)+V(𝑡))−𝛿𝑆(𝑡), (1) 𝑀 0
(6)
1
− V2(𝑡)]𝑑𝑡,
where𝑆(𝑡)isthestore-assistanceservicelevelattime𝑡with 2
𝑆(0) = 𝑆0andtheparameter𝛿representsthedecayrate.The +∞ 1
first term states the store-assistance service investments of 𝐽 =∫ 𝑒−𝜌𝑡[𝑝(𝑡)(𝛼−𝛽𝑝(𝑡))(1−𝑄(𝑡))− 𝑢2(𝑡)
𝐶 2
supply chain members on the store-assistance service level. 0
(7)
Thesecondtermreflectsthedecayofstore-assistanceservice 1
− V2(𝑡)]𝑑𝑡.
level, and the decay of store-assistance service level can
2
characterizeseveralcases,suchas(1)hiringthesalespeople
with less experience or outdated service style and (2) the Remark 1. In this paper, we assume that the manufacturer
requirement of store-assistance service which needs to be provides a false failure full-refund policy to the consumer,
improvedduetotheincreasingoftheproductcategoriesand whichmeansthereturnexistsduetothedifferencebetween
attributes, which can lead to the decreasing of the original theproductpropertyandtheexpectationoftheconsumers
store-assistance service level across the time. The above- rather than the failure of the product. The store-assistance
mentioned cases would result in the decay of the store- considered in this paper can reduce the false failure return
assistanceservicelevel. effectively. On the other hand, we consider the full-refund
The return rate of product 𝑄(𝑡) can be described by an policy,andthentheprofitfunctions(i.e.,(5)and(6))ofthe
inverselyproportionalfunctionofthestore-assistanceservice manufacturerandretailercanbeobtainedtogetherwith𝑐 =
levelas 0.Itisworthwhiletopointoutthatthecoreofthispapercan
beformulatedimmediatelyintermsofthefalsefailurereturn
𝑄(𝑡)=𝑄max(1−𝜆𝑆(𝑡)), (2) andfull-refund.
where𝑄max ∈[0,1)isthemaximumvalueofreturnrateand 3.TheOptimalStrategiesunderthe
𝜆 > 0representstheeffectivenessofstore-assistanceservice DecentralizedDecision
leveltobedeterminedlater.Notethattheproductsoldcannot
becompletelyreturnedinrealityandhenceweassumethat Underthedecentralizeddecision,themanufacturerandthe
𝑄max ≠1. retailermaximizetheirownprofits,respectively.Actually,the
4 MathematicalProblemsinEngineering
contract parameters are often fixed from the outset of the where
game.Oncethecontractisadopted,thecontractparameters
(2𝐵−2𝜆2𝑄2 𝛼2−𝐴)
are fixed throughout the game. Therefore, the supply chain 𝐷= max ,
gamecanbeconceptualizedasatwo-stagegame.Inthefirst (𝛼𝜆2𝑄2 )
max
stage, the retailer decides the consignment price 𝑤. In the
second stage, both the retailer and the manufacturer make 𝐵=[2𝛽(1−𝑄 )(𝜌+𝛿)2+2𝜌𝛽𝜆𝑄 (𝜌+𝛿)𝑆
max max 0
theirdecisions,respectively.Inparticular,theretailerdeter-
(12)
mines the store-assistance service investment 𝑢(𝑡) and the +𝜆2𝑄2 𝛼2],
max
manufacturer determines the retail price 𝑝(𝑡) as well as the
store-assistanceserviceinvestmentV(𝑡).Ontheotherhand, 𝐴=√4𝐵2−𝐶,
asdiscussedin[38],theestablishedmodelisshowntohave
𝐶=3𝛼2𝜆2𝑄2 (2𝐵−𝜆2𝑄2 𝛼2).
thestate-reparabilityproperty,whichimpliesthatthesecond max max
stageofgameadmitsauniquesubgameequilibriuminboth
Furthermore,theoptimalstore-assistanceservicelevelandthe
NashandStackelberggames.Thatis,theresultofthegameis
optimalprofitfunctionsforsupplychainmemberssatisfy
irrespectiveofthefactwhetherthemovesaresimultaneousor
sequential. 𝜆𝑄 (9𝛼2−(𝛼−𝐷)(5𝛼+𝐷))
Now,weareinapositiontoproposetheoptimalstrategies 𝑆𝑑(𝑡)=𝑆 𝑒−𝛿𝑡+ max (1
0 36𝛽𝛿(𝜌+𝛿)
ofsupplychainunderthedecentralizeddecision. (13)
Theorem 2. Under the decentralized decision, the optimal −𝑒−𝛿𝑡),
strategiesofsupplychainmembersaregivenby
(2𝛼+𝐷)(𝛼−𝐷) 1−𝑄 𝜆𝑄 𝑆
𝐽𝑑∗ = ( max + max 0
𝑅 18𝛽 𝜌 𝜌+𝛿
(14)
1 𝜆2𝑄2 𝛼(𝛼−𝐷)
𝑤∗ = (2𝛼+𝐷), + max ),
3𝛽 (8) 12𝜌𝛽(𝜌+𝛿)2
(𝛼−𝐷)2 1−𝑄 𝜆𝑄 𝑆
𝑝𝑑∗ = 1 (5𝛼+𝐷), (9) 𝐽𝑀𝑑∗ = 36𝛽 ( 𝜌max + 𝜌m+ax𝛿0
6𝛽
(15)
𝜆2𝑄2 (3𝛼+𝐷)(𝛼−𝐷)
V𝑑∗ = 𝜆𝑄max(𝛼−𝐷)2, (10) + m2ax4𝜌𝛽(𝜌+𝛿)2 ).
