Research ArticleA Closed-Loop Supply Chain Problem with Retailing andRecycling Competition
Chuanchao Xu Bo Li Yanfei Lan and Yi Tang
College of Management amp Economics Tianjin University Tianjin 300072 China
Correspondence should be addressed to Yanfei Lan yanfei-lan163com
Received 27 March 2014 Accepted 1 July 2014 Published 24 July 2014
Academic Editor Jozef Banas
Copyright copy 2014 Chuanchao Xu et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We investigate a durable product retailing and recycling problem in a closed-loop supply chain consisting of a single manufacturerand two competitive retailers in which the manufacturer collects used products via retailers from the consumers and has sufficientchannel power over the retailers to act as a Stackelberg leader the retailers compete in retail products and recycling used products Inorder to analyze the impact of retailing and recycling competitions on the profits of the manufacturer and the competitive retailerstwo collection models (coordinated collection (Model C) and decentralized collection (Model D)) are established respectivelyThen based on game theory we derive the optimal retail price the optimal repurchase price and the optimal profits of themanufacturer and the retailers The managerial insights demonstrate that more intense retailing competition induces the increaseof the manufacturerrsquos profits in both forward and reverse channels and retailersrsquo profits in the forward channel and the decreaseof retailersrsquo profits in the reverse channel while more intense recycling competition induces the decrease of the profits of themanufacturer and retailers in both forward and reverse channels Finally numerical examples are given to illustrate the effectivenessof the proposed models
1 Introduction
Durable product remanufacturing and processes for sus-tainable manufacturing play an important role in closed-loop supply chain operations Increasinglymanufacturers areestablishing economically viable production and distributionsystems that enable remanufacturing of used products inparallel with the manufacturing of new units Remanufac-tured products are typically upgraded to the quality standardsof new products so that they can be sold in new productmarkets
In current practice HP Xerox and Apple are retail com-petitive products that utilize retailers for collecting their usedproducts and have created a fully integrated manufacturing-remanufacturing strategy around their reusable product lines[1] For example HP on average 70 of the disposedcartridges are reused in the production of a new oneEach time a cartridge is returned to HP the retailers arereimbursed a fixed fee per cartridge and the transportationcosts [2] HP collects its used products via retailers and some
remanufacturers or third parties also collect HPrsquos productwhich results in competition between HP and remanufactur-ersthird partiesThus many researchers began to investigatethe effect of competition on the channel memberrsquos decisionsin closed-loop supply channel problem
In current literature the operations of typical closed-loopsupply chain have been widely studied [3ndash6] FurthermoreChoi et al [7] and Yalabik et al [8] provided an analysisfor calculating the optimal acquisition price Liang et al [9]proposed an option pricing for used products of differentquality Some literature discussed the competition betweenchannel members which affects their behaviors of decisionSavaskan et al [1] provided a problem of choosing a suitablechannel structure for the collection of end-of-life returnsfrom customers and considered that the retailing competi-tion affects channel membersrsquo decision behavior in differentcollection ways and Inderfurth [10] extended the researchand investigated an optimal pricing decision problem of afuzzy closed-loop supply chain with retailing competitionSavaskan and vanWassenhove [11] focused on the interaction
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 509825 14 pageshttpdxdoiorg1011552014509825
2 Abstract and Applied Analysis
between a manufacturerrsquos reverse channel choice to collectused product and the strategic product pricing decisionsunder retailing competition in the forward channel Ofek etal [12] examined how consumer returns affected competitivesellersrsquo prices in-store assistance levels and decisions to offeran online channel in addition to the brick and mortar store
For the research in reverse channel of closed-loop supplychain the contracting of collecting the used products affectsnot only the supply of used products but also the priceof the remanufactured product Hong et al [13] provideda typical recycling competition of manufacturers who havethree reverse collection channel structures to choose (1)manufacturer and retailer collecting used products havecompetition at the same time (2) manufacturer collectingused products have competition with a retailer and a thirdparty and (3) manufacturer and third party collecting usedproducts have competition at the same time Choi et al [14]investigated a closed-loop supply chain (CLSC) which con-sists of a manufacturer a third parties collector and a retailerand examined the performance of different CLSCs underdifferent channel leaderships Moreover they illustrated howboth the serial and parallel CLSCs can be coordinated byusing different kinds of practical contracts Webster andMitra [15] and Ferguson and Toktay [16] investigated thejoint pricing problem faced by a manufacturer in whicha remanufacturable product is derived from heterogeneousconsumers and discussed the market and technology driversof product remanufacturability and identified some phe-nomena of managerial importance that are typical of aremanufacturing environment Wu [17] considered a two-period closed-loop supply chain problem consisting of twosupply chain members an original equipment manufacturerand a remanufacturer and investigated the product designdecision of the original equipment manufacturer and bothchain membersrsquo competitive pricing strategies Tayur andGaneshan [18] considered that a manufacturer used herforesight about the retailerrsquos and the third partyrsquos reactionfunctions in her decision making Majumder and Groen-evelt [19] examined how third party remanufacturing couldinduce competitive behavior when the recycling productscannibalize the demand for the original product Hickey [20]presented that the third parties such as GENCO distributionsystem were also preferred by some consumer goods underwhose experience manufacturers collect the used productJung and Hwang [21] provided a typical example of reman-ufacturing in a reverse channel supply chain with an originalequipment manufacturer and a remanufacturer Chern et al[22] provided a multiobjective master planning problem forreverse supply chains by considering product structures
However among all the literature associated with theclosed-loop supply chain problems under competition mostof them have considered the retailing or recycling compe-tition problems respectively but few of them consider theimpact on the closed loop supply chain members involvedin retailing and recycling competitions together Actuallyretailing and recycling competitions often exist togetherin closed-loop supply chain For example HPrsquos cartridgesApplersquos cellphone the Xeroxrsquos office equipment and so forthhave both retailing and recycling competitions [2] Based on
observations from current practice and the extant literaturewe study a competition of retailing and recycling problemin a closed-loop supply chain in which a manufacturercollects used products via retailers from the consumers andhas sufficient channel power over the retailers to act as aStackelberg leader the retailers compete in a Bertrand pricinggame in the channels of both forward and reverse In order toanalyze the impact of retailing and recycling competitions onthe profits of the manufacturer and the competitive retailerstwo collectionmodels (coordinated collection (model119862) anddecentralized collection (model 119863)) are established respec-tively Then we derive the optimal retail price the optimalrepurchase price and the optimal profits of the manufacturerand the retailers Finally numerical examples are given toillustrate the effectiveness of the proposed models
More specifically we address the following questions
(1) How do the competitive factors of retailing and recy-cling affect the decision of the manufacturer and eachretailer (ie retail price repurchase price wholesaleprice and buy-back payment)
(2) How do the retailing and recycling competitionsaffect the profits of the channel members
The rest of the paper is organized as follows Section 3describes the assumptions and notations of the modelingframework The formulations and analysis are presented inSection 4 Section 5 provides the conclusion of the results andpossible directions for future research
2 Modeling Assumptions and Notations
This paper investigates a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers twocollection ways (coordinated collection and decentralizedcollection) are introduced (see Figure 1)
According to the consumption nature of durable productassume that the variation of retail price has little effect onthe size of retailing market that is the demand of marketsizes for customers has little sensitivity to retail price andthe variation of repurchase price has little effect on the sizeof recycling market Consistent with the extant literature[11] we assume that the manufacturer has sufficient channelpower over the retailers to act as a Stackelberg leader andthe retailers compete in a Bertrand pricing game In orderto get more product sale opportunities the retailers shouldcollect used products for the manufacturer More specificallywe consider a manufacturer who has incorporated reman-ufacturing of used products into hisher original productsystem so that it can manufacture a new product directlyfrom raw materials or remanufacture part or whole of areturned unit into a new product and there is no distinctionbetween a remanufactured and a manufactured product Thereal archetypal example for products is the Kodak single-use camera where the customer knows that the companyutilizes used remanufacturing parts in the production ofsome cameras but does not know whether a specific productcontains used parts or not The manufacturer sells both new
Abstract and Applied Analysis 3
Manufacturer
Retailer 2Retailer 1
Customer
(a) Coordinated collection (model 119862)
Manufacturer
Retailer 2 Retailer 1
Customer
Competing
(b) Decentralized collection (model119863)
Figure 1 Closed-loop supply channel structures
and remanufactured products to the retailer at the samewholesale price119908 who in turn sells both products at the sameprice 119901 on the market [23] Let 119888
119898and 119888119903denote the unit
cost of manufacturing a new product and remanufacturing areturned product into a new one respectively Assume that119888119903
lt 119888119898 which implies that the manufacturer can make
savings from remanufacturing Let Δ = 119888119898
minus 119888119903denote the
unit saving cost by remanufacturingWe also add a Notationssection to help the authors to understand and distinguish thenotations
Consistent with the existing literature [24] we assumethat retailer 119894 faces the following linear demand function inthe forward channel
119863119894(119901119894 119901119895) = Φ
119894minus 119901119894+ 120573119901119895 0 le 120573 lt 1 (1)
where Φ119894represents the market size of retailer 119894 119901
119894is the
price of the product at retailer 119894 119901119895 denotes the competitiveretailerrsquos retail price and 120573 denotes the retailing substitutioneffect
We assume that retailer 119894 faces the following recyclingfunction in the reverse channel
119876119894(119901119877119894 119901119877119895
) = 120601119894+ 119901119877119894
minus 120574119901119877119895
0 le 120574 lt 1 (2)
where 120601119894 represents the market size of retailer 119894 by free recy-
cling119901119877119894 is the repurchase price of the used product at retailer119894 119901119877119895 denotes the competitive retailerrsquos repurchase price and120574 denotes the recycling product recycling substitution effect
Considering the relationship between the demand func-tions and recycling functions we assume 120601119894 = 119904119863119894(119901119894 119901119895)which denotes the market capacity which is recycled by theretailer freely where 119904 is a recycling coefficient and 0 le 119904 lt 1
[25] Ceteris paribus the profit of retailing a new product ismore than recycling obsolete products So we assume that theretailing substitution effect is more intense than the recyclingsubstitution effect that is 0 le 120574 lt 120573 lt 1 and 119904 is lower thanthe retailing substitution effect that is 0 le 119904 lt 120573 lt 1
According to the above assumptions and notations thechannelrsquos profit function in the case of coordinated collectionis
Π119862(119901119894 119901119895 119901119877119894 119901119877119895
)
=
2
sum
119894=1
((119901119894minus 119888119898)119863119894(119901119894 119901119895) + (Δ minus 119901
119877119894) 119876119894(119901119894 119901119877119895
))
(3)
wheresum2
119894=1(119901119894minus119888119898)119863119894(119901119894 119901119895) denotes the profit of the forward
channel Similarlysum2119894=1
(Δminus119901119877119894)119876119894(119901119877119894 119901119877119895
) denotes the profitof the reverse channel
Themanufacturer profit function in the case of decentral-ized collection is
Π119863
119898(119901119894 119901119895 119901119877119894 119901119877119895)
=
2
sum
119894=1
((119908 minus 119888119898)119863119894 (119901119894 119901119895) + (Δ minus 119887)119876119894 (119901119877119894 119901119877119895))
(4)
where 119908 represents the unit wholesale price and 119887 denotesthe unit price of a returned product from the retailerto the manufacturer (119908 minus 119888119898) sum
2
119894=1119863119894(119901119894 119901119895) denotes the
manufacturerrsquos profit in the forward channel Similarly (Δ minus
119887)sum2
119894=1119876119894(119901119877119894 119901119877119895
) represents themanufacturerrsquos profit in thereverse channel
Each retailerrsquos profit function in the case of decentralizedcollection is
Π119863
119894= (119901119894minus 119908)119863
119894(119901119894 119901119895) + (119887 minus 119901
119877119894) 119876119894(119901119877119894 119901119877119895
) (5)
where (119901119894minus 119908)119863
119894(119901119894 119901119895) denotes the profit of retailer 119894 in the
forward channel and (119887 minus 119901119877119894)119876119894(119901119877119894 119901119877119895
) denotes the profitof retailer 119894 in the reverse channel
4 Abstract and Applied Analysis
3 Model Formulation and Solution
In this section two equilibrium models are discussed in thecases of coordinated collection and decentralized collectionrespectively
31 Coordinated Collection According to (3) the coordi-nated collection model can be formulated as follows
max119901119894119901119895119901119877119894119901119877119895
Π119862(119901119894 119901119895 119901119877119894 119901119877119895
)
=
2
sum
119894=1
((119901119894minus 119888119898)119863119894(119901119894 119901119895) + (Δ minus 119901
119877119894) 119876119894(119901119877119894 119901119877119895
))
(6)
Proposition 1 In the case of coordinated collection theoptimal retail 119901119862
119894and the optimal repurchase price 119901
119862
119895satisfy
119901lowast119862
119894=
(Φ119894+ Φ119895120573)
2 (1 minus 1205732)
+4 (Δ119904 minus 2119888
119898) (1 minus 120574) + 119904
2(Φ119894+ Φ3minus119894
)
41198831
minus1199042(Φ119894 minus Φ3minus119894)
41198832
(7)
119901lowast119862
119877119894=
Δ
2+ s[
(Φ119894+ Φ3minus119894
) minus (2119888119898
minus Δs) (1 minus 120573)
21198831
minus(Φ119894minus Φ3minus119894
)
21198832
]
(8)
where1198831 = 1199042(1minus120573)minus4(1minus120574) and1198832 = minus4(1minus120574)minus119904
2(1+120573)
Proof According to (3) the second-order partial derivativesof Π119862(119901
119894 119901119895 119901119877119894 119901119877119895
) are
1205972Π119862
1205971199012
119894
= minus21205972Π119862
120597119901119894120597119901119895
= 21205731205972Π119862
120597119901119894120597119901119877119894
= 119904
1205972Π119862
120597119901119894120597119901119877119895
= minus119904120573
1205972Π119862
120597119901119895120597119901119894
= 21205731205972Π119862
1205971199012
119895
= minus21205972Π119862
120597119901119895120597119901119877119894
= minus119904120573
1205972Π119862
120597119901119895120597119901119877119895
= 119904
1205972Π119862
120597119901119877119894120597119901119894
= 1199041205972Π119862
120597119901119877119894120597119901119895
= minus1199041205731205972Π119862
1205971199012
119877119894
= minus2
1205972Π119862
120597119901119877119894120597119901119877119895
= 2120574
1205972Π119862
120597119901119877119895
120597119901119894
= minus1199041205731205972Π119862
120597119901119877119895
120597119901119895
= 1199041205972Π119862
120597119901119877119895
120597119901119877119894
= 2120574
1205972Π119862
1205971199012
119877119895
= minus2
(9)
Then the Hessian matrix is
|119867| =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus2 2120573 119904 minus119904120573
2120573 minus2 minus119904120573 119904
119904 minus119904120573 minus2 2120574
minus119904120573 119904 2120574 minus2
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(10)
By using the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt
1 we can obtain that |1198631| = minus2 lt 0 |119863
2| = 4(1 minus 120573) gt 0
|1198633| = minus2(1 minus 120573
2)(4 minus 119904
2) lt 0 and |1198634| = 16 + 16(120574
2minus 1205732(1 minus
1205742))+8119904
2(1205732+120574120573(1minus120573
2)minus1)+119904
4(1minus1205732)2gt 0 So119867 is negative
definiteΠ119862 is jointly concavewith respect to119901119862
1198941199011198623minus119894
119901119862119895 and
119901119862
3minus119895 thus the optimal retailing and repurchase prices can be
obtained by the first-order conditions as follows
120597Π119862
120597119901119894
= Φ119894minus 2119901119894+ 2120573119901
119895+ 119904119901119877119894
minus 119904120573119901119877119895
+ (1 minus 120573) (119888119898
minus 119904Δ)
= 0
(11)
120597Π119862
120597119901119895
= Φ119895 minus 2119901119895 + 2120573119901119894 + 119904119901119877119894 minus 119904120573119901119877119895 + (1 minus 120573) (119888119898 minus 119904Δ)
= 0
(12)
120597Π119862
120597119901119877119894
= minus 119904Φ119894+ 119904119901119894minus 119904120573119901
119895minus 2119901119877119894
+ 2120574119901119877119895
+ (1 minus 120574) Δ
= 0
(13)
120597Π119862
120597119901119877119895
= minus 119904Φ119895+ 119904119901119895minus 119904120573119901
119894minus 2119901119877119895
+ 2120574119901119877119894
+ (1 minus 120574) Δ
= 0
(14)
By combining (11) (12) (13) and (14) we can obtainthe optimal retail prices 119901
lowast
119894and 119901
lowast
119895 and optimal repurchase
prices 119901lowast
119877119894and 119901
lowast
119877119895
Corollary 2 In the case of coordinated collection the retailprices 119901
119862
119894and 119901
119862
119895are strictly monotonically decreasing with
respect to the recycling substitution effect