36𝛽(𝜌+𝛿)
Proof. By applying the backward induction, we first derive
the retail price of product and the optimal store-assistance
𝜆𝑄 (𝛼−𝐷)(2𝛼+𝐷)
𝑢𝑑∗ = max , (11) serviceinvestmentsofsupplychainmembersforthesecond
18𝛽(𝜌+𝛿)
stageofgame.Theoptimizationproblemofmanufactureris
∞ 1
𝑝m>0a,Vx>0 𝐽𝑀 =∫0 𝑒−𝜌𝑡[(𝑝(𝑡)−𝑤)(𝛼−𝛽𝑝(𝑡))(1−𝑄(𝑡))− 2V2(𝑡)]𝑑𝑡 (16)
s.t. 𝑆̇(𝑡)=(𝑢(𝑡)+V(𝑡))−𝛿𝑆(𝑡), 𝑆(0)=𝑆0.
Subsequently, the Hamilton function is adopted to address level.Substituting(2)into(17),wehave
theestablishedoptimizationmodel.Byusingtheresultsfrom 𝐻
𝑀
the differential game theory [38], we introduce the costate
variable𝑞=𝑞(𝑡)toconstructthecurrent-valueHamiltonian =(𝑝(𝑡)−𝑤)(𝛼−𝛽𝑝(𝑡))(1−𝑄max+𝜆𝑄max𝑆(𝑡)) (18)
forthemanufacturerasfollows: 1
− V2(𝑡)+𝑞(𝑡)(𝑢(𝑡)+V(𝑡)−𝛿𝑆(𝑡)).
2
1
𝐻 =(𝑝(𝑡)−𝑤)(𝛼−𝛽𝑝(𝑡))(1−𝑄(𝑡))− V2(𝑡)
𝑀 2 Applying the necessary conditions for maximum principle,
(17) weobtain
+𝑞(𝑡)(𝑢(𝑡)+V(𝑡)−𝛿𝑆(𝑡)), 𝜕𝐻
𝑀 =0, (19)
𝜕V
where the costate variable 𝑞(𝑡) characterizes the shadow 𝜕𝐻
𝑀 =𝜌𝑞−𝑞̇; (20)
prices of an additionalunit of the store-assistance service 𝜕𝑆
MathematicalProblemsinEngineering 5
the first-order condition of the manufacturer’s profit with Subsequently, substituting 𝑝, 𝑢, and V into (5) and (6),
respecttotheretailprice𝑝(𝑡)iswrittenas𝜕𝐻𝑀/𝜕𝑝=0;that respectively, the profits of retailer and manufacturer are as
is,𝑝=(𝛼+𝛽𝑤)/2𝛽.Equation(19)impliesthat follows:
(1−𝑄 )𝑤(𝛼−𝛽𝑤) 𝜆𝑄 𝑆 𝑤(𝛼−𝛽𝑤)
V=𝑞. (21) 𝐽 = max + max 0
𝑅 2𝜌 2(𝜌+𝛿)
Substituting𝑝into(20)andsolvingthedifferentialequation, 𝜆2𝑄2 𝛼𝑤(𝛼−𝛽𝑤)2
onehas + max ,
8𝜌𝛽(𝜌+𝛿)2
𝜆𝑄 (𝛼−𝛽𝑤)2
𝑞(𝑡)=𝑐𝑒(𝜌+𝛿)𝑡+ m4𝛽ax(𝜌+𝛿) . (22) (1−𝑄 )(𝛼−𝛽𝑤)2 𝜆𝑄 𝑆 (𝛼−𝛽𝑤)2 (29)
𝐽 = max + max 0
𝑀 4𝜌𝛽 4𝛽(𝜌+𝛿)
Thus, the optimal store-assistance service investment of
manufactureris 𝜆2𝑄2 (𝛼+3𝛽𝑤)(𝛼−𝛽𝑤)3
+ max .
𝜆𝑄 (𝛼−𝛽𝑤)2 32𝜌𝛽2(𝜌+𝛿)2
V(𝑡)=𝑐𝑒(𝜌+𝛿)𝑡+ max . (23)
4𝛽(𝜌+𝛿) In the first stage of game, retailer maximizes its profit
𝐽𝑅 with respect to consignment price of product 𝑤 and
Once𝑐 ≠ 0,thevalueofmanufacturer’sserviceinvestment optimizationproblemisasfollows:
given in (23) will tend to be infinite when 𝑡 → ∞, which
does not fit the current situation. Therefore, we obtain that m𝑤>a0x 𝐽𝑅
𝑐 = 0 holds. Combining the result in (23), the optimal
(1−𝑄 )𝑤(𝛼−𝛽𝑤)
store-assistance service investment of manufacturer under = max
thedecentralizeddecisionisgivenasfollows: 2𝜌
𝜆𝑄 (𝛼−𝛽𝑤)2 + 𝜆𝑄max𝑆0𝑤(𝛼−𝛽𝑤) (30)
V(𝑡)= max . (24) 2(𝜌+𝛿)
4𝛽(𝜌+𝛿)
𝜆2𝑄2 𝛼𝑤(𝛼−𝛽𝑤)2
Ontheotherhand,theretailer’soptimizationproblemis + max .