Proof According to (7) the first-order derivative of 119901119862119894with
respect to 120574 is
d119901119862119894
d120574= (1199042[1198832
1(Φ119894 minus Φ3minus119894)
minus1198832
2(Φ119894+ Φ3minus119894
+ (1 minus 120573) (Δ119904 minus 2119888119898))])
times (1198832
11198832
2)minus1
(15)
where1198831= 1199042(1minus120573)minus4(1minus120574) and119883
2= minus4(1minus120574)minus119904
2(1+120573)
It follows from the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt
120573 lt 1 that d119901119862119894d120574 lt 0 thus the retail prices 119901
119862
119894and 119901
119862
119895are
strictly decreasing with respect to the recycling substitutioneffect
Abstract and Applied Analysis 5
Corollary 3 In the case of coordinated collection the repur-chase prices 119901
119862
119877119894and 119901
119862
119877119895are strictly increasing with respect to
the recycling substitution effect
Proof According to (8) the first-order derivative of 119901119862119877119894is
d119901119862119877119894
d120574= 2119904 (119883
2
2[Φ119894+ Φ3minus119894
minus (2119888119898
minus Δ119904) (1 minus 120573)]
minus1198832
1(Φ119894minus Φ3minus119894
)) times (1198832
11198832
2)minus1
(16)
where1198831 = 1199042(1minus120573)minus4(1minus120574) and1198832 = minus4(1minus120574)minus119904
2(1+120573)
By assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1 wecan obtain that d119901119862119862
119895d120574 gt 0 thus the repurchase prices 119901
119862
119877119894
and 119901119862
119877119895are strictly increasing with respect to the recycling
substitution effect
Corollary 2 implies how the recycling substitution effectimpacts the profit of forward channel Specially themanufac-turer can obtain more profit from forward channel by induc-ing recycling substitution effect (select the lower retail priceto induce more recycling substitution effect) Corollary 3implies how the recycling substitution effect impacts theprofit of reverse channel themanufacturer can select optimalrepurchase price by inducing recycling substitution effect(inducing more recycling substitution effect to select thehigher retail price) in order to obtain more profit of reversechannel Corollaries 2 and 3 illustrate that the recyclingsubstitution effectrsquos impact on the profit of forward andreverse is converse so themanufacturer can induce a suitablerecycling substitution effect between the retailers to obtainthe maximum profit in forward and reverse supply chain
Proposition 4 In the case of coordinated collection the profitof forward channel is strictly increasing with respect to theretailing substitution effect and decreasing with respect to therecycling substitution effect While the profit of reverse channelis strictly decreasing with respect to the retailing substitutioneffect and increasing with respect to the recycling substitutioneffect
Proof First for the analysis of the profit in forward channelfor any given 119901119895 according to (1) the demand119863119894 is invariablefor the durable product and Φ119894 is constant for retailer 119894the retail price is increasing with respect to the retailingsubstitution effect Then it follows from (3) that the profitof forward channel sum2
119894=1(119901119894minus 119888119898)119863119894(119901119894 119901119895) is increasing with
respect to 119901119894and 119901
119895 Furthermore by Corollary 2 the retail
prices 119901119894and 119901
119895are both strictly decreasing with respect to
the recycling substitution effect Hence the profit of forwardchannel function sum
2
119894=1(119901119894minus 119888119898)119863119894(119901119894 119901119895) is decreasing with
respect to the recycling substitution effectSecond for the analysis of the profit in reverse channel
for any given 119901119877119894 since the recycling function119876
119894is invariable
for the durable product and 120601119894= 119904119863119894and 119901
119877119894are invariable
for retailer 119894 by (2) the repurchase price is increasing withrespect to recycling substitution effect Then the profit ofreverse channelsum2
119894=1(Δ minus 119901
119877119894)119876119894(119901119877119894 119901119877119895
) in (3) is decreasing
with respect to 119901119877119894
and 119901119877119895 According to Corollary 3 the
repurchase prices 119901119877119894
and 119901119877119895
are strictly increasing withrespect to the recycling substitution effect Therefore theprofit of reverse channel function sum
2
119894=1(Δ minus 119901
119877119894)119876119894(119901119877119894 119901119877119895
)
is decreasing with the recycling substitution effect
From what is discussed above in the case of coordi-nated collection we can obtain some managerial insights inpractice It is illustrated that the variation tendency of theforward channel profit is decided by the difference betweenthe increase of forward channel profit from the retailingsubstitution effect and the decrease of forward channelrsquos profitstemmed from the recycling substitution effect Similarly thevariation of reverse channel profit is decided by the differencebetween the increase of reverse channel profit for the retail-ing substitution effect and the decrease of reverse channelprofit for the recycling substitution effect Additionally thevariation of the total profit of forward and reverse channel isuncertain Furthermore the manufacturer can induce lowerretailing substitution effect and higher recovery substitutioneffect to collect more obsolete products and increase theprofit of reverse channel In other words the manufacturercan easily achieve the goal of remanufacturing strategy byrecovering the residual value of used products
32 Decentralized Collection In the case of decentralizedcollection the manufacturer decides on the wholesale price119908 and reimburses each retailer a fixed fee 119887 per unit returnedand the retailers compete with each other and decide on theretail price of the product as well as the repurchase price ofcollection used product
We can obtain retailer 119894rsquos reaction function by solving thefollowing optimization problem
max119901119894 119901119895
Π119863
119894= (119901119894 minus 119908)119863119894 (119901119894 119901119895) + (119887 minus 119901119877119895)119876119877119894 (119901119877119894 119901119877119895)
(17)
Proposition 5 In the case of decentralized collection theretailerrsquos optimal reaction retail price 119901
lowast119863
119894and the optimal
reaction repurchase price 119901lowast119863
119877119894satisfy
119901lowast119863
119894=
119887119904 (1 minus 120574) minus 119908 (2 minus 120574)
11988411
+(Φ119894+ Φ3minus119894
) 1198841
211988411
+(Φ119894minus Φ3minus119894
) 1198851
211988511
(18)
119901lowast119863
119877119894=
119904 (Φ119894+ Φ3minus119894
)
211988411
minus119904 (Φ119894minus Φ3minus119894
)
211988511
+ 119887[1199042(1 minus 120574)
119884111988411
+
(1 minus 1199042)1198851
119879]
minus 119908119904 [1198851
119879+ (2 minus 120574)]
(19)
where 1198841
= 1199042+ 120574 minus 2 119884
11= 1199042minus 2(2 minus 120574) + 120573(2 minus 119904
2minus 120574)
1198851 = 2 + 120574 minus 119904
2 11988511 = 2(2 + 120574) minus 120573(2 + 1199042minus 120574) minus 119904
2 and119879 = 120574
2minus 1199044minus 4(1 minus 119904
2)
6 Abstract and Applied Analysis
Proof The second-order partial derivatives of retailerrsquos profitΠ119863
119894are
1205972Π119863
119894
1205971199012
119894
= minus21205972Π119863
119894
120597119901119894120597119901119895
= 119904
1205972Π119863
119894
120597119901119895120597119901119894
= 1199041205972Π119863
119894
1205971199012
119895
= minus119904120573
(20)
Then the Hessian matrix is
|119867| =
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1205972Π119863
119894
1205971199012
119894
1205972Π119863
119894
120597119901119894120597119901119895
1205972Π119863
119894
120597119901119895120597119901119894
1205972Π119863
119894
1205971199012
119895
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(21)
By using the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt
120573 lt 1 we can obtain that |1198631| = 120597
2Π119863
1198941205971199012
119894= minus2 lt 0
and |1198632| = 2119904(120573 minus 119904) gt 0 So 119867 is negative definite Π119863 is
jointly concave with respect to 119901119894and 119901
119895 Thus the optimal
retailing and repurchase price can be obtained by the first-order condition as follows
According to (17) for retailer 1 the first-order conditionsof Π119863119894are
120597Π119863
119894
120597119901119894
= Φ119894 minus 2119901119894 + 120573119901119895 + 119908 minus 119904 (119887 minus 119901119877119894) = 0 (22)
120597Π119863
119894
120597119901119877119894
= minus119904 (Φ119894 minus 119901119894 + 120573119901119895) minus 119901119877119894 + 120574119901119877119895 + 119887 minus 119901119877119894 = 0
(23)
Similarly for retailer 2 the first-order partial conditions ofΠ119863
119895are
120597Π119863
119895
120597119901119895
= Φ119895minus 2119901119895+ 120573119901119894+ 119908 minus 119904 (119887 minus 119901
119877119895) = 0 (24)
120597Π119863
119895
120597119901119877119895
= minus119904 (Φ119895 minus 119901119895+ 120573119901119894) minus 119901
119877119895+ 120574119901119877119894
+ 119887 minus 119901119877119895
= 0
(25)
By combining (22) (23) (24) and (25) we can obtain theoptimal retail prices 119901119894 and 119901119895 and optimal repurchase prices119901119877119894 and 119901119877119895
Corollary 6 The retail price is increasing with respect to thewholesale price 119908 and decreasing with respect to the buy-backpayment 119887 But the repurchase price is decreasing with respectto the wholesale price119908 and increasing with respect to the buy-back payment 119887
Proof By the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1we can easily obtain119884
1= 1199042+120574minus2 lt 0119884
11= 1199042minus2(2minus120574)+120573(2minus
1199042minus120574) lt 0119885
1= 2+120574minus119904
2gt 0 and 119879 = 120574
2minus1199044minus4(1minus119904
2) lt 0
Thus according to (18) we can obtain that the retail prices119901119894and 119901
119895are increasing with respect to the wholesale price
119908 while decreasing with respect to the buy-back payment 119887And it follows from (19) that the repurchase prices 119901
119877119894and
119901119877119895are decreasingwith respect to thewholesale price119908 while
increasing with respect to the buy-back payment 119887
Substituting (18) and (19) into (17) yields
max119908119887
Π119863
119898= (119908 minus 119888119898)
2
sum
119894=1
119863119894 (119901lowast119863
119894 119901lowast119863
119895)
+ (Δ minus 119887)
2
sum
119894=1
119876119894(119901lowast119863
119877119894 119901lowast119863
119877119895)
(26)
Proposition 7 In the case of decentralized collection themanufacturerrsquos optimal wholesale price 119908
lowast and optimal buy-back payment 119887lowast satisfy
119908lowast
=(Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701 [1198701 (120573 + 1) + 4 (1 minus 120574)]
+1198703(1 minus 120574)
1198701(2 minus 120574)
(27)
119887lowast
=2Δ1198731+ 1205741198732+ 120573120574 (2119888
119898119904 minus 4Δ)
4120573 (2 minus 1199042) + 2120574 (8 minus 119904
2) minus 2 (2 + 120573120574) (4 minus 119904
2) (28)
where 1198701 = 4(120574 minus 1) + 1199042(2 minus 120574) 1198702 = (2119888119898 minus Δ119904)(1 minus 120574)
1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873
1= 120573(2 minus 119904
2) minus (4 minus 119904
2) and
1198732= 8Δ + (Φ
1+ Φ2minus 2119888119898)119904
Proof The second-order partial derivatives of Π119863119898are
1205972Π119863
119898
1205971199082
=41199042(2 minus 120574) (1 minus 120573) (1 minus 120574)
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
1205972Π119863
119898
1205971198872
=41199042(2 minus 120574) (1 minus 120574)
2
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
1205972Π119863
119898
120597119908120597119887
=1205972Π119863
119898
120597119887120597119908
=41199042(2 minus 120574)
2[1 minus 120573 + 2119904 (1 minus 120574) minus (2 minus 120573) (2 minus 120574)]
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
10038161003816100381610038161198671198981003816100381610038161003816 =
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1205972Π119863
119898
1205971199082
1205972Π119863
119898
120597119908120597119887
1205972Π119863
119898
120597119887120597119908
1205972Π119863119862
119898
1205971198872
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(29)
By the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1 wecan easily obtain that119867
119898is negative definite soΠ
119863
119898is jointly
concave with respect to the wholesale price 119908 and buy-back
Abstract and Applied Analysis 7
payment 119887Thus the optimal wholesale price119908 and buy-backpayment 119887 can be obtained by the first-order condition
According to (26) the first-order conditions of Π119863119898with
respect to 119908 and 119887 are as follows
120597Π119863
119898
120597119908
=2 (2 minus 120574) (2119908 minus 119887119904 + 119888
119898) + 2Δ119904
1199042+ 120574 minus 2
+ ((120574 minus 2) [2119887119904 (1 minus 120574) + (Φ1+ Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)
+ 2 (119888119898
minus 119908) (2 minus 120574 + 2119904 (119887 minus Δ)) ])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(30)
120597Π119863
119898
120597119887
=2 (1 minus 120574) (2119887 minus Δ + 119888
119898119904) minus 2119908119904 (2 minus 120574)
1199042+ 120574 minus 2
+ (119904 [2119887119904 (1 minus 120574) + (Φ1 + Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)]
+ 2119904 (1 minus 120574) [(119888119898
minus 119908) (2 minus 120574) + 119904 (119887 minus Δ)])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(31)
By combining (30) and (31) we can obtain the optimal retailwholesale price119908lowast and buy-back payment 119887lowast that is (27) and(28) hold
Proposition 8 In the case of decentralized collection theoptimal retail 119901lowast119863
119894and optimal repurchase price 119901
lowast119863
119877119894of retailer
119894 satisfy
119901lowast119863
119894
=(Φ119894 minus Φ3minus119894)
2[
1198851
11988511
minus1198841
11988411
]
+119904 (1 minus 120574)
211988411
[2Δ1198721 + 1205741198722 + 2120573120574 (119888119898119904 minus 2Δ)
21198721+ 120574119872
2
]
minus(2 minus 120574)
11988411
[(Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(1 + 120573) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
]
119901lowast119863
119877119894
= ((1 minus 120574) 119904
2
119884111988411
minus
(1 minus 1199042)1198851
119879)
times (2Δ1198721 + 1205741198723 + 120573120574 (2119888119898119904 minus 4Δ)
21198721+ 120574119872
2
) minus 119904 [(2 minus 120574) +1198851
119879]
times ((Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(120573 + 1) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
) + [119904 (Φ119894 + Φ3minus119894)
2](
1
11988411
minus1
11988511
)
(32)
where 1198721
= 120573(2 minus 1199042) minus (4 minus 119904
2) 1198722
= (8 minus 1199042) minus 120573(4 minus 119904
2)
1198723
= 8Δ + (Φ1+ Φ2minus 2119888119898)119904 1198701
= 4(120574 minus 1) + 1199042(2 minus 120574)
1198702= (2119888119898
minus Δ119904)(1 minus 120574) 1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873
1=
120573(2minus1199042)minus(4minus119904
2)1198732= 8Δ+(Φ
1+Φ2minus2119888119898)1199041198841= 1199042+120574minus2
11988411
= 1199042minus 2(2 minus 120574) + 120573(2 minus 119904
2minus 120574) 119885
1= 2 + 120574 minus 119904
2 11988511
=
2(2 + 120574) minus 120573(2 + 1199042minus 120574) minus 119904
2 and 119879 = 1205742minus 1199044minus 4(1 minus 119904
2)
Proof By substituting (27) and (28) into (18) and (19) respec-tively we can obtain the optimal retail price and repurchaseprice of retailer 119894
Proposition 9 In the case of decentralized collection themanufacturerrsquos profit of forward channel is strictly increasingwith respect to the retailing substitution effect and decreasingwith respect to the recycling substitution effect while themanufacturerrsquos reverse channelrsquos profit is increasingwith respectto the retailing substitution effect and decreasing with respect tothe recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in forward channelAccording to (1) the demand of retailing 119863
119894is invariable
for the durable product and Φ119894and 119901
119895are invariable for
retailer 119894 thus the retail price is increasing with respectto the retailing substitution effect that is increasing theretailing substitution effect 120573 will induce the increase ofretail price 119901119894 and according to Corollary 6 it will inducethe increase of wholesale price 119908 According to (4) themanufacturerrsquos profit in forward channel (119908minus119888119898)(119863119894(119901119894 119901119895)+
119863119895(119901119895 119901119894)) is increasing with respect to retailing substitutioneffect Similarly we can easily obtain that the manufacturerrsquosprofit in forward channel (119908 minus 119888119898)(119863119894(119901119894 119901119895) + 119863119895(119901119895 119901119894)) isdecreasing with respect to recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in reverse channelAccording to (2) the recycling function 119876
119894is invariable for
the durable product and 120601119894
= 119904119863119894and 119901
119877119895are invariable
8 Abstract and Applied Analysis
for retailer 119894 thus the repurchase price is increasing withrespect to recycling substitution effect Then we can easilyobtain that the increase of retailing substitution effect 120573
will induce the increase of retail price 119901119894 and according
to Corollary 6 it will induce the decrease of buy-backpayment 119887 According to (4) the manufacturerrsquos reversechannel (Δ minus 119887)(119876119894(119901119877119894 119901119877119895) + 119876
119895(119901119877119895
119901119877119894)) is increasing
with respect to retailing substitution effect Similarly we caneasily get that the manufacturerrsquos reverse channel profit (Δ minus
119887)(119876119894(119901119877119894 119901119877119895) + 119876119895(119901119877119895 119901119877119894)) is decreasing with respect torecycling substitution effect
It is illustrated that in the case of decentralized collectionthe variation of manufacturerrsquos profit in forward or reversechannel is decided by the difference between the increase offorward or reverse channel profit for the retailing substitutioneffect and the decrease of forward and reverse channelprofit for the recycling substitution effect Additionally thechange of closed-loop supply chainrsquos profit with respect tothe retailing and recycling substitution effects is uncertainin the case of decentralized collection On the other handthe manufacturer can induce higher retailing substitutioneffect and lower recovery substitution effect via choosingsuitable wholesale price and buy-back payment to collectmore obsolete products and increase the profit of reversechannel It means that the manufacturers can achieve theirgoal of remanufacturing strategy by recovering the residualvalue of used products