8𝜌𝛽(𝜌+𝛿)2
m𝑢>a0x 𝐽𝑅 Thefirst-orderconditionis𝜕𝐽𝑅/𝜕𝑤=0;thatis,
=∫∞𝑒−𝜌𝑡[𝑤(𝛼−𝛽𝑝(𝑡))(1−𝑄(𝑡))− 1𝑢2(𝑡)]𝑑𝑡 (25) 𝜕𝐽𝑅
2
0 𝜕𝑤
s.t. 𝑆̇(𝑡)=(𝑢(𝑡)+V(𝑡))−𝛿𝑆(𝑡), 𝑆(0)=𝑆0. 3𝛼𝛽𝜆2𝑄2
= max𝑤2
8𝜌(𝜌+𝛿)2
Andthecurrent-valueHamiltonianfortheretailerisgivenby
𝐻 =𝑤(𝛼−𝛽𝑝(𝑡))(1−𝑄 +𝜆𝑄 𝑆(𝑡)) (1−𝑄 ) 𝜆𝑄 𝑆 𝜆2𝑄2 𝛼2
𝑅 max max −[ max + max 0 + max ]𝛽𝑤 (31)
1 (26) 𝜌 𝜌+𝛿 2𝜌𝛽(𝜌+𝛿)2
− 𝑢2(𝑡)+𝜂(𝑡)(𝑢(𝑡)+V(𝑡)−𝛿𝑆(𝑡)),
2 (1−𝑄 )𝛼 𝜆𝑄 𝑆 𝛼 𝜆2𝑄2 𝛼3
+ max + max 0 + max
where𝜂(𝑡)isacostatevariableandsatisfies 2𝜌 2(𝜌+𝛿) 8𝜌𝛽(𝜌+𝛿)2
𝜕𝐻 𝜆𝑄 𝑤(𝛼−𝛽𝑤) =0.
𝜂̇ =𝜌𝜂− 𝑀 =(𝜌+𝛿)𝜂− max . (27)
𝜕𝑆 2
Then,
2𝛼
Similarly, we solve the optimal control problem of retailer. 𝑤∗ =
By utilizing the maximization principle, the optimal store- 1 3𝛽
assistanceserviceinvestmentofretaileris𝑢(𝑡)=𝜆𝑄max𝑤(𝛼− 4𝛽(𝜌+𝛿)[(𝜌+𝛿)(1−𝑄 )+𝜆𝑄 𝜌𝑆 ]+𝐴
𝛽𝑤)/2(𝜌+𝛿).Next,substituting𝑢(𝑡)andV(𝑡)intodifferential + max max 0 ,
3𝛼𝛽𝜆2𝑄2
equation(1)andusingtheinitialconditionsof(1),thegeneral max
(32)
solution of differential equation for store-assistance service 2𝛼
level𝑆(𝑡)is 𝑤∗ =
2 3𝛽
𝑆(𝑡)=𝑆 𝑒−𝛿𝑡+ 𝜆𝑄max(𝛼2−𝛽2𝑤2)(1−𝑒−𝛿𝑡). (28) + 4𝛽(𝜌+𝛿)[(𝜌+𝛿)(1−𝑄max)+𝜆𝑄max𝜌𝑆0]−𝐴,
0 4𝛽𝛿(𝜌+𝛿) 3𝛼𝛽𝜆2𝑄2
max
6 MathematicalProblemsinEngineering
where 𝐴 is defined as in Theorem 2. Note that (𝜕2∏ / asymmetric,whichareconsistentwiththeasymmetryofthe
𝑅
𝜕𝑤2)|𝑤=𝑤∗ = (1/4𝜌(𝜌 + 𝛿)2)𝐴 > 0; (𝜕2∏𝑅/𝜕𝑤2)|𝑤=𝑤∗ = profitmargins[39].
1 2
−(1/4𝜌(𝜌+𝛿)2)𝐴 < 0;thustheoptimalconsignmentprices
ofretailersare 4.TheOptimalStrategiesin
theCentralizedSystem
1
𝑤∗ =𝑤∗ = (2𝛼
2 3𝛽 In this section, we examine the performance of the supply
(33) chaininthecentralizedsystem.Supplychainmembersinte-
+ 4𝛽(𝜌+𝛿)[(𝜌+𝛿)(1−𝑄max)+𝜆𝑄max𝜌𝑆0]−𝐴). gratetosettheoptimalpricingstrategyandstore-assistance
𝛼𝜆2𝑄2 serviceinvestmentsinviewofmaximizingthetotalprofitof
max
supply chain. In this game, 𝑝(𝑡),𝑢(𝑡), and V(𝑡) are decision
Setting𝐷=(2𝐵−2𝜆2𝑄2 𝛼2−𝐴)/(𝛼𝜆2𝑄2 ),itiseasytosee variables.
max max
that−𝛼 < 𝐷 < 0.Finally,bysubstituting𝑤∗into𝑝,V,and𝑢, Theorem3. Inthecentralizedsystem,theoptimalstrategiesof
wecanobtaintheoptimaldecisions.
supplychainsystemsaregivenby
𝛼
𝑝𝑐∗ = ,
It is worth mentioning that the optimal strategies (i.e., 2𝛽
optimalpricing𝑤∗,𝑝𝑑∗,andserviceinvestments𝑢𝑑∗andV𝑑∗) (34)
𝜆𝑄 𝛼2
areconstants,whichareeasytoimplementfromapractical 𝑢𝑐∗ =V𝑐∗ = max .
perspective.Inaddition,wecanverifythat𝑤∗,𝑝𝑑∗,𝑢𝑑∗,and 4𝛽(𝜌+𝛿)
V𝑑∗ aredecreasingfunctionswithrespectto𝛽,whichmean Furthermore, the optimal store-assistance service level and
thattheretailersshouldsetalowerconsignmentpricewhen optimalprofitofthetotalsupplychainare
theimpact𝛽fromthepriceontothemarketdemandbecomes
𝜆𝑄 𝛼2
higher.Accordingly,themanufacturershouldsetlowerretail 𝑆𝑐(𝑡)=𝑆 𝑒−𝛿𝑡+ max (1−𝑒−𝛿𝑡),
0 2𝛽𝛿(𝜌+𝛿)
price and the supply chain members need to reduce the
store-assistanceserviceinvestments.Ontheotherhand,from
(1−𝑄 )𝛼2 𝜆𝑄 𝑆 𝛼2
expressions(10),(11),(14),and(15),itisnotdifficulttoshow 𝐽∗ = max + max 0
that 𝑢𝑑∗ > V𝑑∗ and 𝐽𝑑∗ > 𝐽𝑑∗, which depict that the 𝐶 4𝜌𝛽 4𝛽(𝜌+𝛿) (35)
𝑅 𝑀
retailer makes more investment than the manufacturer and
𝜆2𝑄2 𝛼4
the retailer’s profit is greater than the manufacturer’s profit, + max .
respectively.Inotherwords,theproposedassertioninThe- 16𝜌𝛽2(𝜌+𝛿)2
orem 2 implies that both the profit and the store-assistance
service investments from the retailer and manufacturer are Proof. Theoptimizationproblemofsupplychainmembersis
∞ 1 1
𝑝>0m,𝑢>a0x,V>0 𝐽𝐶 =∫0 𝑒−𝜌𝑡[𝑝(𝑡)(𝛼−𝛽𝑝(𝑡))(1−𝑄(𝑡))− 2𝑢2(𝑡)− 2V2(𝑡)]𝑑𝑡 (36)
s.t. 𝑆̇(𝑡)=(𝑢(𝑡)+V(𝑡))−𝛿𝑆(𝑡), 𝑆(0)=𝑆0.
Thecurrent-valueHamiltonianisgivenby andthefirst-orderconditionofthesupplychain’sprofitwith
1 respecttotheretailprice𝑝(𝑡)iswrittenas𝜕𝐻𝐶/𝜕𝑝 = 0;that
𝐻 =𝑝(𝑡)(𝛼−𝛽𝑝(𝑡))(1−𝑄(𝑡))− 𝑢2(𝑡) is,𝑝=𝛼/2𝛽.Equations(38)and(39)implythat
𝐶 2
1 (37) 𝑢=V=𝜇. (41)
− V2(𝑡)+𝜇(𝑡)(𝑢(𝑡)+V(𝑡)−𝛿𝑆(𝑡)).
2
Substituting𝑝into(40)andsolvingthedifferentialequation,
Applying the necessary conditions for maximum principle, onehas
weobtain 𝜆𝑄 𝛼2
𝜕𝐻 𝜇(𝑡)=𝑐𝑒(𝜌+𝛿)𝑡+ max . (42)
𝐶 =0, (38) 4𝛽(𝜌+𝛿)
𝜕𝑢
𝜕𝐻 Similar to proof of Theorem 2, the optimal store-assistance
𝐶 =0, (39) serviceinvestmentsofsupplychainare
𝜕V
𝜆𝑄 𝛼2
𝜕𝜕𝐻𝑆𝐶 =𝜌𝜇−𝜇̇, (40) 𝑢𝑐 =V𝑐 = 4𝛽(m𝜌a+x 𝛿), (43)
MathematicalProblemsinEngineering 7
and then 𝑆̇(𝑡) = 𝜆𝑄max𝛼2/2𝛽(𝜌 + 𝛿) − 𝛿𝑆(𝑡). According to themanufacturerdecidestheretailpriceandstore-assistance
the initial condition 𝑆(0) = 𝑆0, we have 𝑆𝑐(𝑡) = 𝑆0𝑒−𝛿𝑡 + service investment by maximizing its own profit, while the
(𝜆𝑄max𝛼2/2𝛽𝛿(𝜌+𝛿))(1−𝑒−𝛿𝑡).Therefore,thesupplychain retailer controls the store-assistance service investment by
profitisgivenby maximizingtheprofitofthetotalsupplychain.