Proposition 10 In the case of decentralized collection theretailerrsquos profit of forward channel is strictly increasing withrespect to the retailing substitution effect and decreasing withrespect to the recycling substitution effect while his reversechannel profit is decreasing with respect to the retailing substi-tution effect and recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in forward channel Accordingto (1) the demand of retailing119863
119894is invariable for the durable
product and Φ119894and 119901
119895are invariable for retailer 119894 thus
the retail price is increasing with respect to the retailingsubstitution effect that is the increase of 120573 will induce theincrease of retail price 119901
119894 According to (5) each retailerrsquos
profit in forward channel (119901119894minus 119908)119863
119894(119901119894 119901119895) is increasing
with respect to the retailing substitution effect similarly forretailer 119895 We can easily obtain that each retailerrsquos profit inreverse channel (119901119894 minus 119908)119863119894(119901119894 119901119895) is increasing with respectto recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in reverse channel Accordingto (2) the recycling function119876119877119894 is invariable for the durableproduct and 120601119894 = 119904119863119894 and 119901119877119895 are invariable for retailer119894 thus the repurchase price is increasing with respect torecycling substitution effect Then we can easily obtain thatthe repurchase price is increasing with respect to recyclingsubstitution effect So the increase of retailing substitutioneffect 120573 will induce the increase of retail price 119901
119894 induce
the increase of buy-back payment 119887 and then will inducethe decrease of 119901
119877119894 According to (5) the retailer 119894rsquos profit in
reverse channel (119887minus119901119877119894)119876119894(119901119877119894 119901119877119895
) is decreasingwith respectto retailing substitution effect Similarly we can easily get thatthe retailer 119894rsquos profit in reverse channel (119887 minus119901
119877119894)119876119894(119901119877119894 119901119877119895
) isincreasing with respect to recycling substitution effect
Proposition 10 illustrates that in the case of decentralizedcollection the variation of retailerrsquos profit in forward channelis decided by the difference between the increase of forwardchannelrsquos profit for the retailing substitution effect and thedecrease of forward channel profit for the recycling substitu-tion effect While for the retailerrsquos profit in reverse channelit is decreasing with respect to retailing substitution effectand recycling substitution effect Additionally because thechange of forward channel profit is uncertain the variationof the closed-loop supply chainrsquos profit is also uncertain Inthis case the retailer can induce lower retailing substitutioneffect and recovery substitution effect via choosing suitablewholesale price and buy-back payment to collect moreobsolete products and increase the profit of reverse channel
In a word the variations of manufacturer and retailerrsquosrespective profit with the retailing substitution effect inforward channel are uniform but in the reverse channelthey are inconsistent For example the higher recyclingsubstitution effect will induce the manufacturer to get moreprofits in reverse channel (ie augments the scale of theman-ufacturerrsquos reverse supply chains) But the higher recyclingsubstitution effect will cause the retailer to obtain less profitsin reverse channel (ie cut down the scale of the retailerrsquosreverse supply chains) There is a contradiction betweenthe profits of the manufacturer and the retailer in reversechannel because the manufacturer has sufficient channelpower over the retailers to act as a Stackelberg leader thus themanufacturer can choose the appropriate wholesale price andbuy-back payment to achieve the remanufacturing strategyand balance the retailerrsquos profit And we make a comparisonwith the coordinated collection we can easily know thatthe coordinated collection is more effective to augment thescale of the whole reverse supply chains than decentralizedcollection
4 Numerical Examples
In this section numerical examples are provided to comparethe results obtained in the above two different collectionmodels and to analyze the impacts of retailing and recyclingsubstitution effects on the decisions and profits of chainmembers Some managerial insights can be derived throughthe numerical analysis The numerical analysis is performedfor Φ1= 100 Φ
2= 50 119888
119898= 20 Δ = 10 120574 = 04 or 120574 = 02
and 119904 = 01
41 Numerical Analysis of Model 119862 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofcoordinated collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect to retail-ing substitution effect that is the higher retailing substitution
Abstract and Applied Analysis 9
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
120574 = 02
120574 = 04
pi
(a) Impact on the retail price
0
5
10
15
20
25
30
35
40
45
50
pRi
0 02 04 06 08 1120573
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 2 Impact of the retailing and recycling substitution effects on price in model 119862
effect will induce the increase of retail price Moreover thehigher recycling substitution effect will induce the increaseof retail price the higher recycling substitution effect willinduce the decrease of repurchase price In addition thehigher retailing substitution effect will induce the decrease ofretail price The green and blue lines in Figures 2(a) and 2(b)give us a visual presentation
From the previous assumptions and analysis we knowthat the profit of forward channel is strictly increasing withrespect to retailing substitution effect and decreasing withrespect to recycling substitution effect that is the manu-facturer can induce higher retailing competition and lowerrecycling competition to obtain more profit from forwardchannel
Andwe known that the profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andincreasing with respect to recycling substitution effect thatis the manufacturer can induce lower recycling substitutioneffect to obtain more profit in the reverse channel In otherwords themanufacturer can induce higher retailing substitu-tion effect and lower recycling substitution effect could obtainmore profit in closed-loop supply chain The green and bluelines in Figures 3(a) and 3(b) give us a visual presentation
42 Numerical Analysis of Model 119863 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofdecentralized collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect toretailing substitution effect that is the higher retailing substi-tution effect will induce the increase of retail price And the
higher recycling substitution effect will induce the decreaseof retail price And we known that the repurchase priceis strictly decreasing with respect to recycling substitutioneffect that is the higher recycling substitution effect willinduce the increase of repurchase price while the higherretailing substitution effect will induce the decrease of retailprice The green and blue lines in Figures 4(a) and 4(b) giveus a visual presentation
According to the assumptions and the analysis weknow that the manufacturerrsquos profit of forward channel isstrictly increasing with respect to retailing substitution effectand decreasing with respect to recycling substitution effectAnd the manufacturerrsquos profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andrecycling substitution effect In other words in the case ofdecentralized collection the manufacturer can induce higherretailing competition and lower recycling competition toobtain more profit in the forward channel and induce lowerrecycling substitution effect to obtain more profit in thethe reverse channel So the manufacturer can select higherretailing substitution effect and lower recycling substitutioneffect to obtain more profit in their closed-loop supply chainMoreover the higher recycling competition will induce thelower profit of forward channel and induce lower profit ofreverse channel
For the forward channel profit of retailers is strictlyincreasing with respect to retailing substitution effect anddecreasing with respect to recycling substitution effect Andthe retailersrsquo profit of reverse channel is strictly decreasingwith respect to retailing substitution effect and increasingwith respect to recycling substitution effect In other wordsin the case of decentralized collection for the retailers thehigher retailing competition and lower recycling competition
10 Abstract and Applied Analysis
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(a) Impact on the profit in forward channel
0 02 04 06 08 10
200
400
600
800
1000
1200
1400
1600
1800
2000
120573
120574 = 02
120574 = 04120587
(b) Impact on the profit in reverse channel
Figure 3 Impact of the retailing and recycling substitution effects on channel profit in model 119862
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
pi
120574 = 02
120574 = 04
(a) Impact on the retail price
0 02 04 06 08 1minus20
minus15
minus10
minus5
0
5
10
15
20
25
30
120573
pi
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 4 Impact of the retailing and recycling substitution effects on price in model 119863
can make them obtain more profit in their forward channelbut the lower retailing substitution effect and higher recyclingsubstitution effect can make them obtain more profit in thereverse channel So the retailers tend to select suitable retailprice and repurchase price to induce retailing and recyclingcompetitions (not too high or too low the higher retailsubstitution effect will make the profit of reverse channel
negative or the lower recycling substitution effect will makethe profit of forward channel decrease) to obtain more profitin their closed-loop supply chain The lines in Figure 5 giveus a visual presentation
43 Comparison of Model119862 andModel119863 In this subsectionwe discuss the impact of retailing substitution effect on the
Abstract and Applied Analysis 11
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120587
(a) Impact on the manufacturerrsquos profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120587
(b) Impact on the manufacturerrsquos profit in reverse channel
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(c) Impact on the retailersrsquo profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120574 = 02
120574 = 04
120587
(d) Impact on the retailersrsquo profit in reverse channel
Figure 5 Impact of the retailing and recycling substitution effects on channel profit in model 119863119862
optimal prices and optimal channel profits and comparethem in the case of coordinated collection with that ofdecentralized collection
From the previous assumptions and analysis we knowthat the retail price in the case of coordinated collection islower than that in decentralized collection that is facingthe same circumstances the retail price in the case ofdecentralized collection is higher than that in coordinatedcollection And the repurchase price in the case of decentral-ized collection is lower than that in coordinated collectionIt implies that for the retailer the coordinated collectionhas more price advantages (lower retail price and higherrepurchase price) than the decentralized collection in the
closed-loop channel The lines in Figure 6 give us a visualpresentation
In what is discussed above we know that the channelprofit in model 119862 is higher than that in model 119863 Theprofit of reverse channel in model 119863 is very low even thevalue is negative In this case the manufacturer and retailerstend to select coordinated collection to obtain more profitsfor example HP Xerox and Apple select the coordinatedcollection to collect their used product The lines in Figure 6give us a visual presentation
44 Impact of Retailing and Recycling Substitution Effects onClosed Supply Chainrsquos Profit In this subsection we discuss
12 Abstract and Applied Analysis
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
Model CCModel DC
120573
pi
(a) Comparison of retail price
minus100
minus50
0
50
100
150
200
pRi
0 02 04 06 08 1
Model CCModel DC
120573
(b) Comparison of repurchase price
Figure 6 Pricing decisions of coordinated collection versus decentralized collection
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
Model CCModel DC
times104
120573
120587
(a) Comparison of forward channel profits
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
Model CCModel DC
120573
120587
(b) Comparison of reverse channel profits
Figure 7 The profit of coordinated collection versus decentralized collection
the impact of retailing and recycling substitution effects onthe optimal forward channel profits and optimal reversechannel profits in the cases of coordinated collection anddecentralized collection
From the previous assumptions and analysis weknow that the recycling substitution effectrsquos impact on theclosed-loop supply channel profit in the case of coordinated
collection is more obvious than that in decentralizedcollection In other words the manufacturer can obtainmore profit than in decentralized collection more easily bychanging the recycling substitution effect The reason maybe as follows in the case of coordinated collection the retailprice and repurchase price are made by the center planner(ie manufacturer) this collection model can adjust the
Abstract and Applied Analysis 13
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(a) Impact on the closed-loop supply channelrsquos profit in model 119862119862
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(b) Impact on the closed-loop supply channelrsquos profit in model119863119862
Figure 8 Impact of the retailing and recycling substitution effects on closed-Loop supply chainrsquos profit
retail and repurchase price fast and accurately But in caseof decentralized collection there is a pricing game betweenmanufacturer and retailers The manufacturer firstly selectsoptimal wholesale price and buy-back payment then theretailers select their optimal reaction retail price and repur-chase price that is the pricing game and pricing decision areunsynchronized between the manufacturer and retailer sothe manufacturer can obtain more profits in the case of coor-dinated collection than the case of decentralized collectionThe lines in Figures 7 and 8 give us a visual presentation
5 Conclusion
In this paper we investigated a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers Theresults demonstrate that more intense retailing competitioninduces the increase of the forward and reverse profitsof manufacturer and the forward channel of retailers anddecrease of the retailers profit in reverse channel while moreintense recycling competition induces the decrease of theprofits of the manufacturer and retailers in forward andreverse channels while the whole supply chainrsquos profit ofmodel 119862 is increasing with the respect to the free recyclingproportional coefficient but the effect on model 119863 is uncer-tain The results have important managerial insights for themanufacturer and retailers
This paper makes a contribution to the literature onreverse channel choice and coordination by drawing atten-tion to closed-loop supply chains with competition of retail-ing and recycling Our future research will be as follows ourmodel does not consider any fixed costs of retailingrecyclinga collection system If such costs were incurred a minimumnumber of retailingreturns would be required to justify thecost of operations Last but not least our cost formulation
does not include any operationalcapacity constraints inthe channel of products collection Operationalcapacityconstraint on the channel of collection can be incorporatedinto the model by defining an upper bound on the numbercollected via each channel Another possibility is to modelthe collection cost as a quadratic function of the quantityreturned
Notations
120574 Recycling substitution effect120573 Retailing substitution effectΦ Market size of the retailers in the forward
channel120601 Market size of the retailers by free recovering
in the reverse channel120575 Unit saving cost by remanufacturing119904 Recovering coefficient119888119903 Unit cost of remanufacturing a returned
product into a new one119888119898 Unit cost of manufacturing a new product
119908 New productrsquos wholesale price of themanufacturer
119887 Obsolete productrsquos repurchase price from theretailers to the manufacturer
119863119894(119901119894 119901119895) Demand function of retailer 119894
119876119894(119901119877119894 119901119877119895
) Recycling function of retailer 119894
119901119862
119894 119901119863
119895 Retailer 119894 and 119895rsquos respective retail price in
models 119862 and 119863
119901119862
119877119894 119901119863
119877119895 Retailer 119894 and 119895rsquos repurchase price in models
119862 and 119863
Π119870
119894 Profit function for channel member 119894 in
model 119896 Superscript 119896 takes the values of 119862and 119863 Subscript 119894 takes the values of 119898 119903and 119890 denoting the manufacturer theretailer and the 119890-tailer respectively
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
between a manufacturerrsquos reverse channel choice to collectused product and the strategic product pricing decisionsunder retailing competition in the forward channel Ofek etal [12] examined how consumer returns affected competitivesellersrsquo prices in-store assistance levels and decisions to offeran online channel in addition to the brick and mortar store
For the research in reverse channel of closed-loop supplychain the contracting of collecting the used products affectsnot only the supply of used products but also the priceof the remanufactured product Hong et al [13] provideda typical recycling competition of manufacturers who havethree reverse collection channel structures to choose (1)manufacturer and retailer collecting used products havecompetition at the same time (2) manufacturer collectingused products have competition with a retailer and a thirdparty and (3) manufacturer and third party collecting usedproducts have competition at the same time Choi et al [14]investigated a closed-loop supply chain (CLSC) which con-sists of a manufacturer a third parties collector and a retailerand examined the performance of different CLSCs underdifferent channel leaderships Moreover they illustrated howboth the serial and parallel CLSCs can be coordinated byusing different kinds of practical contracts Webster andMitra [15] and Ferguson and Toktay [16] investigated thejoint pricing problem faced by a manufacturer in whicha remanufacturable product is derived from heterogeneousconsumers and discussed the market and technology driversof product remanufacturability and identified some phe-nomena of managerial importance that are typical of aremanufacturing environment Wu [17] considered a two-period closed-loop supply chain problem consisting of twosupply chain members an original equipment manufacturerand a remanufacturer and investigated the product designdecision of the original equipment manufacturer and bothchain membersrsquo competitive pricing strategies Tayur andGaneshan [18] considered that a manufacturer used herforesight about the retailerrsquos and the third partyrsquos reactionfunctions in her decision making Majumder and Groen-evelt [19] examined how third party remanufacturing couldinduce competitive behavior when the recycling productscannibalize the demand for the original product Hickey [20]presented that the third parties such as GENCO distributionsystem were also preferred by some consumer goods underwhose experience manufacturers collect the used productJung and Hwang [21] provided a typical example of reman-ufacturing in a reverse channel supply chain with an originalequipment manufacturer and a remanufacturer Chern et al[22] provided a multiobjective master planning problem forreverse supply chains by considering product structures