(1−𝑄 )𝛼2 𝜆𝑄 𝑆 𝛼2 𝜆2𝑄2 𝛼4 Theorem6. Thecommitteddynamicconsignmentpricecon-
𝐽𝐶∗ = 4𝜌m𝛽ax + 4𝛽m(𝜌ax+0𝛿) + 16𝜌𝛽2(m𝜌ax+𝛿)2. (44) tractcancoordinatesupplychain.Thatis,underthecommitted
dynamicconsignmentpricecontract,theequilibriumstrategies
areconsistentwiththecentralizedsolutions.
Proof. Under the committed dynamic consignment price
Remark 4. It follows from Theorems 2 and 3 that 𝑆𝑑(𝑡) is
contract,thecurrent-valueHamiltonianofmanufactureris
the weighted sum of 𝑆0 and 𝜆𝑄max(9𝛼2 − (𝛼 − 𝐷)(5𝛼 +
𝐷))/36𝛽𝛿(𝜌 + 𝛿), and 𝑆𝑐(𝑡) is the weighted sum of 𝑆0 and 𝐻𝑀𝐸 =𝑝𝐸(𝑡)(𝛼−𝛽𝑝𝐸(𝑡))(1−𝑄max+𝜆𝑄max𝑆(𝑡))
𝜆𝑄max𝛼2/2𝛽𝛿(𝜌+𝛿).Moreover,−2𝛼 < 𝐷 < 0impliesthat 1 (46)
𝑆𝑑(𝑡) < 𝑆𝑐(𝑡). Hence, 1 − 𝜆𝑆(𝑡) ≥ 0 holds (i.e., 𝜆𝑆(𝑡) ≤ −𝐾− V2(𝑡)+𝑞𝐸(𝑡)(𝑢(𝑡)+V(𝑡)−𝛿𝑆(𝑡)).
1) if 𝜆𝑆𝑐(𝑡) ≤ 1. Therefore, we can choose the parameter 2
𝜆 obeying 𝜆 ≤ min{1/𝑆0,√2𝛽𝛿(𝜌+𝛿)/𝑄max𝛼2}; that is, Frersopmectthteofithrset-roetradielrprciocned𝑝it(i𝑡o)n,woefhmaavneufacturer’sprofitwith
the effectiveness on decreasing the return rate of the store-
assistanceservicelevelwillnotbetoohigh. 𝜕𝐻𝐸
𝑀 =(1−𝑄 +𝜆𝑄 𝑆(𝑡))(−2𝛽𝑝𝐸(𝑡)+𝛼)
𝜕𝑝 max max
Based on the Theorem 3, three viewpoints can be (47)
obtained. (1) The optimal pricing and service investments =0;
areindependentofthetimeandconstants.Maintainingthe
constant pricing and service strategies are relatively easy to then
implementfromanadministrativeperspective.(2)Whenthe 𝛼
𝑝𝐸∗ =𝑝𝑐∗ = .
effectivenessofservicelevelishigher(forlarger𝜆),theman- 2𝛽 (48)
ufacturerandtheretailerwillinvestmoreinstore-assistance
service investment. Meanwhile, the higher maximum value Themanufacturer’scurrent-valueHamiltonianis
ofreturnratewillinducemorestore-assistanceservicelevel. 𝛼2 1
(3)Discountrateanddecayratehavenegativeimpactsonthe 𝐻𝑀𝐸 = 4𝛽(1−𝑄max+𝜆𝑄max𝑆(𝑡))−𝐾− 2𝑢2(𝑡)
correspondingserviceinvestments. (49)
1
Proposition 5. Compared with optimal strategies and profit − V2(𝑡)+𝑞𝐸(𝑡)(𝑢(𝑡)+V(𝑡)−𝛿𝑆(𝑡)).
2
functionsinthedecentralizedandcentralizedsystems,onehas
Andtheretailer’scurrent-valueHamiltonianis
𝑝𝑑∗ >𝑝𝑐∗, 𝛼2 1
𝐻𝐸 = (1−𝑄 +𝜆𝑄 𝑆(𝑡))− 𝑈2(𝑡)
𝑀 4𝛽 max max 2
𝑢𝑐∗ >𝑢𝑑∗,
(50)
(45) 1
V𝑐∗ >V𝑑∗, − 𝑉2(𝑡)+𝜂𝐸(𝑡)(𝑢(𝑡)+V(𝑡)−𝛿𝑆(𝑡)).
2
𝐽∗ >𝐽𝑑∗+𝐽𝑑∗. SimilartoproofofTheorem2,wehave
𝐶 𝑅 𝑀
𝜆𝑄 𝛼2
Proposition 5 shows that the retail price is higher and 𝑢𝐸∗ =V𝐸∗ =𝑢𝑐∗ =V𝑐∗ = max . (51)
4𝛽(𝜌+𝛿)
store-assistanceserviceinvestmentsarelowerinthedecen-
tralized system, which induces that the total profit is lower
Thus,thecontractcancoordinatethesupplychain.
inthedecentralizedsystem.Hence,thereisaneedtodesign
theappropriatecontractsoastoimprovetheefficiencyofthe Whenthesupplychainiscoordinated,theprofitsofboth
system. membersareasfollows:
𝐾 𝜆2𝑄2 𝛼4
𝐽𝐸 = − max ,
5.CoordinationContract 𝑅 𝜌 32𝜌𝛽2(𝜌+𝛿)2
Motivated by the coordination method in [39], a com-
(1−𝑄 )𝛼2 𝜆𝑄 𝛼2
mitted dynamic consignment price contract is provided to 𝐽𝐸 = max + max 𝑆 (52)
𝑀 4𝜌𝛽 4𝛽(𝜌+𝛿) 0
coordinate the supply chain and improve the performance
of the decentralized supply chain. Contract provisions are
3𝜆2𝑄2 𝛼4 𝐾
structuredasfollows.First,theretaileroffersaconsignment + max − .
price𝑤𝐸 =𝐾/𝐷(𝑡)(1−𝑄(𝑡)),where𝐾isaconstant.Second, 32𝜌𝛽2(𝜌+𝛿)2 𝜌
8 MathematicalProblemsinEngineering
Inordertoensurethatbothpartymembersareinvolved AsshowninTable3,anincreasingeffectivenessofstore-
inthissupplychaincoordinationcontract,thefollowingcon- assistanceservicelevel𝜆leadstoalowerconsignmentprice
ditions should be satisfied simultaneously: 𝐽𝐸 > 𝐽𝑑∗, 𝐽𝐸 > and the retail price in decentralized channel, but there is
𝑅 𝑅 𝑀
𝐽𝑑∗;thenwecangetthefollowingconclusions. no impact on the retail price in the centralized channel.
𝑀 The higher 𝜆 will lead to more store-assistance service
Proposition 7. When 𝑘1 < 𝐾 < 𝑘2, both of supply chain investmentsandchannelprofitsundercentralizedanddecen-
tralizedchannels.Thisresultindicatesthat,regardlessofthe
members are willing to be involved in the implementation
o𝜌f𝐽𝑑t∗h,is𝑘con=tra(c1t,−w𝑄here)𝛼𝑘12/4=𝛽 +𝜆2𝜌𝑄𝜆m2𝑄ax𝛼4𝑆/3𝛼22𝛽/24(𝛽𝜌(𝜌++𝛿𝛿)2) ++ cmeonttirvaalitzeadtiboynhainghdedffeceecntitvreanliezsastioofnth,tehsetochrea-nansseilsmtanemcebseerrsvaicree
𝑅 2 max max 0 level to invest more in the store-assistance service, which
3𝜆2𝑄m2ax𝛼4/32𝛽2(𝜌 + 𝛿)2 − 𝜌𝐽𝑀𝑑∗ and the interval (𝑘1,𝑘2) is result in a higher store-assistance service level or a lower
called the feasible region of committed dynamic consignment return rate. Furthermore, the lower return rate and higher
pricecontract. (unchanged)demandexpandedbylower(unchanged)retail
price can jointly contribute to higher supply chain profits.
Remark8. Fortheaddressedproblem,themanufacturerpays Comparedwiththedecentralizedequilibria,thecentralized
theconsignmentfeefortheretailerattheendoftheperiod. onesarerelativelyhigher,whichisconsistentwiththeresults
For this case, the retail price 𝑝(𝑡) is fixed when signing the in Proposition 5. Moreover, the gaps of investments/profits
contactmadeamongthesupplychainmembers,andhence among two-channel structures enlarge with increasing of
the market demand quantity 𝐷(𝑡) is deterministic based on effectiveness of the store-assistance service level, which
(3)atendoftheclearingperiod;thatis,themarketdemand means that the investments/profits in both centralized and
quantity 𝐷(𝑡) is equal to the whole sales of the products. decentralized settings increase with 𝜆; the improvements
In fact, the dynamics of the proposed consignment price in centralized case are faster relative to the decentralized
contract is reflected by the return rate 𝑄(𝑡) rather than the one.
marketdemandquantity𝐷(𝑡).Althoughthereturnrate𝑄(𝑡) Similarly, according to Table 4, the decay rate 𝛿 of the
maybedifferentamongdifferentperiods,𝑄(𝑡)isdeterministic store-assistance service level will not affect the retail price
at the end of the period. Therefore, the developed dynamic under the centralized channel, and the effects from 𝛿 onto
consignmentpricecontractcanbeimplementedinreality. other equilibria are opposite to the effects from 𝜆 onto
other equilibria. In addition, higher 𝛿 induces lower store-
assistance service investments and profits under central-
ized and decentralized channel. This result indicates that,
6.NumericalAnalysis regardlessofthecentralizationanddecentralization,higher
𝛿 induces a lower (unchanged) demand, store-assistance
Based on the results proposed in the above sections, a service investments, and the higher return rate, which
simulationisgivenheretoillustratethemanagerialinsights, results in the lower supply chain profits. Compared with
such as (1) the effects from the price sensitivity 𝛽 and the the decentralized equilibria, the investments and profits in
effectiveness of the store-assistance service level 𝜆 and the thecentralizedstructurearerelativelyhigher.Moreover,the
delayrate𝛿ontothedecisionsandprofitsoftherelatedsupply gaps of investments/profits among two-channel structures
chain, (2) the changes of store-assistance service level and shrink with increasing of decay rate. It means that the
thereturnratewithtime,and(3)theimpactsfromtheprice investments/profits in both centralized and decentralized
sensitivity𝛽andtheeffectivenessofstore-assistanceservice settingsincreasewith𝛿;thedeclinesincentralizedcaseare
investment 𝜆 and the delay rate 𝛿 onto the feasible region fasterrelativetothedecentralizedcase.