However among all the literature associated with theclosed-loop supply chain problems under competition mostof them have considered the retailing or recycling compe-tition problems respectively but few of them consider theimpact on the closed loop supply chain members involvedin retailing and recycling competitions together Actuallyretailing and recycling competitions often exist togetherin closed-loop supply chain For example HPrsquos cartridgesApplersquos cellphone the Xeroxrsquos office equipment and so forthhave both retailing and recycling competitions [2] Based on
observations from current practice and the extant literaturewe study a competition of retailing and recycling problemin a closed-loop supply chain in which a manufacturercollects used products via retailers from the consumers andhas sufficient channel power over the retailers to act as aStackelberg leader the retailers compete in a Bertrand pricinggame in the channels of both forward and reverse In order toanalyze the impact of retailing and recycling competitions onthe profits of the manufacturer and the competitive retailerstwo collectionmodels (coordinated collection (model119862) anddecentralized collection (model 119863)) are established respec-tively Then we derive the optimal retail price the optimalrepurchase price and the optimal profits of the manufacturerand the retailers Finally numerical examples are given toillustrate the effectiveness of the proposed models
More specifically we address the following questions
(1) How do the competitive factors of retailing and recy-cling affect the decision of the manufacturer and eachretailer (ie retail price repurchase price wholesaleprice and buy-back payment)
(2) How do the retailing and recycling competitionsaffect the profits of the channel members
The rest of the paper is organized as follows Section 3describes the assumptions and notations of the modelingframework The formulations and analysis are presented inSection 4 Section 5 provides the conclusion of the results andpossible directions for future research
2 Modeling Assumptions and Notations
This paper investigates a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers twocollection ways (coordinated collection and decentralizedcollection) are introduced (see Figure 1)
According to the consumption nature of durable productassume that the variation of retail price has little effect onthe size of retailing market that is the demand of marketsizes for customers has little sensitivity to retail price andthe variation of repurchase price has little effect on the sizeof recycling market Consistent with the extant literature[11] we assume that the manufacturer has sufficient channelpower over the retailers to act as a Stackelberg leader andthe retailers compete in a Bertrand pricing game In orderto get more product sale opportunities the retailers shouldcollect used products for the manufacturer More specificallywe consider a manufacturer who has incorporated reman-ufacturing of used products into hisher original productsystem so that it can manufacture a new product directlyfrom raw materials or remanufacture part or whole of areturned unit into a new product and there is no distinctionbetween a remanufactured and a manufactured product Thereal archetypal example for products is the Kodak single-use camera where the customer knows that the companyutilizes used remanufacturing parts in the production ofsome cameras but does not know whether a specific productcontains used parts or not The manufacturer sells both new
Abstract and Applied Analysis 3
Manufacturer
Retailer 2Retailer 1
Customer
(a) Coordinated collection (model 119862)
Manufacturer
Retailer 2 Retailer 1
Customer
Competing
(b) Decentralized collection (model119863)
Figure 1 Closed-loop supply channel structures
and remanufactured products to the retailer at the samewholesale price119908 who in turn sells both products at the sameprice 119901 on the market [23] Let 119888
119898and 119888119903denote the unit
cost of manufacturing a new product and remanufacturing areturned product into a new one respectively Assume that119888119903
lt 119888119898 which implies that the manufacturer can make
savings from remanufacturing Let Δ = 119888119898
minus 119888119903denote the
unit saving cost by remanufacturingWe also add a Notationssection to help the authors to understand and distinguish thenotations
Consistent with the existing literature [24] we assumethat retailer 119894 faces the following linear demand function inthe forward channel
119863119894(119901119894 119901119895) = Φ
119894minus 119901119894+ 120573119901119895 0 le 120573 lt 1 (1)
where Φ119894represents the market size of retailer 119894 119901
119894is the
price of the product at retailer 119894 119901119895 denotes the competitiveretailerrsquos retail price and 120573 denotes the retailing substitutioneffect
We assume that retailer 119894 faces the following recyclingfunction in the reverse channel
119876119894(119901119877119894 119901119877119895
) = 120601119894+ 119901119877119894
minus 120574119901119877119895
0 le 120574 lt 1 (2)
where 120601119894 represents the market size of retailer 119894 by free recy-
cling119901119877119894 is the repurchase price of the used product at retailer119894 119901119877119895 denotes the competitive retailerrsquos repurchase price and120574 denotes the recycling product recycling substitution effect
Considering the relationship between the demand func-tions and recycling functions we assume 120601119894 = 119904119863119894(119901119894 119901119895)which denotes the market capacity which is recycled by theretailer freely where 119904 is a recycling coefficient and 0 le 119904 lt 1
[25] Ceteris paribus the profit of retailing a new product ismore than recycling obsolete products So we assume that theretailing substitution effect is more intense than the recyclingsubstitution effect that is 0 le 120574 lt 120573 lt 1 and 119904 is lower thanthe retailing substitution effect that is 0 le 119904 lt 120573 lt 1
According to the above assumptions and notations thechannelrsquos profit function in the case of coordinated collectionis
Π119862(119901119894 119901119895 119901119877119894 119901119877119895
)
=
2
sum
119894=1
((119901119894minus 119888119898)119863119894(119901119894 119901119895) + (Δ minus 119901
119877119894) 119876119894(119901119894 119901119877119895
))
(3)
wheresum2
119894=1(119901119894minus119888119898)119863119894(119901119894 119901119895) denotes the profit of the forward
channel Similarlysum2119894=1
(Δminus119901119877119894)119876119894(119901119877119894 119901119877119895
) denotes the profitof the reverse channel
Themanufacturer profit function in the case of decentral-ized collection is
Π119863
119898(119901119894 119901119895 119901119877119894 119901119877119895)
=
2
sum
119894=1
((119908 minus 119888119898)119863119894 (119901119894 119901119895) + (Δ minus 119887)119876119894 (119901119877119894 119901119877119895))
(4)
where 119908 represents the unit wholesale price and 119887 denotesthe unit price of a returned product from the retailerto the manufacturer (119908 minus 119888119898) sum
2
119894=1119863119894(119901119894 119901119895) denotes the
manufacturerrsquos profit in the forward channel Similarly (Δ minus
119887)sum2
119894=1119876119894(119901119877119894 119901119877119895
) represents themanufacturerrsquos profit in thereverse channel
Each retailerrsquos profit function in the case of decentralizedcollection is
Π119863
119894= (119901119894minus 119908)119863
119894(119901119894 119901119895) + (119887 minus 119901
119877119894) 119876119894(119901119877119894 119901119877119895
) (5)
where (119901119894minus 119908)119863
119894(119901119894 119901119895) denotes the profit of retailer 119894 in the
forward channel and (119887 minus 119901119877119894)119876119894(119901119877119894 119901119877119895
) denotes the profitof retailer 119894 in the reverse channel
4 Abstract and Applied Analysis
3 Model Formulation and Solution
In this section two equilibrium models are discussed in thecases of coordinated collection and decentralized collectionrespectively
31 Coordinated Collection According to (3) the coordi-nated collection model can be formulated as follows
max119901119894119901119895119901119877119894119901119877119895
Π119862(119901119894 119901119895 119901119877119894 119901119877119895
)
=
2
sum
119894=1
((119901119894minus 119888119898)119863119894(119901119894 119901119895) + (Δ minus 119901
119877119894) 119876119894(119901119877119894 119901119877119895
))
(6)
Proposition 1 In the case of coordinated collection theoptimal retail 119901119862
119894and the optimal repurchase price 119901
119862
119895satisfy
119901lowast119862
119894=
(Φ119894+ Φ119895120573)
2 (1 minus 1205732)
+4 (Δ119904 minus 2119888
119898) (1 minus 120574) + 119904
2(Φ119894+ Φ3minus119894
)
41198831
minus1199042(Φ119894 minus Φ3minus119894)
41198832
(7)
119901lowast119862
119877119894=
Δ
2+ s[
(Φ119894+ Φ3minus119894
) minus (2119888119898
minus Δs) (1 minus 120573)
21198831
minus(Φ119894minus Φ3minus119894
)
21198832
]
(8)
where1198831 = 1199042(1minus120573)minus4(1minus120574) and1198832 = minus4(1minus120574)minus119904
2(1+120573)
Proof According to (3) the second-order partial derivativesof Π119862(119901
119894 119901119895 119901119877119894 119901119877119895
) are
1205972Π119862
1205971199012
119894
= minus21205972Π119862
120597119901119894120597119901119895
= 21205731205972Π119862
120597119901119894120597119901119877119894
= 119904
1205972Π119862
120597119901119894120597119901119877119895
= minus119904120573
1205972Π119862
120597119901119895120597119901119894
= 21205731205972Π119862
1205971199012
119895
= minus21205972Π119862
120597119901119895120597119901119877119894
= minus119904120573
1205972Π119862
120597119901119895120597119901119877119895
= 119904
1205972Π119862
120597119901119877119894120597119901119894
= 1199041205972Π119862
120597119901119877119894120597119901119895
= minus1199041205731205972Π119862
1205971199012
119877119894
= minus2
1205972Π119862
120597119901119877119894120597119901119877119895
= 2120574
1205972Π119862
120597119901119877119895
120597119901119894
= minus1199041205731205972Π119862
120597119901119877119895
120597119901119895
= 1199041205972Π119862
120597119901119877119895
120597119901119877119894
= 2120574
1205972Π119862
1205971199012
119877119895
= minus2
(9)
Then the Hessian matrix is
|119867| =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus2 2120573 119904 minus119904120573
2120573 minus2 minus119904120573 119904
119904 minus119904120573 minus2 2120574
minus119904120573 119904 2120574 minus2
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(10)
By using the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt
1 we can obtain that |1198631| = minus2 lt 0 |119863
2| = 4(1 minus 120573) gt 0
|1198633| = minus2(1 minus 120573
2)(4 minus 119904
2) lt 0 and |1198634| = 16 + 16(120574
2minus 1205732(1 minus
1205742))+8119904
2(1205732+120574120573(1minus120573
2)minus1)+119904
4(1minus1205732)2gt 0 So119867 is negative
definiteΠ119862 is jointly concavewith respect to119901119862
1198941199011198623minus119894
119901119862119895 and
119901119862
3minus119895 thus the optimal retailing and repurchase prices can be
obtained by the first-order conditions as follows
120597Π119862
120597119901119894
= Φ119894minus 2119901119894+ 2120573119901
119895+ 119904119901119877119894
minus 119904120573119901119877119895
+ (1 minus 120573) (119888119898
minus 119904Δ)
= 0
(11)
120597Π119862
120597119901119895
= Φ119895 minus 2119901119895 + 2120573119901119894 + 119904119901119877119894 minus 119904120573119901119877119895 + (1 minus 120573) (119888119898 minus 119904Δ)
= 0
(12)
120597Π119862
120597119901119877119894
= minus 119904Φ119894+ 119904119901119894minus 119904120573119901
119895minus 2119901119877119894
+ 2120574119901119877119895
+ (1 minus 120574) Δ
= 0
(13)
120597Π119862
120597119901119877119895
= minus 119904Φ119895+ 119904119901119895minus 119904120573119901
119894minus 2119901119877119895
+ 2120574119901119877119894
+ (1 minus 120574) Δ
= 0
(14)
By combining (11) (12) (13) and (14) we can obtainthe optimal retail prices 119901
lowast
119894and 119901
lowast
119895 and optimal repurchase
prices 119901lowast
119877119894and 119901
lowast
119877119895
Corollary 2 In the case of coordinated collection the retailprices 119901
119862
119894and 119901
119862
119895are strictly monotonically decreasing with
respect to the recycling substitution effect
Proof According to (7) the first-order derivative of 119901119862119894with
respect to 120574 is
d119901119862119894
d120574= (1199042[1198832
1(Φ119894 minus Φ3minus119894)
minus1198832
2(Φ119894+ Φ3minus119894
+ (1 minus 120573) (Δ119904 minus 2119888119898))])
times (1198832
11198832
2)minus1
(15)
where1198831= 1199042(1minus120573)minus4(1minus120574) and119883
2= minus4(1minus120574)minus119904
2(1+120573)
It follows from the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt
120573 lt 1 that d119901119862119894d120574 lt 0 thus the retail prices 119901
119862
119894and 119901
119862
119895are
strictly decreasing with respect to the recycling substitutioneffect
Abstract and Applied Analysis 5
Corollary 3 In the case of coordinated collection the repur-chase prices 119901
119862
119877119894and 119901
119862
119877119895are strictly increasing with respect to
the recycling substitution effect
Proof According to (8) the first-order derivative of 119901119862119877119894is
d119901119862119877119894
d120574= 2119904 (119883
2
2[Φ119894+ Φ3minus119894
minus (2119888119898
minus Δ119904) (1 minus 120573)]
minus1198832
1(Φ119894minus Φ3minus119894
)) times (1198832
11198832
2)minus1
(16)
where1198831 = 1199042(1minus120573)minus4(1minus120574) and1198832 = minus4(1minus120574)minus119904
2(1+120573)
By assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1 wecan obtain that d119901119862119862
119895d120574 gt 0 thus the repurchase prices 119901
119862
119877119894
and 119901119862
119877119895are strictly increasing with respect to the recycling
substitution effect
Corollary 2 implies how the recycling substitution effectimpacts the profit of forward channel Specially themanufac-turer can obtain more profit from forward channel by induc-ing recycling substitution effect (select the lower retail priceto induce more recycling substitution effect) Corollary 3implies how the recycling substitution effect impacts theprofit of reverse channel themanufacturer can select optimalrepurchase price by inducing recycling substitution effect(inducing more recycling substitution effect to select thehigher retail price) in order to obtain more profit of reversechannel Corollaries 2 and 3 illustrate that the recyclingsubstitution effectrsquos impact on the profit of forward andreverse is converse so themanufacturer can induce a suitablerecycling substitution effect between the retailers to obtainthe maximum profit in forward and reverse supply chain
Proposition 4 In the case of coordinated collection the profitof forward channel is strictly increasing with respect to theretailing substitution effect and decreasing with respect to therecycling substitution effect While the profit of reverse channelis strictly decreasing with respect to the retailing substitutioneffect and increasing with respect to the recycling substitutioneffect
Proof First for the analysis of the profit in forward channelfor any given 119901119895 according to (1) the demand119863119894 is invariablefor the durable product and Φ119894 is constant for retailer 119894the retail price is increasing with respect to the retailingsubstitution effect Then it follows from (3) that the profitof forward channel sum2
119894=1(119901119894minus 119888119898)119863119894(119901119894 119901119895) is increasing with
respect to 119901119894and 119901
119895 Furthermore by Corollary 2 the retail
prices 119901119894and 119901
119895are both strictly decreasing with respect to
the recycling substitution effect Hence the profit of forwardchannel function sum
2
119894=1(119901119894minus 119888119898)119863119894(119901119894 119901119895) is decreasing with
respect to the recycling substitution effectSecond for the analysis of the profit in reverse channel
for any given 119901119877119894 since the recycling function119876
119894is invariable
for the durable product and 120601119894= 119904119863119894and 119901
119877119894are invariable
for retailer 119894 by (2) the repurchase price is increasing withrespect to recycling substitution effect Then the profit ofreverse channelsum2
119894=1(Δ minus 119901
119877119894)119876119894(119901119877119894 119901119877119895
) in (3) is decreasing
with respect to 119901119877119894
and 119901119877119895 According to Corollary 3 the
repurchase prices 119901119877119894
and 119901119877119895
are strictly increasing withrespect to the recycling substitution effect Therefore theprofit of reverse channel function sum
2
119894=1(Δ minus 119901
119877119894)119876119894(119901119877119894 119901119877119895
)
is decreasing with the recycling substitution effect
From what is discussed above in the case of coordi-nated collection we can obtain some managerial insights inpractice It is illustrated that the variation tendency of theforward channel profit is decided by the difference betweenthe increase of forward channel profit from the retailingsubstitution effect and the decrease of forward channelrsquos profitstemmed from the recycling substitution effect Similarly thevariation of reverse channel profit is decided by the differencebetween the increase of reverse channel profit for the retail-ing substitution effect and the decrease of reverse channelprofit for the recycling substitution effect Additionally thevariation of the total profit of forward and reverse channel isuncertain Furthermore the manufacturer can induce lowerretailing substitution effect and higher recovery substitutioneffect to collect more obsolete products and increase theprofit of reverse channel In other words the manufacturercan easily achieve the goal of remanufacturing strategy byrecovering the residual value of used products
32 Decentralized Collection In the case of decentralizedcollection the manufacturer decides on the wholesale price119908 and reimburses each retailer a fixed fee 119887 per unit returnedand the retailers compete with each other and decide on theretail price of the product as well as the repurchase price ofcollection used product
We can obtain retailer 119894rsquos reaction function by solving thefollowing optimization problem
max119901119894 119901119895
Π119863
119894= (119901119894 minus 119908)119863119894 (119901119894 119901119895) + (119887 minus 119901119877119895)119876119877119894 (119901119877119894 119901119877119895)
(17)
Proposition 5 In the case of decentralized collection theretailerrsquos optimal reaction retail price 119901
lowast119863
119894and the optimal
reaction repurchase price 119901lowast119863
119877119894satisfy
119901lowast119863
119894=
119887119904 (1 minus 120574) minus 119908 (2 minus 120574)
11988411
+(Φ119894+ Φ3minus119894
) 1198841
211988411
+(Φ119894minus Φ3minus119894
) 1198851
211988511
(18)
119901lowast119863
119877119894=
119904 (Φ119894+ Φ3minus119894
)
211988411
minus119904 (Φ119894minus Φ3minus119894
)
211988511
+ 119887[1199042(1 minus 120574)
119884111988411
+
(1 minus 1199042)1198851
119879]
minus 119908119904 [1198851
119879+ (2 minus 120574)]
(19)
where 1198841
= 1199042+ 120574 minus 2 119884
11= 1199042minus 2(2 minus 120574) + 120573(2 minus 119904
2minus 120574)
1198851 = 2 + 120574 minus 119904
2 11988511 = 2(2 + 120574) minus 120573(2 + 1199042minus 120574) minus 119904
2 and119879 = 120574
2minus 1199044minus 4(1 minus 119904
2)
6 Abstract and Applied Analysis
Proof The second-order partial derivatives of retailerrsquos profitΠ119863
119894are
1205972Π119863
119894
1205971199012
119894
= minus21205972Π119863
119894
120597119901119894120597119901119895
= 119904
1205972Π119863
119894
120597119901119895120597119901119894
= 1199041205972Π119863
119894
1205971199012
119895
= minus119904120573
(20)
Then the Hessian matrix is
|119867| =
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1205972Π119863
119894
1205971199012
119894
1205972Π119863
119894
120597119901119894120597119901119895
1205972Π119863
119894
120597119901119895120597119901119894
1205972Π119863
119894
1205971199012
119895
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(21)
By using the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt
120573 lt 1 we can obtain that |1198631| = 120597
2Π119863
1198941205971199012
119894= minus2 lt 0
and |1198632| = 2119904(120573 minus 119904) gt 0 So 119867 is negative definite Π119863 is
jointly concave with respect to 119901119894and 119901
119895 Thus the optimal
retailing and repurchase price can be obtained by the first-order condition as follows
According to (17) for retailer 1 the first-order conditionsof Π119863119894are
120597Π119863
119894
120597119901119894
= Φ119894 minus 2119901119894 + 120573119901119895 + 119908 minus 119904 (119887 minus 119901119877119894) = 0 (22)
120597Π119863
119894
120597119901119877119894
= minus119904 (Φ119894 minus 119901119894 + 120573119901119895) minus 119901119877119894 + 120574119901119877119895 + 119887 minus 119901119877119894 = 0
(23)
Similarly for retailer 2 the first-order partial conditions ofΠ119863
119895are
120597Π119863
119895
120597119901119895
= Φ119895minus 2119901119895+ 120573119901119894+ 119908 minus 119904 (119887 minus 119901
119877119895) = 0 (24)
120597Π119863
119895
120597119901119877119895
= minus119904 (Φ119895 minus 119901119895+ 120573119901119894) minus 119901
119877119895+ 120574119901119877119894
+ 119887 minus 119901119877119895
= 0
(25)
By combining (22) (23) (24) and (25) we can obtain theoptimal retail prices 119901119894 and 119901119895 and optimal repurchase prices119901119877119894 and 119901119877119895
Corollary 6 The retail price is increasing with respect to thewholesale price 119908 and decreasing with respect to the buy-backpayment 119887 But the repurchase price is decreasing with respectto the wholesale price119908 and increasing with respect to the buy-back payment 119887
Proof By the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1we can easily obtain119884
1= 1199042+120574minus2 lt 0119884
11= 1199042minus2(2minus120574)+120573(2minus
1199042minus120574) lt 0119885
1= 2+120574minus119904
2gt 0 and 119879 = 120574
2minus1199044minus4(1minus119904
2) lt 0
Thus according to (18) we can obtain that the retail prices119901119894and 119901
119895are increasing with respect to the wholesale price
119908 while decreasing with respect to the buy-back payment 119887And it follows from (19) that the repurchase prices 119901
119877119894and
119901119877119895are decreasingwith respect to thewholesale price119908 while
increasing with respect to the buy-back payment 119887
Substituting (18) and (19) into (17) yields
max119908119887
Π119863
119898= (119908 minus 119888119898)
2
sum
119894=1
119863119894 (119901lowast119863
119894 119901lowast119863
119895)
+ (Δ minus 119887)
2
sum
119894=1
119876119894(119901lowast119863
119877119894 119901lowast119863
119877119895)
(26)
Proposition 7 In the case of decentralized collection themanufacturerrsquos optimal wholesale price 119908
lowast and optimal buy-back payment 119887lowast satisfy
119908lowast
=(Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701 [1198701 (120573 + 1) + 4 (1 minus 120574)]
+1198703(1 minus 120574)
1198701(2 minus 120574)
(27)
119887lowast
=2Δ1198731+ 1205741198732+ 120573120574 (2119888
119898119904 minus 4Δ)
4120573 (2 minus 1199042) + 2120574 (8 minus 119904
2) minus 2 (2 + 120573120574) (4 minus 119904
2) (28)
where 1198701 = 4(120574 minus 1) + 1199042(2 minus 120574) 1198702 = (2119888119898 minus Δ119904)(1 minus 120574)
1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873
1= 120573(2 minus 119904
2) minus (4 minus 119904
2) and
1198732= 8Δ + (Φ
1+ Φ2minus 2119888119898)119904
Proof The second-order partial derivatives of Π119863119898are
1205972Π119863
119898
1205971199082
=41199042(2 minus 120574) (1 minus 120573) (1 minus 120574)
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
1205972Π119863
119898
1205971198872
=41199042(2 minus 120574) (1 minus 120574)
2
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
1205972Π119863
119898
120597119908120597119887
=1205972Π119863
119898
120597119887120597119908
=41199042(2 minus 120574)
2[1 minus 120573 + 2119904 (1 minus 120574) minus (2 minus 120573) (2 minus 120574)]
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
10038161003816100381610038161198671198981003816100381610038161003816 =
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1205972Π119863
119898
1205971199082
1205972Π119863
119898
120597119908120597119887
1205972Π119863
119898
120597119887120597119908
1205972Π119863119862
119898
1205971198872
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(29)
By the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1 wecan easily obtain that119867
119898is negative definite soΠ
119863
119898is jointly
concave with respect to the wholesale price 119908 and buy-back
Abstract and Applied Analysis 7
payment 119887Thus the optimal wholesale price119908 and buy-backpayment 119887 can be obtained by the first-order condition
According to (26) the first-order conditions of Π119863119898with
respect to 119908 and 119887 are as follows
120597Π119863
119898
120597119908
=2 (2 minus 120574) (2119908 minus 119887119904 + 119888
119898) + 2Δ119904
1199042+ 120574 minus 2
+ ((120574 minus 2) [2119887119904 (1 minus 120574) + (Φ1+ Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)
+ 2 (119888119898
minus 119908) (2 minus 120574 + 2119904 (119887 minus Δ)) ])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(30)
120597Π119863
119898
120597119887
=2 (1 minus 120574) (2119887 minus Δ + 119888
119898119904) minus 2119908119904 (2 minus 120574)
1199042+ 120574 minus 2
+ (119904 [2119887119904 (1 minus 120574) + (Φ1 + Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)]
+ 2119904 (1 minus 120574) [(119888119898
minus 119908) (2 minus 120574) + 119904 (119887 minus Δ)])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(31)
By combining (30) and (31) we can obtain the optimal retailwholesale price119908lowast and buy-back payment 119887lowast that is (27) and(28) hold
Proposition 8 In the case of decentralized collection theoptimal retail 119901lowast119863
119894and optimal repurchase price 119901
lowast119863
119877119894of retailer
119894 satisfy
119901lowast119863
119894
=(Φ119894 minus Φ3minus119894)
2[
1198851
11988511
minus1198841
11988411
]
+119904 (1 minus 120574)
211988411
[2Δ1198721 + 1205741198722 + 2120573120574 (119888119898119904 minus 2Δ)
21198721+ 120574119872
2
]
minus(2 minus 120574)
11988411
[(Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(1 + 120573) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
]
119901lowast119863
119877119894
= ((1 minus 120574) 119904
2
119884111988411
minus
(1 minus 1199042)1198851
119879)
times (2Δ1198721 + 1205741198723 + 120573120574 (2119888119898119904 minus 4Δ)
21198721+ 120574119872
2
) minus 119904 [(2 minus 120574) +1198851
119879]
times ((Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(120573 + 1) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
) + [119904 (Φ119894 + Φ3minus119894)
2](
1
11988411
minus1
11988511
)
(32)
where 1198721
= 120573(2 minus 1199042) minus (4 minus 119904
2) 1198722
= (8 minus 1199042) minus 120573(4 minus 119904
2)
1198723
= 8Δ + (Φ1+ Φ2minus 2119888119898)119904 1198701
= 4(120574 minus 1) + 1199042(2 minus 120574)
1198702= (2119888119898
minus Δ119904)(1 minus 120574) 1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873
1=
120573(2minus1199042)minus(4minus119904
2)1198732= 8Δ+(Φ
1+Φ2minus2119888119898)1199041198841= 1199042+120574minus2
11988411
= 1199042minus 2(2 minus 120574) + 120573(2 minus 119904
2minus 120574) 119885
1= 2 + 120574 minus 119904
2 11988511
=
2(2 + 120574) minus 120573(2 + 1199042minus 120574) minus 119904
2 and 119879 = 1205742minus 1199044minus 4(1 minus 119904
2)
Proof By substituting (27) and (28) into (18) and (19) respec-tively we can obtain the optimal retail price and repurchaseprice of retailer 119894
Proposition 9 In the case of decentralized collection themanufacturerrsquos profit of forward channel is strictly increasingwith respect to the retailing substitution effect and decreasingwith respect to the recycling substitution effect while themanufacturerrsquos reverse channelrsquos profit is increasingwith respectto the retailing substitution effect and decreasing with respect tothe recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in forward channelAccording to (1) the demand of retailing 119863
119894is invariable
for the durable product and Φ119894and 119901
119895are invariable for
retailer 119894 thus the retail price is increasing with respectto the retailing substitution effect that is increasing theretailing substitution effect 120573 will induce the increase ofretail price 119901119894 and according to Corollary 6 it will inducethe increase of wholesale price 119908 According to (4) themanufacturerrsquos profit in forward channel (119908minus119888119898)(119863119894(119901119894 119901119895)+
119863119895(119901119895 119901119894)) is increasing with respect to retailing substitutioneffect Similarly we can easily obtain that the manufacturerrsquosprofit in forward channel (119908 minus 119888119898)(119863119894(119901119894 119901119895) + 119863119895(119901119895 119901119894)) isdecreasing with respect to recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in reverse channelAccording to (2) the recycling function 119876
119894is invariable for
the durable product and 120601119894
= 119904119863119894and 119901
119877119895are invariable
8 Abstract and Applied Analysis
for retailer 119894 thus the repurchase price is increasing withrespect to recycling substitution effect Then we can easilyobtain that the increase of retailing substitution effect 120573
will induce the increase of retail price 119901119894 and according
to Corollary 6 it will induce the decrease of buy-backpayment 119887 According to (4) the manufacturerrsquos reversechannel (Δ minus 119887)(119876119894(119901119877119894 119901119877119895) + 119876
119895(119901119877119895
119901119877119894)) is increasing
with respect to retailing substitution effect Similarly we caneasily get that the manufacturerrsquos reverse channel profit (Δ minus
119887)(119876119894(119901119877119894 119901119877119895) + 119876119895(119901119877119895 119901119877119894)) is decreasing with respect torecycling substitution effect
It is illustrated that in the case of decentralized collectionthe variation of manufacturerrsquos profit in forward or reversechannel is decided by the difference between the increase offorward or reverse channel profit for the retailing substitutioneffect and the decrease of forward and reverse channelprofit for the recycling substitution effect Additionally thechange of closed-loop supply chainrsquos profit with respect tothe retailing and recycling substitution effects is uncertainin the case of decentralized collection On the other handthe manufacturer can induce higher retailing substitutioneffect and lower recovery substitution effect via choosingsuitable wholesale price and buy-back payment to collectmore obsolete products and increase the profit of reversechannel It means that the manufacturers can achieve theirgoal of remanufacturing strategy by recovering the residualvalue of used products
Proposition 10 In the case of decentralized collection theretailerrsquos profit of forward channel is strictly increasing withrespect to the retailing substitution effect and decreasing withrespect to the recycling substitution effect while his reversechannel profit is decreasing with respect to the retailing substi-tution effect and recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in forward channel Accordingto (1) the demand of retailing119863
119894is invariable for the durable
product and Φ119894and 119901
119895are invariable for retailer 119894 thus
the retail price is increasing with respect to the retailingsubstitution effect that is the increase of 120573 will induce theincrease of retail price 119901
119894 According to (5) each retailerrsquos
profit in forward channel (119901119894minus 119908)119863
119894(119901119894 119901119895) is increasing
with respect to the retailing substitution effect similarly forretailer 119895 We can easily obtain that each retailerrsquos profit inreverse channel (119901119894 minus 119908)119863119894(119901119894 119901119895) is increasing with respectto recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in reverse channel Accordingto (2) the recycling function119876119877119894 is invariable for the durableproduct and 120601119894 = 119904119863119894 and 119901119877119895 are invariable for retailer119894 thus the repurchase price is increasing with respect torecycling substitution effect Then we can easily obtain thatthe repurchase price is increasing with respect to recyclingsubstitution effect So the increase of retailing substitutioneffect 120573 will induce the increase of retail price 119901
119894 induce
the increase of buy-back payment 119887 and then will inducethe decrease of 119901
119877119894 According to (5) the retailer 119894rsquos profit in
reverse channel (119887minus119901119877119894)119876119894(119901119877119894 119901119877119895
) is decreasingwith respectto retailing substitution effect Similarly we can easily get thatthe retailer 119894rsquos profit in reverse channel (119887 minus119901
119877119894)119876119894(119901119877119894 119901119877119895
) isincreasing with respect to recycling substitution effect
Proposition 10 illustrates that in the case of decentralizedcollection the variation of retailerrsquos profit in forward channelis decided by the difference between the increase of forwardchannelrsquos profit for the retailing substitution effect and thedecrease of forward channel profit for the recycling substitu-tion effect While for the retailerrsquos profit in reverse channelit is decreasing with respect to retailing substitution effectand recycling substitution effect Additionally because thechange of forward channel profit is uncertain the variationof the closed-loop supply chainrsquos profit is also uncertain Inthis case the retailer can induce lower retailing substitutioneffect and recovery substitution effect via choosing suitablewholesale price and buy-back payment to collect moreobsolete products and increase the profit of reverse channel
In a word the variations of manufacturer and retailerrsquosrespective profit with the retailing substitution effect inforward channel are uniform but in the reverse channelthey are inconsistent For example the higher recyclingsubstitution effect will induce the manufacturer to get moreprofits in reverse channel (ie augments the scale of theman-ufacturerrsquos reverse supply chains) But the higher recyclingsubstitution effect will cause the retailer to obtain less profitsin reverse channel (ie cut down the scale of the retailerrsquosreverse supply chains) There is a contradiction betweenthe profits of the manufacturer and the retailer in reversechannel because the manufacturer has sufficient channelpower over the retailers to act as a Stackelberg leader thus themanufacturer can choose the appropriate wholesale price andbuy-back payment to achieve the remanufacturing strategyand balance the retailerrsquos profit And we make a comparisonwith the coordinated collection we can easily know thatthe coordinated collection is more effective to augment thescale of the whole reverse supply chains than decentralizedcollection
4 Numerical Examples
In this section numerical examples are provided to comparethe results obtained in the above two different collectionmodels and to analyze the impacts of retailing and recyclingsubstitution effects on the decisions and profits of chainmembers Some managerial insights can be derived throughthe numerical analysis The numerical analysis is performedfor Φ1= 100 Φ
2= 50 119888
119898= 20 Δ = 10 120574 = 04 or 120574 = 02
and 119904 = 01
41 Numerical Analysis of Model 119862 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofcoordinated collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect to retail-ing substitution effect that is the higher retailing substitution
Abstract and Applied Analysis 9
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
120574 = 02
120574 = 04
pi
(a) Impact on the retail price
0
5
10
15
20
25
30
35
40
45
50
pRi
0 02 04 06 08 1120573
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 2 Impact of the retailing and recycling substitution effects on price in model 119862
effect will induce the increase of retail price Moreover thehigher recycling substitution effect will induce the increaseof retail price the higher recycling substitution effect willinduce the decrease of repurchase price In addition thehigher retailing substitution effect will induce the decrease ofretail price The green and blue lines in Figures 2(a) and 2(b)give us a visual presentation
From the previous assumptions and analysis we knowthat the profit of forward channel is strictly increasing withrespect to retailing substitution effect and decreasing withrespect to recycling substitution effect that is the manu-facturer can induce higher retailing competition and lowerrecycling competition to obtain more profit from forwardchannel
Andwe known that the profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andincreasing with respect to recycling substitution effect thatis the manufacturer can induce lower recycling substitutioneffect to obtain more profit in the reverse channel In otherwords themanufacturer can induce higher retailing substitu-tion effect and lower recycling substitution effect could obtainmore profit in closed-loop supply chain The green and bluelines in Figures 3(a) and 3(b) give us a visual presentation