(𝑘1,𝑘2)ofdynamiccontract.Toaddressthesequestions,the Next, from (2), Theorems 2 and 3, and Proposition 7,
following parameter values are set in the example: 𝛼 = 1, we have Figures 1–5. Figures 1 and 2 show the changes of
𝛽=0.1,𝛿=0.05,𝑄max =0.3,𝜌=0.1,𝑆0 =0.2,and𝜆=0.06. store-assistanceservicelevel andreturnratewithtime.The
DefineΔ𝐽 = 𝐽∗ −𝐽𝑑∗ −𝐽𝑑∗ (𝐽∗ > 𝐽𝑑∗ +𝐽𝑑∗).According impacts from the effectiveness of store-assistance service
𝐶 𝑅 𝑀 𝐶 𝑅 𝑀
toTheorems2and3,wecanobtainTables2–4.FromTable2, level and decay rate onto the feasible region of committed
we can see that the price sensitivity 𝛽 negatively affects the dynamicconsignmentpricecontractaregiveninFigures3–
consignment price and the retail price. The result indicates 5.Inparticular,inFigure1,wecanseethattheoptimalstore-
that,regardlessofthecentralizationand/ordecentralization, assistanceservicelevelsinthecentralizeddecisionsystemare
thehigherpricesensitivity𝛽inducesalowerretailprice,the higherthantheonesindecentralizeddecisionsystemdueto
store-assistance service investments, and the higher return thefactthattheinvestmentsofstore-assistanceservicelevels
rate, which leads to lower supply chain profits. Compared are higher in the centralized decision system. Accordingly,
withthedecentralizedequilibria,thestore-assistanceservice the return rate in the centralized decision system is lower
investments and profits in the centralized structure are rel- thantheoneindecentralizeddecisionsystemfromFigure2,
ativelyhigher.Moreover,thegapsoftheinvestments/profits whichshowsthatthelowerretailpriceandreturnrateinthe
amongtwo-channelstructuresshrinkwiththeincreasingof centralizeddecisionsystemcanimprovetheprofitofsupply
𝛽. It means that the investments/profits in both centralized chain.
anddecentralizedsettingsdecreasewith𝛽,andthedeclines Figure 3 shows that when 𝐾 locates between 𝑘1 and
incentralizedcasearemorefaster. 𝑘2,thecommitteddynamicconsignmentpricecontractcan
MathematicalProblemsinEngineering 9
Table2:Theimpactofthepricesensitivity𝛽ondecisionsandprofits.
𝛽 𝑤∗ 𝑝𝑑∗ 𝑝𝑐∗ 𝑢𝑑∗ 𝑢𝑐∗(V𝑐∗) V𝑑∗ 𝐽𝑑∗ 𝐽𝑑∗ 𝐽∗ Δ𝐽
𝑀 𝑅 𝐶
0.10 4.891 7.445 5.000 0.1999 0.400 0.1044 0.710 9.194 19.18 9.275
0.30 1.654 2.493 1.666 0.0667 0.133 0.0338 0.684 2.974 6.037 2.378
0.50 0.995 1.497 1.000 0.0400 0.080 0.0202 0.679 1.774 3.580 1.126
0.70 0.712 1.070 0.714 0.0286 0.057 0.0144 0.677 1.263 2.544 0.602
0.90 0.554 0.832 0.555 0.0222 0.044 0.0112 0.676 0.981 1.973 0.315
Table3:Theimpactofeffectivenessofstore-assistanceservicelevel𝜆ondecisionsandprofits.
𝜆 𝑤∗ 𝑝𝑑∗ 𝑝𝑐∗ 𝑢𝑑∗ 𝑢𝑐∗(V𝑐∗) V𝑑∗ 𝐽𝑑∗ 𝐽𝑑∗ 𝐽∗ Δ𝐽
𝑀 𝑅 𝐶
0.005 4.999 7.499 5.0 0.0125 0.025 0.0063 4.378 8.754 17.511 4.379
0.020 4.992 7.496 5.0 0.0500 0.100 0.0251 4.408 8.785 17.620 4.426
0.035 4.978 7.489 5.0 0.0875 0.175 0.0441 4.470 8.844 17.841 4.526
0.050 4.956 7.478 5.0 0.1250 0.250 0.0636 4.564 8.931 18.175 4.678
0.065 4.927 7.463 5.0 0.0812 0.325 0.0836 4.691 9.048 18.621 4.881
0.080 4.891 7.445 5.0 0.1000 0.400 0.1044 4.851 9.194 19.180 5.134
Table4:Theimpactofdecayrateofstore-assistanceservicelevel𝛿ondecisionsandprofits.