42 Numerical Analysis of Model 119863 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofdecentralized collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect toretailing substitution effect that is the higher retailing substi-tution effect will induce the increase of retail price And the
higher recycling substitution effect will induce the decreaseof retail price And we known that the repurchase priceis strictly decreasing with respect to recycling substitutioneffect that is the higher recycling substitution effect willinduce the increase of repurchase price while the higherretailing substitution effect will induce the decrease of retailprice The green and blue lines in Figures 4(a) and 4(b) giveus a visual presentation
According to the assumptions and the analysis weknow that the manufacturerrsquos profit of forward channel isstrictly increasing with respect to retailing substitution effectand decreasing with respect to recycling substitution effectAnd the manufacturerrsquos profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andrecycling substitution effect In other words in the case ofdecentralized collection the manufacturer can induce higherretailing competition and lower recycling competition toobtain more profit in the forward channel and induce lowerrecycling substitution effect to obtain more profit in thethe reverse channel So the manufacturer can select higherretailing substitution effect and lower recycling substitutioneffect to obtain more profit in their closed-loop supply chainMoreover the higher recycling competition will induce thelower profit of forward channel and induce lower profit ofreverse channel
For the forward channel profit of retailers is strictlyincreasing with respect to retailing substitution effect anddecreasing with respect to recycling substitution effect Andthe retailersrsquo profit of reverse channel is strictly decreasingwith respect to retailing substitution effect and increasingwith respect to recycling substitution effect In other wordsin the case of decentralized collection for the retailers thehigher retailing competition and lower recycling competition
10 Abstract and Applied Analysis
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(a) Impact on the profit in forward channel
0 02 04 06 08 10
200
400
600
800
1000
1200
1400
1600
1800
2000
120573
120574 = 02
120574 = 04120587
(b) Impact on the profit in reverse channel
Figure 3 Impact of the retailing and recycling substitution effects on channel profit in model 119862
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
pi
120574 = 02
120574 = 04
(a) Impact on the retail price
0 02 04 06 08 1minus20
minus15
minus10
minus5
0
5
10
15
20
25
30
120573
pi
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 4 Impact of the retailing and recycling substitution effects on price in model 119863
can make them obtain more profit in their forward channelbut the lower retailing substitution effect and higher recyclingsubstitution effect can make them obtain more profit in thereverse channel So the retailers tend to select suitable retailprice and repurchase price to induce retailing and recyclingcompetitions (not too high or too low the higher retailsubstitution effect will make the profit of reverse channel
negative or the lower recycling substitution effect will makethe profit of forward channel decrease) to obtain more profitin their closed-loop supply chain The lines in Figure 5 giveus a visual presentation
43 Comparison of Model119862 andModel119863 In this subsectionwe discuss the impact of retailing substitution effect on the
Abstract and Applied Analysis 11
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120587
(a) Impact on the manufacturerrsquos profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120587
(b) Impact on the manufacturerrsquos profit in reverse channel
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(c) Impact on the retailersrsquo profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120574 = 02
120574 = 04
120587
(d) Impact on the retailersrsquo profit in reverse channel
Figure 5 Impact of the retailing and recycling substitution effects on channel profit in model 119863119862
optimal prices and optimal channel profits and comparethem in the case of coordinated collection with that ofdecentralized collection
From the previous assumptions and analysis we knowthat the retail price in the case of coordinated collection islower than that in decentralized collection that is facingthe same circumstances the retail price in the case ofdecentralized collection is higher than that in coordinatedcollection And the repurchase price in the case of decentral-ized collection is lower than that in coordinated collectionIt implies that for the retailer the coordinated collectionhas more price advantages (lower retail price and higherrepurchase price) than the decentralized collection in the
closed-loop channel The lines in Figure 6 give us a visualpresentation
In what is discussed above we know that the channelprofit in model 119862 is higher than that in model 119863 Theprofit of reverse channel in model 119863 is very low even thevalue is negative In this case the manufacturer and retailerstend to select coordinated collection to obtain more profitsfor example HP Xerox and Apple select the coordinatedcollection to collect their used product The lines in Figure 6give us a visual presentation
44 Impact of Retailing and Recycling Substitution Effects onClosed Supply Chainrsquos Profit In this subsection we discuss
12 Abstract and Applied Analysis
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
Model CCModel DC
120573
pi
(a) Comparison of retail price
minus100
minus50
0
50
100
150
200
pRi
0 02 04 06 08 1
Model CCModel DC
120573
(b) Comparison of repurchase price
Figure 6 Pricing decisions of coordinated collection versus decentralized collection
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
Model CCModel DC
times104
120573
120587
(a) Comparison of forward channel profits
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
Model CCModel DC
120573
120587
(b) Comparison of reverse channel profits
Figure 7 The profit of coordinated collection versus decentralized collection
the impact of retailing and recycling substitution effects onthe optimal forward channel profits and optimal reversechannel profits in the cases of coordinated collection anddecentralized collection
From the previous assumptions and analysis weknow that the recycling substitution effectrsquos impact on theclosed-loop supply channel profit in the case of coordinated
collection is more obvious than that in decentralizedcollection In other words the manufacturer can obtainmore profit than in decentralized collection more easily bychanging the recycling substitution effect The reason maybe as follows in the case of coordinated collection the retailprice and repurchase price are made by the center planner(ie manufacturer) this collection model can adjust the
Abstract and Applied Analysis 13
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(a) Impact on the closed-loop supply channelrsquos profit in model 119862119862
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(b) Impact on the closed-loop supply channelrsquos profit in model119863119862
Figure 8 Impact of the retailing and recycling substitution effects on closed-Loop supply chainrsquos profit
retail and repurchase price fast and accurately But in caseof decentralized collection there is a pricing game betweenmanufacturer and retailers The manufacturer firstly selectsoptimal wholesale price and buy-back payment then theretailers select their optimal reaction retail price and repur-chase price that is the pricing game and pricing decision areunsynchronized between the manufacturer and retailer sothe manufacturer can obtain more profits in the case of coor-dinated collection than the case of decentralized collectionThe lines in Figures 7 and 8 give us a visual presentation
5 Conclusion
In this paper we investigated a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers Theresults demonstrate that more intense retailing competitioninduces the increase of the forward and reverse profitsof manufacturer and the forward channel of retailers anddecrease of the retailers profit in reverse channel while moreintense recycling competition induces the decrease of theprofits of the manufacturer and retailers in forward andreverse channels while the whole supply chainrsquos profit ofmodel 119862 is increasing with the respect to the free recyclingproportional coefficient but the effect on model 119863 is uncer-tain The results have important managerial insights for themanufacturer and retailers
This paper makes a contribution to the literature onreverse channel choice and coordination by drawing atten-tion to closed-loop supply chains with competition of retail-ing and recycling Our future research will be as follows ourmodel does not consider any fixed costs of retailingrecyclinga collection system If such costs were incurred a minimumnumber of retailingreturns would be required to justify thecost of operations Last but not least our cost formulation
does not include any operationalcapacity constraints inthe channel of products collection Operationalcapacityconstraint on the channel of collection can be incorporatedinto the model by defining an upper bound on the numbercollected via each channel Another possibility is to modelthe collection cost as a quadratic function of the quantityreturned
Notations
120574 Recycling substitution effect120573 Retailing substitution effectΦ Market size of the retailers in the forward
channel120601 Market size of the retailers by free recovering
in the reverse channel120575 Unit saving cost by remanufacturing119904 Recovering coefficient119888119903 Unit cost of remanufacturing a returned
product into a new one119888119898 Unit cost of manufacturing a new product
119908 New productrsquos wholesale price of themanufacturer
119887 Obsolete productrsquos repurchase price from theretailers to the manufacturer
119863119894(119901119894 119901119895) Demand function of retailer 119894
119876119894(119901119877119894 119901119877119895
) Recycling function of retailer 119894
119901119862
119894 119901119863
119895 Retailer 119894 and 119895rsquos respective retail price in
models 119862 and 119863
119901119862
119877119894 119901119863
119877119895 Retailer 119894 and 119895rsquos repurchase price in models
119862 and 119863
Π119870
119894 Profit function for channel member 119894 in
model 119896 Superscript 119896 takes the values of 119862and 119863 Subscript 119894 takes the values of 119898 119903and 119890 denoting the manufacturer theretailer and the 119890-tailer respectively
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
Manufacturer
Retailer 2Retailer 1
Customer
(a) Coordinated collection (model 119862)
Manufacturer
Retailer 2 Retailer 1
Customer
Competing
(b) Decentralized collection (model119863)
Figure 1 Closed-loop supply channel structures
and remanufactured products to the retailer at the samewholesale price119908 who in turn sells both products at the sameprice 119901 on the market [23] Let 119888
119898and 119888119903denote the unit
cost of manufacturing a new product and remanufacturing areturned product into a new one respectively Assume that119888119903
lt 119888119898 which implies that the manufacturer can make
savings from remanufacturing Let Δ = 119888119898
minus 119888119903denote the
unit saving cost by remanufacturingWe also add a Notationssection to help the authors to understand and distinguish thenotations
Consistent with the existing literature [24] we assumethat retailer 119894 faces the following linear demand function inthe forward channel
119863119894(119901119894 119901119895) = Φ
119894minus 119901119894+ 120573119901119895 0 le 120573 lt 1 (1)
where Φ119894represents the market size of retailer 119894 119901
119894is the
price of the product at retailer 119894 119901119895 denotes the competitiveretailerrsquos retail price and 120573 denotes the retailing substitutioneffect
We assume that retailer 119894 faces the following recyclingfunction in the reverse channel
119876119894(119901119877119894 119901119877119895
) = 120601119894+ 119901119877119894
minus 120574119901119877119895
0 le 120574 lt 1 (2)
where 120601119894 represents the market size of retailer 119894 by free recy-
cling119901119877119894 is the repurchase price of the used product at retailer119894 119901119877119895 denotes the competitive retailerrsquos repurchase price and120574 denotes the recycling product recycling substitution effect
Considering the relationship between the demand func-tions and recycling functions we assume 120601119894 = 119904119863119894(119901119894 119901119895)which denotes the market capacity which is recycled by theretailer freely where 119904 is a recycling coefficient and 0 le 119904 lt 1
[25] Ceteris paribus the profit of retailing a new product ismore than recycling obsolete products So we assume that theretailing substitution effect is more intense than the recyclingsubstitution effect that is 0 le 120574 lt 120573 lt 1 and 119904 is lower thanthe retailing substitution effect that is 0 le 119904 lt 120573 lt 1
According to the above assumptions and notations thechannelrsquos profit function in the case of coordinated collectionis
Π119862(119901119894 119901119895 119901119877119894 119901119877119895
)
=
2
sum
119894=1
((119901119894minus 119888119898)119863119894(119901119894 119901119895) + (Δ minus 119901
119877119894) 119876119894(119901119894 119901119877119895
))
(3)
wheresum2
119894=1(119901119894minus119888119898)119863119894(119901119894 119901119895) denotes the profit of the forward
channel Similarlysum2119894=1
(Δminus119901119877119894)119876119894(119901119877119894 119901119877119895
) denotes the profitof the reverse channel
Themanufacturer profit function in the case of decentral-ized collection is
Π119863
119898(119901119894 119901119895 119901119877119894 119901119877119895)
=
2
sum
119894=1
((119908 minus 119888119898)119863119894 (119901119894 119901119895) + (Δ minus 119887)119876119894 (119901119877119894 119901119877119895))
(4)
where 119908 represents the unit wholesale price and 119887 denotesthe unit price of a returned product from the retailerto the manufacturer (119908 minus 119888119898) sum
2
119894=1119863119894(119901119894 119901119895) denotes the
manufacturerrsquos profit in the forward channel Similarly (Δ minus
119887)sum2
119894=1119876119894(119901119877119894 119901119877119895
) represents themanufacturerrsquos profit in thereverse channel
Each retailerrsquos profit function in the case of decentralizedcollection is
Π119863
119894= (119901119894minus 119908)119863
119894(119901119894 119901119895) + (119887 minus 119901
119877119894) 119876119894(119901119877119894 119901119877119895
) (5)
where (119901119894minus 119908)119863
119894(119901119894 119901119895) denotes the profit of retailer 119894 in the
forward channel and (119887 minus 119901119877119894)119876119894(119901119877119894 119901119877119895
) denotes the profitof retailer 119894 in the reverse channel
4 Abstract and Applied Analysis
3 Model Formulation and Solution
In this section two equilibrium models are discussed in thecases of coordinated collection and decentralized collectionrespectively
31 Coordinated Collection According to (3) the coordi-nated collection model can be formulated as follows
max119901119894119901119895119901119877119894119901119877119895
Π119862(119901119894 119901119895 119901119877119894 119901119877119895
)
=
2
sum
119894=1
((119901119894minus 119888119898)119863119894(119901119894 119901119895) + (Δ minus 119901
119877119894) 119876119894(119901119877119894 119901119877119895
))
(6)
Proposition 1 In the case of coordinated collection theoptimal retail 119901119862
119894and the optimal repurchase price 119901
119862
119895satisfy
119901lowast119862
119894=
(Φ119894+ Φ119895120573)
2 (1 minus 1205732)
+4 (Δ119904 minus 2119888
119898) (1 minus 120574) + 119904
2(Φ119894+ Φ3minus119894
)
41198831
minus1199042(Φ119894 minus Φ3minus119894)
41198832
(7)
119901lowast119862
119877119894=
Δ
2+ s[
(Φ119894+ Φ3minus119894
) minus (2119888119898
minus Δs) (1 minus 120573)
21198831
minus(Φ119894minus Φ3minus119894
)
21198832
]
(8)
where1198831 = 1199042(1minus120573)minus4(1minus120574) and1198832 = minus4(1minus120574)minus119904
2(1+120573)
Proof According to (3) the second-order partial derivativesof Π119862(119901
119894 119901119895 119901119877119894 119901119877119895
) are
1205972Π119862
1205971199012
119894
= minus21205972Π119862
120597119901119894120597119901119895
= 21205731205972Π119862
120597119901119894120597119901119877119894
= 119904
1205972Π119862
120597119901119894120597119901119877119895
= minus119904120573
1205972Π119862
120597119901119895120597119901119894
= 21205731205972Π119862
1205971199012
119895
= minus21205972Π119862
120597119901119895120597119901119877119894
= minus119904120573
1205972Π119862
120597119901119895120597119901119877119895
= 119904
1205972Π119862
120597119901119877119894120597119901119894
= 1199041205972Π119862
120597119901119877119894120597119901119895
= minus1199041205731205972Π119862
1205971199012
119877119894
= minus2
1205972Π119862
120597119901119877119894120597119901119877119895
= 2120574
1205972Π119862
120597119901119877119895
120597119901119894
= minus1199041205731205972Π119862
120597119901119877119895
120597119901119895
= 1199041205972Π119862
120597119901119877119895
120597119901119877119894
= 2120574
1205972Π119862
1205971199012
119877119895
= minus2
(9)
Then the Hessian matrix is
|119867| =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus2 2120573 119904 minus119904120573
2120573 minus2 minus119904120573 119904
119904 minus119904120573 minus2 2120574
minus119904120573 119904 2120574 minus2
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(10)
By using the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt
1 we can obtain that |1198631| = minus2 lt 0 |119863
2| = 4(1 minus 120573) gt 0
|1198633| = minus2(1 minus 120573
2)(4 minus 119904
2) lt 0 and |1198634| = 16 + 16(120574
2minus 1205732(1 minus
1205742))+8119904
2(1205732+120574120573(1minus120573
2)minus1)+119904
4(1minus1205732)2gt 0 So119867 is negative
definiteΠ119862 is jointly concavewith respect to119901119862
1198941199011198623minus119894
119901119862119895 and
119901119862
3minus119895 thus the optimal retailing and repurchase prices can be
obtained by the first-order conditions as follows
120597Π119862
120597119901119894
= Φ119894minus 2119901119894+ 2120573119901
119895+ 119904119901119877119894
minus 119904120573119901119877119895
+ (1 minus 120573) (119888119898
minus 119904Δ)
= 0
(11)
120597Π119862
120597119901119895
= Φ119895 minus 2119901119895 + 2120573119901119894 + 119904119901119877119894 minus 119904120573119901119877119895 + (1 minus 120573) (119888119898 minus 119904Δ)
= 0
(12)
120597Π119862
120597119901119877119894
= minus 119904Φ119894+ 119904119901119894minus 119904120573119901
119895minus 2119901119877119894
+ 2120574119901119877119895
+ (1 minus 120574) Δ
= 0
(13)
120597Π119862
120597119901119877119895
= minus 119904Φ119895+ 119904119901119895minus 119904120573119901
119894minus 2119901119877119895
+ 2120574119901119877119894
+ (1 minus 120574) Δ
= 0
(14)
By combining (11) (12) (13) and (14) we can obtainthe optimal retail prices 119901
lowast
119894and 119901
lowast
119895 and optimal repurchase
prices 119901lowast
119877119894and 119901
lowast
119877119895
Corollary 2 In the case of coordinated collection the retailprices 119901
119862
119894and 119901
119862
119895are strictly monotonically decreasing with
respect to the recycling substitution effect
Proof According to (7) the first-order derivative of 119901119862119894with
respect to 120574 is
d119901119862119894
d120574= (1199042[1198832
1(Φ119894 minus Φ3minus119894)
minus1198832
2(Φ119894+ Φ3minus119894
+ (1 minus 120573) (Δ119904 minus 2119888119898))])
times (1198832
11198832
2)minus1
(15)
where1198831= 1199042(1minus120573)minus4(1minus120574) and119883
2= minus4(1minus120574)minus119904
2(1+120573)
It follows from the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt
120573 lt 1 that d119901119862119894d120574 lt 0 thus the retail prices 119901
119862
119894and 119901
119862
119895are
strictly decreasing with respect to the recycling substitutioneffect
Abstract and Applied Analysis 5
Corollary 3 In the case of coordinated collection the repur-chase prices 119901
119862
119877119894and 119901
119862
119877119895are strictly increasing with respect to
the recycling substitution effect
Proof According to (8) the first-order derivative of 119901119862119877119894is
d119901119862119877119894
d120574= 2119904 (119883
2
2[Φ119894+ Φ3minus119894
minus (2119888119898
minus Δ119904) (1 minus 120573)]
minus1198832
1(Φ119894minus Φ3minus119894
)) times (1198832
11198832
2)minus1
(16)
where1198831 = 1199042(1minus120573)minus4(1minus120574) and1198832 = minus4(1minus120574)minus119904
2(1+120573)
By assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1 wecan obtain that d119901119862119862
119895d120574 gt 0 thus the repurchase prices 119901
119862
119877119894
and 119901119862
119877119895are strictly increasing with respect to the recycling
substitution effect
Corollary 2 implies how the recycling substitution effectimpacts the profit of forward channel Specially themanufac-turer can obtain more profit from forward channel by induc-ing recycling substitution effect (select the lower retail priceto induce more recycling substitution effect) Corollary 3implies how the recycling substitution effect impacts theprofit of reverse channel themanufacturer can select optimalrepurchase price by inducing recycling substitution effect(inducing more recycling substitution effect to select thehigher retail price) in order to obtain more profit of reversechannel Corollaries 2 and 3 illustrate that the recyclingsubstitution effectrsquos impact on the profit of forward andreverse is converse so themanufacturer can induce a suitablerecycling substitution effect between the retailers to obtainthe maximum profit in forward and reverse supply chain
Proposition 4 In the case of coordinated collection the profitof forward channel is strictly increasing with respect to theretailing substitution effect and decreasing with respect to therecycling substitution effect While the profit of reverse channelis strictly decreasing with respect to the retailing substitutioneffect and increasing with respect to the recycling substitutioneffect
Proof First for the analysis of the profit in forward channelfor any given 119901119895 according to (1) the demand119863119894 is invariablefor the durable product and Φ119894 is constant for retailer 119894the retail price is increasing with respect to the retailingsubstitution effect Then it follows from (3) that the profitof forward channel sum2
119894=1(119901119894minus 119888119898)119863119894(119901119894 119901119895) is increasing with
respect to 119901119894and 119901
119895 Furthermore by Corollary 2 the retail
prices 119901119894and 119901
119895are both strictly decreasing with respect to
the recycling substitution effect Hence the profit of forwardchannel function sum
2
119894=1(119901119894minus 119888119898)119863119894(119901119894 119901119895) is decreasing with
respect to the recycling substitution effectSecond for the analysis of the profit in reverse channel
for any given 119901119877119894 since the recycling function119876
119894is invariable
for the durable product and 120601119894= 119904119863119894and 119901
119877119894are invariable
for retailer 119894 by (2) the repurchase price is increasing withrespect to recycling substitution effect Then the profit ofreverse channelsum2
119894=1(Δ minus 119901
119877119894)119876119894(119901119877119894 119901119877119895
) in (3) is decreasing
with respect to 119901119877119894
and 119901119877119895 According to Corollary 3 the
repurchase prices 119901119877119894
and 119901119877119895
are strictly increasing withrespect to the recycling substitution effect Therefore theprofit of reverse channel function sum
2
119894=1(Δ minus 119901
119877119894)119876119894(119901119877119894 119901119877119895
)
is decreasing with the recycling substitution effect
From what is discussed above in the case of coordi-nated collection we can obtain some managerial insights inpractice It is illustrated that the variation tendency of theforward channel profit is decided by the difference betweenthe increase of forward channel profit from the retailingsubstitution effect and the decrease of forward channelrsquos profitstemmed from the recycling substitution effect Similarly thevariation of reverse channel profit is decided by the differencebetween the increase of reverse channel profit for the retail-ing substitution effect and the decrease of reverse channelprofit for the recycling substitution effect Additionally thevariation of the total profit of forward and reverse channel isuncertain Furthermore the manufacturer can induce lowerretailing substitution effect and higher recovery substitutioneffect to collect more obsolete products and increase theprofit of reverse channel In other words the manufacturercan easily achieve the goal of remanufacturing strategy byrecovering the residual value of used products
32 Decentralized Collection In the case of decentralizedcollection the manufacturer decides on the wholesale price119908 and reimburses each retailer a fixed fee 119887 per unit returnedand the retailers compete with each other and decide on theretail price of the product as well as the repurchase price ofcollection used product
We can obtain retailer 119894rsquos reaction function by solving thefollowing optimization problem
max119901119894 119901119895
Π119863
119894= (119901119894 minus 119908)119863119894 (119901119894 119901119895) + (119887 minus 119901119877119895)119876119877119894 (119901119877119894 119901119877119895)
(17)
Proposition 5 In the case of decentralized collection theretailerrsquos optimal reaction retail price 119901
lowast119863
119894and the optimal
reaction repurchase price 119901lowast119863
119877119894satisfy
119901lowast119863
119894=
119887119904 (1 minus 120574) minus 119908 (2 minus 120574)
11988411
+(Φ119894+ Φ3minus119894
) 1198841
211988411
+(Φ119894minus Φ3minus119894
) 1198851
211988511
(18)
119901lowast119863
119877119894=
119904 (Φ119894+ Φ3minus119894
)
211988411
minus119904 (Φ119894minus Φ3minus119894
)
211988511
+ 119887[1199042(1 minus 120574)
119884111988411
+
(1 minus 1199042)1198851
119879]
minus 119908119904 [1198851
119879+ (2 minus 120574)]
(19)
where 1198841
= 1199042+ 120574 minus 2 119884
11= 1199042minus 2(2 minus 120574) + 120573(2 minus 119904
2minus 120574)
1198851 = 2 + 120574 minus 119904
2 11988511 = 2(2 + 120574) minus 120573(2 + 1199042minus 120574) minus 119904
2 and119879 = 120574
2minus 1199044minus 4(1 minus 119904
2)
6 Abstract and Applied Analysis
Proof The second-order partial derivatives of retailerrsquos profitΠ119863
119894are
1205972Π119863
119894
1205971199012
119894
= minus21205972Π119863
119894
120597119901119894120597119901119895
= 119904
1205972Π119863
119894
120597119901119895120597119901119894
= 1199041205972Π119863
119894
1205971199012
119895
= minus119904120573
(20)
Then the Hessian matrix is
|119867| =
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1205972Π119863
119894
1205971199012
119894
1205972Π119863
119894
120597119901119894120597119901119895
1205972Π119863
119894
120597119901119895120597119901119894
1205972Π119863
119894
1205971199012
119895
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(21)
By using the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt
120573 lt 1 we can obtain that |1198631| = 120597
2Π119863
1198941205971199012
119894= minus2 lt 0
and |1198632| = 2119904(120573 minus 119904) gt 0 So 119867 is negative definite Π119863 is
jointly concave with respect to 119901119894and 119901
119895 Thus the optimal
retailing and repurchase price can be obtained by the first-order condition as follows
According to (17) for retailer 1 the first-order conditionsof Π119863119894are
120597Π119863
119894
120597119901119894
= Φ119894 minus 2119901119894 + 120573119901119895 + 119908 minus 119904 (119887 minus 119901119877119894) = 0 (22)
120597Π119863
119894
120597119901119877119894
= minus119904 (Φ119894 minus 119901119894 + 120573119901119895) minus 119901119877119894 + 120574119901119877119895 + 119887 minus 119901119877119894 = 0
(23)
Similarly for retailer 2 the first-order partial conditions ofΠ119863
119895are
120597Π119863
119895
120597119901119895
= Φ119895minus 2119901119895+ 120573119901119894+ 119908 minus 119904 (119887 minus 119901
119877119895) = 0 (24)
120597Π119863
119895
120597119901119877119895
= minus119904 (Φ119895 minus 119901119895+ 120573119901119894) minus 119901
119877119895+ 120574119901119877119894
+ 119887 minus 119901119877119895
= 0
(25)
By combining (22) (23) (24) and (25) we can obtain theoptimal retail prices 119901119894 and 119901119895 and optimal repurchase prices119901119877119894 and 119901119877119895
Corollary 6 The retail price is increasing with respect to thewholesale price 119908 and decreasing with respect to the buy-backpayment 119887 But the repurchase price is decreasing with respectto the wholesale price119908 and increasing with respect to the buy-back payment 119887
Proof By the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1we can easily obtain119884
1= 1199042+120574minus2 lt 0119884
11= 1199042minus2(2minus120574)+120573(2minus
1199042minus120574) lt 0119885
1= 2+120574minus119904
2gt 0 and 119879 = 120574
2minus1199044minus4(1minus119904
2) lt 0
Thus according to (18) we can obtain that the retail prices119901119894and 119901
119895are increasing with respect to the wholesale price
119908 while decreasing with respect to the buy-back payment 119887And it follows from (19) that the repurchase prices 119901
119877119894and
119901119877119895are decreasingwith respect to thewholesale price119908 while
increasing with respect to the buy-back payment 119887
Substituting (18) and (19) into (17) yields
max119908119887
Π119863
119898= (119908 minus 119888119898)
2
sum
119894=1
119863119894 (119901lowast119863
119894 119901lowast119863
119895)
+ (Δ minus 119887)
2
sum
119894=1
119876119894(119901lowast119863
119877119894 119901lowast119863
119877119895)
(26)
Proposition 7 In the case of decentralized collection themanufacturerrsquos optimal wholesale price 119908
lowast and optimal buy-back payment 119887lowast satisfy
119908lowast
=(Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701 [1198701 (120573 + 1) + 4 (1 minus 120574)]
+1198703(1 minus 120574)
1198701(2 minus 120574)
(27)
119887lowast
=2Δ1198731+ 1205741198732+ 120573120574 (2119888
119898119904 minus 4Δ)
4120573 (2 minus 1199042) + 2120574 (8 minus 119904
2) minus 2 (2 + 120573120574) (4 minus 119904
2) (28)
where 1198701 = 4(120574 minus 1) + 1199042(2 minus 120574) 1198702 = (2119888119898 minus Δ119904)(1 minus 120574)
1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873
1= 120573(2 minus 119904
2) minus (4 minus 119904
2) and
1198732= 8Δ + (Φ
1+ Φ2minus 2119888119898)119904
Proof The second-order partial derivatives of Π119863119898are
1205972Π119863
119898
1205971199082
=41199042(2 minus 120574) (1 minus 120573) (1 minus 120574)
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
1205972Π119863
119898
1205971198872
=41199042(2 minus 120574) (1 minus 120574)
2
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
1205972Π119863
119898
120597119908120597119887
=1205972Π119863
119898
120597119887120597119908
=41199042(2 minus 120574)
2[1 minus 120573 + 2119904 (1 minus 120574) minus (2 minus 120573) (2 minus 120574)]
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
10038161003816100381610038161198671198981003816100381610038161003816 =
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1205972Π119863
119898
1205971199082
1205972Π119863
119898
120597119908120597119887
1205972Π119863
119898
120597119887120597119908
1205972Π119863119862
119898
1205971198872
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(29)
By the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1 wecan easily obtain that119867
119898is negative definite soΠ
119863
119898is jointly
concave with respect to the wholesale price 119908 and buy-back
Abstract and Applied Analysis 7
payment 119887Thus the optimal wholesale price119908 and buy-backpayment 119887 can be obtained by the first-order condition
According to (26) the first-order conditions of Π119863119898with
respect to 119908 and 119887 are as follows
120597Π119863
119898
120597119908
=2 (2 minus 120574) (2119908 minus 119887119904 + 119888
119898) + 2Δ119904
1199042+ 120574 minus 2
+ ((120574 minus 2) [2119887119904 (1 minus 120574) + (Φ1+ Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)
+ 2 (119888119898
minus 119908) (2 minus 120574 + 2119904 (119887 minus Δ)) ])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(30)
120597Π119863
119898
120597119887
=2 (1 minus 120574) (2119887 minus Δ + 119888
119898119904) minus 2119908119904 (2 minus 120574)
1199042+ 120574 minus 2
+ (119904 [2119887119904 (1 minus 120574) + (Φ1 + Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)]
+ 2119904 (1 minus 120574) [(119888119898
minus 119908) (2 minus 120574) + 119904 (119887 minus Δ)])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(31)
By combining (30) and (31) we can obtain the optimal retailwholesale price119908lowast and buy-back payment 119887lowast that is (27) and(28) hold
Proposition 8 In the case of decentralized collection theoptimal retail 119901lowast119863
119894and optimal repurchase price 119901
lowast119863
119877119894of retailer
119894 satisfy
119901lowast119863
119894
=(Φ119894 minus Φ3minus119894)
2[
1198851
11988511
minus1198841
11988411
]
+119904 (1 minus 120574)
211988411
[2Δ1198721 + 1205741198722 + 2120573120574 (119888119898119904 minus 2Δ)
21198721+ 120574119872
2
]
minus(2 minus 120574)
11988411
[(Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(1 + 120573) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
]
119901lowast119863
119877119894
= ((1 minus 120574) 119904
2
119884111988411
minus
(1 minus 1199042)1198851
119879)
times (2Δ1198721 + 1205741198723 + 120573120574 (2119888119898119904 minus 4Δ)
21198721+ 120574119872
2
) minus 119904 [(2 minus 120574) +1198851
119879]
times ((Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(120573 + 1) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
) + [119904 (Φ119894 + Φ3minus119894)
2](
1
11988411
minus1
11988511
)
(32)
where 1198721
= 120573(2 minus 1199042) minus (4 minus 119904
2) 1198722
= (8 minus 1199042) minus 120573(4 minus 119904
2)
1198723
= 8Δ + (Φ1+ Φ2minus 2119888119898)119904 1198701
= 4(120574 minus 1) + 1199042(2 minus 120574)
1198702= (2119888119898
minus Δ119904)(1 minus 120574) 1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873
1=
120573(2minus1199042)minus(4minus119904
2)1198732= 8Δ+(Φ
1+Φ2minus2119888119898)1199041198841= 1199042+120574minus2
11988411
= 1199042minus 2(2 minus 120574) + 120573(2 minus 119904
2minus 120574) 119885
1= 2 + 120574 minus 119904
2 11988511
=
2(2 + 120574) minus 120573(2 + 1199042minus 120574) minus 119904
2 and 119879 = 1205742minus 1199044minus 4(1 minus 119904
2)
Proof By substituting (27) and (28) into (18) and (19) respec-tively we can obtain the optimal retail price and repurchaseprice of retailer 119894
Proposition 9 In the case of decentralized collection themanufacturerrsquos profit of forward channel is strictly increasingwith respect to the retailing substitution effect and decreasingwith respect to the recycling substitution effect while themanufacturerrsquos reverse channelrsquos profit is increasingwith respectto the retailing substitution effect and decreasing with respect tothe recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in forward channelAccording to (1) the demand of retailing 119863
119894is invariable
for the durable product and Φ119894and 119901
119895are invariable for
retailer 119894 thus the retail price is increasing with respectto the retailing substitution effect that is increasing theretailing substitution effect 120573 will induce the increase ofretail price 119901119894 and according to Corollary 6 it will inducethe increase of wholesale price 119908 According to (4) themanufacturerrsquos profit in forward channel (119908minus119888119898)(119863119894(119901119894 119901119895)+
119863119895(119901119895 119901119894)) is increasing with respect to retailing substitutioneffect Similarly we can easily obtain that the manufacturerrsquosprofit in forward channel (119908 minus 119888119898)(119863119894(119901119894 119901119895) + 119863119895(119901119895 119901119894)) isdecreasing with respect to recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in reverse channelAccording to (2) the recycling function 119876
119894is invariable for
the durable product and 120601119894
= 119904119863119894and 119901
119877119895are invariable
8 Abstract and Applied Analysis
for retailer 119894 thus the repurchase price is increasing withrespect to recycling substitution effect Then we can easilyobtain that the increase of retailing substitution effect 120573
will induce the increase of retail price 119901119894 and according
to Corollary 6 it will induce the decrease of buy-backpayment 119887 According to (4) the manufacturerrsquos reversechannel (Δ minus 119887)(119876119894(119901119877119894 119901119877119895) + 119876
119895(119901119877119895
119901119877119894)) is increasing
with respect to retailing substitution effect Similarly we caneasily get that the manufacturerrsquos reverse channel profit (Δ minus
119887)(119876119894(119901119877119894 119901119877119895) + 119876119895(119901119877119895 119901119877119894)) is decreasing with respect torecycling substitution effect
It is illustrated that in the case of decentralized collectionthe variation of manufacturerrsquos profit in forward or reversechannel is decided by the difference between the increase offorward or reverse channel profit for the retailing substitutioneffect and the decrease of forward and reverse channelprofit for the recycling substitution effect Additionally thechange of closed-loop supply chainrsquos profit with respect tothe retailing and recycling substitution effects is uncertainin the case of decentralized collection On the other handthe manufacturer can induce higher retailing substitutioneffect and lower recovery substitution effect via choosingsuitable wholesale price and buy-back payment to collectmore obsolete products and increase the profit of reversechannel It means that the manufacturers can achieve theirgoal of remanufacturing strategy by recovering the residualvalue of used products