𝜆 𝑤∗ 𝑝𝑑∗ 𝑝𝑐∗ 𝑢𝑑∗ 𝑢𝑐∗(V𝑐∗) V𝑑∗ 𝐽𝑑∗ 𝐽𝑑∗ 𝐽∗ Δ𝐽
𝑀 𝑅 𝐶
0 4.864 7.432 5.0 0.224 0.450 0.118 4.976 9.308 19.615 5.331
0.01 4.886 7.443 5.0 0.204 0.409 0.107 4.873 9.214 19.255 5.168
0.02 4.903 7.452 5.0 0.187 0.375 0.097 4.794 9.142 18.981 5.045
0.03 4.917 7.458 5.0 0.173 0.346 0.089 4.732 9.086 18.767 4.949
0.04 4.928 7.464 5.0 0.160 0.321 0.082 4.684 9.042 18.597 4.871
0.05 4.937 7.468 5.0 0.150 0.300 0.076 4.645 9.006 18.460 4.809
7 0.32
0.3
6
0.28
5
0.26
4 0.24
S(t) Q(t) 0.22
3
0.2
2
0.18
1
0.16
0
0 2 4 6 8 10 0 2 4 6 8 10
t t
Centralized Centralized
Decentralized Decentralized
Figure1:Thedynamicchangeofstore-assistanceservicelevel. Figure2:Thedynamicchangeofreturnrate.
coordinatethesupplychainwellandbothmembersarebetter
off. Specifically, when the price sensitivity 𝛽 increases, the of the effectiveness of the store-assistance service level 𝜆.
win-win region becomes smaller. This implies that larger 𝛽 Thisimpliesthatlarger𝜆willprovideretailerwithagreater
will provide retailer with a smaller degree of flexibility to degreeofflexibilitytocoordinatethesupplychain.Assuch,
coordinate the supply chain. Figure 4 depicts that the win- theretailershouldanalyzethemainreasonofproductreturn
win region becomes large accompanied with the increasing and choose a more effective way to reduce the return rate.
10 MathematicalProblemsinEngineering
1.5 1.5
1.4
1 1.3
1.2
0.5 1.1
1
0 0.9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
𝛽 𝛿
k1 k1
k2 k2
Figure3:Theimpactof𝛽onthefeasibleregionofcontract. Figure5:Theimpactof𝛿onthefeasibleregionofcontract.
1.8
havebeengivenunderthecentralizedanddecentralizedset-
1.6
tings,andacommitteddynamicconsignmentpricecontract
has been designed to coordinate the decentralized supply
1.4
chain. Finally, a numerical analysis has been conducted
to illustrate the effectiveness of the store-assistance service
1.2
level and price sensitivity on equilibria and coordination.
Inparticular,wehaveobtainedthefollowingresults:(1)the
1
optimalretailpriceislowerandtheoptimalstore-assistance
serviceinvestmentsarehigherinthecentralizedsetting.As
0.8
a result, a lower return rate arises in the centralized one.
0.6 (2) Compared with the store-assistance service investments
from the retailer and the manufacturer, it can be seen that
0.4 theinvestmentfromtheretailerishigherthantheoneofthe
0.02 0.03 0.04 0.05 0.06 0.07 0.08 manufacturer in decentralized setting and the effectiveness
𝜆 ofthestore-assistanceservicelevel,thepricesensitivity,and
thedecayrateofstore-assistanceservicelevelwouldaffectthe
k
1
optimalstrategies.(3)Thecommitteddynamicconsignment
k
2
price contract effectively improves the performance of the
Figure4:Theimpactof𝜆onthefeasibleregionofcontract. decentralized supply chain. Future research topics include
theextensionoftheproposedstrategiestotheperformance
analysis and scheme design with multiobjective constraints
Figure5showsthattheeffectsfromthedecayrate𝛿ontowin- in[40]andtoathree-layersupplychainmodelwithmultiple
winregionaresimilarto𝛽. manufacturersasin[41].
7.Conclusions
CompetingInterests
Inthispaper,wehavediscussedthedecisionproblemsofsup-
The authors declare that there are no competing interests
plychainsubjecttotheconsumerreturnunderconsignment
regardingthepublicationofthispaper.
contract.Byapplyingtheoptimalcontroltheory,thestrate-
gies of optimal pricing and service have been presented in
thedecentralizedandcentralizeddecisionsystems.Themain Acknowledgments
noveltyliesinthefollowing:(i)thedynamicevolutionfeature
ofthestore-assistanceservicelevelhasbeenconsideredand This work was supported in part by the National Natural
thestore-assistanceservicelevelhasbeensetasastatevari- Science Foundation of China (Grant 11271103), the Fok
able; (ii) by constructing the differential game model, both Ying Tung Education Foundation of China (Grant 151004),
thepricingandstore-assistanceserviceinvestmentstrategies the University Nursing Program for Young Scholars with