Proposition 10 In the case of decentralized collection theretailerrsquos profit of forward channel is strictly increasing withrespect to the retailing substitution effect and decreasing withrespect to the recycling substitution effect while his reversechannel profit is decreasing with respect to the retailing substi-tution effect and recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in forward channel Accordingto (1) the demand of retailing119863
119894is invariable for the durable
product and Φ119894and 119901
119895are invariable for retailer 119894 thus
the retail price is increasing with respect to the retailingsubstitution effect that is the increase of 120573 will induce theincrease of retail price 119901
119894 According to (5) each retailerrsquos
profit in forward channel (119901119894minus 119908)119863
119894(119901119894 119901119895) is increasing
with respect to the retailing substitution effect similarly forretailer 119895 We can easily obtain that each retailerrsquos profit inreverse channel (119901119894 minus 119908)119863119894(119901119894 119901119895) is increasing with respectto recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in reverse channel Accordingto (2) the recycling function119876119877119894 is invariable for the durableproduct and 120601119894 = 119904119863119894 and 119901119877119895 are invariable for retailer119894 thus the repurchase price is increasing with respect torecycling substitution effect Then we can easily obtain thatthe repurchase price is increasing with respect to recyclingsubstitution effect So the increase of retailing substitutioneffect 120573 will induce the increase of retail price 119901
119894 induce
the increase of buy-back payment 119887 and then will inducethe decrease of 119901
119877119894 According to (5) the retailer 119894rsquos profit in
reverse channel (119887minus119901119877119894)119876119894(119901119877119894 119901119877119895
) is decreasingwith respectto retailing substitution effect Similarly we can easily get thatthe retailer 119894rsquos profit in reverse channel (119887 minus119901
119877119894)119876119894(119901119877119894 119901119877119895
) isincreasing with respect to recycling substitution effect
Proposition 10 illustrates that in the case of decentralizedcollection the variation of retailerrsquos profit in forward channelis decided by the difference between the increase of forwardchannelrsquos profit for the retailing substitution effect and thedecrease of forward channel profit for the recycling substitu-tion effect While for the retailerrsquos profit in reverse channelit is decreasing with respect to retailing substitution effectand recycling substitution effect Additionally because thechange of forward channel profit is uncertain the variationof the closed-loop supply chainrsquos profit is also uncertain Inthis case the retailer can induce lower retailing substitutioneffect and recovery substitution effect via choosing suitablewholesale price and buy-back payment to collect moreobsolete products and increase the profit of reverse channel
In a word the variations of manufacturer and retailerrsquosrespective profit with the retailing substitution effect inforward channel are uniform but in the reverse channelthey are inconsistent For example the higher recyclingsubstitution effect will induce the manufacturer to get moreprofits in reverse channel (ie augments the scale of theman-ufacturerrsquos reverse supply chains) But the higher recyclingsubstitution effect will cause the retailer to obtain less profitsin reverse channel (ie cut down the scale of the retailerrsquosreverse supply chains) There is a contradiction betweenthe profits of the manufacturer and the retailer in reversechannel because the manufacturer has sufficient channelpower over the retailers to act as a Stackelberg leader thus themanufacturer can choose the appropriate wholesale price andbuy-back payment to achieve the remanufacturing strategyand balance the retailerrsquos profit And we make a comparisonwith the coordinated collection we can easily know thatthe coordinated collection is more effective to augment thescale of the whole reverse supply chains than decentralizedcollection
4 Numerical Examples
In this section numerical examples are provided to comparethe results obtained in the above two different collectionmodels and to analyze the impacts of retailing and recyclingsubstitution effects on the decisions and profits of chainmembers Some managerial insights can be derived throughthe numerical analysis The numerical analysis is performedfor Φ1= 100 Φ
2= 50 119888
119898= 20 Δ = 10 120574 = 04 or 120574 = 02
and 119904 = 01
41 Numerical Analysis of Model 119862 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofcoordinated collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect to retail-ing substitution effect that is the higher retailing substitution
Abstract and Applied Analysis 9
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
120574 = 02
120574 = 04
pi
(a) Impact on the retail price
0
5
10
15
20
25
30
35
40
45
50
pRi
0 02 04 06 08 1120573
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 2 Impact of the retailing and recycling substitution effects on price in model 119862
effect will induce the increase of retail price Moreover thehigher recycling substitution effect will induce the increaseof retail price the higher recycling substitution effect willinduce the decrease of repurchase price In addition thehigher retailing substitution effect will induce the decrease ofretail price The green and blue lines in Figures 2(a) and 2(b)give us a visual presentation
From the previous assumptions and analysis we knowthat the profit of forward channel is strictly increasing withrespect to retailing substitution effect and decreasing withrespect to recycling substitution effect that is the manu-facturer can induce higher retailing competition and lowerrecycling competition to obtain more profit from forwardchannel
Andwe known that the profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andincreasing with respect to recycling substitution effect thatis the manufacturer can induce lower recycling substitutioneffect to obtain more profit in the reverse channel In otherwords themanufacturer can induce higher retailing substitu-tion effect and lower recycling substitution effect could obtainmore profit in closed-loop supply chain The green and bluelines in Figures 3(a) and 3(b) give us a visual presentation
42 Numerical Analysis of Model 119863 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofdecentralized collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect toretailing substitution effect that is the higher retailing substi-tution effect will induce the increase of retail price And the
higher recycling substitution effect will induce the decreaseof retail price And we known that the repurchase priceis strictly decreasing with respect to recycling substitutioneffect that is the higher recycling substitution effect willinduce the increase of repurchase price while the higherretailing substitution effect will induce the decrease of retailprice The green and blue lines in Figures 4(a) and 4(b) giveus a visual presentation
According to the assumptions and the analysis weknow that the manufacturerrsquos profit of forward channel isstrictly increasing with respect to retailing substitution effectand decreasing with respect to recycling substitution effectAnd the manufacturerrsquos profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andrecycling substitution effect In other words in the case ofdecentralized collection the manufacturer can induce higherretailing competition and lower recycling competition toobtain more profit in the forward channel and induce lowerrecycling substitution effect to obtain more profit in thethe reverse channel So the manufacturer can select higherretailing substitution effect and lower recycling substitutioneffect to obtain more profit in their closed-loop supply chainMoreover the higher recycling competition will induce thelower profit of forward channel and induce lower profit ofreverse channel
For the forward channel profit of retailers is strictlyincreasing with respect to retailing substitution effect anddecreasing with respect to recycling substitution effect Andthe retailersrsquo profit of reverse channel is strictly decreasingwith respect to retailing substitution effect and increasingwith respect to recycling substitution effect In other wordsin the case of decentralized collection for the retailers thehigher retailing competition and lower recycling competition
10 Abstract and Applied Analysis
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(a) Impact on the profit in forward channel
0 02 04 06 08 10
200
400
600
800
1000
1200
1400
1600
1800
2000
120573
120574 = 02
120574 = 04120587
(b) Impact on the profit in reverse channel
Figure 3 Impact of the retailing and recycling substitution effects on channel profit in model 119862
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
pi
120574 = 02
120574 = 04
(a) Impact on the retail price
0 02 04 06 08 1minus20
minus15
minus10
minus5
0
5
10
15
20
25
30
120573
pi
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 4 Impact of the retailing and recycling substitution effects on price in model 119863
can make them obtain more profit in their forward channelbut the lower retailing substitution effect and higher recyclingsubstitution effect can make them obtain more profit in thereverse channel So the retailers tend to select suitable retailprice and repurchase price to induce retailing and recyclingcompetitions (not too high or too low the higher retailsubstitution effect will make the profit of reverse channel
negative or the lower recycling substitution effect will makethe profit of forward channel decrease) to obtain more profitin their closed-loop supply chain The lines in Figure 5 giveus a visual presentation
43 Comparison of Model119862 andModel119863 In this subsectionwe discuss the impact of retailing substitution effect on the
Abstract and Applied Analysis 11
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120587
(a) Impact on the manufacturerrsquos profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120587
(b) Impact on the manufacturerrsquos profit in reverse channel
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(c) Impact on the retailersrsquo profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120574 = 02
120574 = 04
120587
(d) Impact on the retailersrsquo profit in reverse channel
Figure 5 Impact of the retailing and recycling substitution effects on channel profit in model 119863119862
optimal prices and optimal channel profits and comparethem in the case of coordinated collection with that ofdecentralized collection
From the previous assumptions and analysis we knowthat the retail price in the case of coordinated collection islower than that in decentralized collection that is facingthe same circumstances the retail price in the case ofdecentralized collection is higher than that in coordinatedcollection And the repurchase price in the case of decentral-ized collection is lower than that in coordinated collectionIt implies that for the retailer the coordinated collectionhas more price advantages (lower retail price and higherrepurchase price) than the decentralized collection in the
closed-loop channel The lines in Figure 6 give us a visualpresentation
In what is discussed above we know that the channelprofit in model 119862 is higher than that in model 119863 Theprofit of reverse channel in model 119863 is very low even thevalue is negative In this case the manufacturer and retailerstend to select coordinated collection to obtain more profitsfor example HP Xerox and Apple select the coordinatedcollection to collect their used product The lines in Figure 6give us a visual presentation
44 Impact of Retailing and Recycling Substitution Effects onClosed Supply Chainrsquos Profit In this subsection we discuss
12 Abstract and Applied Analysis
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
Model CCModel DC
120573
pi
(a) Comparison of retail price
minus100
minus50
0
50
100
150
200
pRi
0 02 04 06 08 1
Model CCModel DC
120573
(b) Comparison of repurchase price
Figure 6 Pricing decisions of coordinated collection versus decentralized collection
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
Model CCModel DC
times104
120573
120587
(a) Comparison of forward channel profits
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
Model CCModel DC
120573
120587
(b) Comparison of reverse channel profits
Figure 7 The profit of coordinated collection versus decentralized collection
the impact of retailing and recycling substitution effects onthe optimal forward channel profits and optimal reversechannel profits in the cases of coordinated collection anddecentralized collection
From the previous assumptions and analysis weknow that the recycling substitution effectrsquos impact on theclosed-loop supply channel profit in the case of coordinated
collection is more obvious than that in decentralizedcollection In other words the manufacturer can obtainmore profit than in decentralized collection more easily bychanging the recycling substitution effect The reason maybe as follows in the case of coordinated collection the retailprice and repurchase price are made by the center planner(ie manufacturer) this collection model can adjust the
Abstract and Applied Analysis 13
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(a) Impact on the closed-loop supply channelrsquos profit in model 119862119862
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(b) Impact on the closed-loop supply channelrsquos profit in model119863119862
Figure 8 Impact of the retailing and recycling substitution effects on closed-Loop supply chainrsquos profit
retail and repurchase price fast and accurately But in caseof decentralized collection there is a pricing game betweenmanufacturer and retailers The manufacturer firstly selectsoptimal wholesale price and buy-back payment then theretailers select their optimal reaction retail price and repur-chase price that is the pricing game and pricing decision areunsynchronized between the manufacturer and retailer sothe manufacturer can obtain more profits in the case of coor-dinated collection than the case of decentralized collectionThe lines in Figures 7 and 8 give us a visual presentation
5 Conclusion
In this paper we investigated a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers Theresults demonstrate that more intense retailing competitioninduces the increase of the forward and reverse profitsof manufacturer and the forward channel of retailers anddecrease of the retailers profit in reverse channel while moreintense recycling competition induces the decrease of theprofits of the manufacturer and retailers in forward andreverse channels while the whole supply chainrsquos profit ofmodel 119862 is increasing with the respect to the free recyclingproportional coefficient but the effect on model 119863 is uncer-tain The results have important managerial insights for themanufacturer and retailers
This paper makes a contribution to the literature onreverse channel choice and coordination by drawing atten-tion to closed-loop supply chains with competition of retail-ing and recycling Our future research will be as follows ourmodel does not consider any fixed costs of retailingrecyclinga collection system If such costs were incurred a minimumnumber of retailingreturns would be required to justify thecost of operations Last but not least our cost formulation
does not include any operationalcapacity constraints inthe channel of products collection Operationalcapacityconstraint on the channel of collection can be incorporatedinto the model by defining an upper bound on the numbercollected via each channel Another possibility is to modelthe collection cost as a quadratic function of the quantityreturned
Notations
120574 Recycling substitution effect120573 Retailing substitution effectΦ Market size of the retailers in the forward
channel120601 Market size of the retailers by free recovering
in the reverse channel120575 Unit saving cost by remanufacturing119904 Recovering coefficient119888119903 Unit cost of remanufacturing a returned
product into a new one119888119898 Unit cost of manufacturing a new product
119908 New productrsquos wholesale price of themanufacturer
119887 Obsolete productrsquos repurchase price from theretailers to the manufacturer
119863119894(119901119894 119901119895) Demand function of retailer 119894
119876119894(119901119877119894 119901119877119895
) Recycling function of retailer 119894
119901119862
119894 119901119863
119895 Retailer 119894 and 119895rsquos respective retail price in
models 119862 and 119863
119901119862
119877119894 119901119863
119877119895 Retailer 119894 and 119895rsquos repurchase price in models
119862 and 119863
Π119870
119894 Profit function for channel member 119894 in
model 119896 Superscript 119896 takes the values of 119862and 119863 Subscript 119894 takes the values of 119898 119903and 119890 denoting the manufacturer theretailer and the 119890-tailer respectively
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
3 Model Formulation and Solution
In this section two equilibrium models are discussed in thecases of coordinated collection and decentralized collectionrespectively
31 Coordinated Collection According to (3) the coordi-nated collection model can be formulated as follows
max119901119894119901119895119901119877119894119901119877119895
Π119862(119901119894 119901119895 119901119877119894 119901119877119895
)
=
2
sum
119894=1
((119901119894minus 119888119898)119863119894(119901119894 119901119895) + (Δ minus 119901
119877119894) 119876119894(119901119877119894 119901119877119895
))
(6)
Proposition 1 In the case of coordinated collection theoptimal retail 119901119862
119894and the optimal repurchase price 119901
119862
119895satisfy
119901lowast119862
119894=
(Φ119894+ Φ119895120573)
2 (1 minus 1205732)
+4 (Δ119904 minus 2119888
119898) (1 minus 120574) + 119904
2(Φ119894+ Φ3minus119894
)
41198831
minus1199042(Φ119894 minus Φ3minus119894)
41198832
(7)
119901lowast119862
119877119894=
Δ
2+ s[
(Φ119894+ Φ3minus119894
) minus (2119888119898
minus Δs) (1 minus 120573)
21198831
minus(Φ119894minus Φ3minus119894
)
21198832
]
(8)
where1198831 = 1199042(1minus120573)minus4(1minus120574) and1198832 = minus4(1minus120574)minus119904
2(1+120573)
Proof According to (3) the second-order partial derivativesof Π119862(119901
119894 119901119895 119901119877119894 119901119877119895
) are
1205972Π119862
1205971199012
119894
= minus21205972Π119862
120597119901119894120597119901119895
= 21205731205972Π119862
120597119901119894120597119901119877119894
= 119904
1205972Π119862
120597119901119894120597119901119877119895
= minus119904120573
1205972Π119862
120597119901119895120597119901119894
= 21205731205972Π119862
1205971199012
119895
= minus21205972Π119862
120597119901119895120597119901119877119894
= minus119904120573
1205972Π119862
120597119901119895120597119901119877119895
= 119904
1205972Π119862
120597119901119877119894120597119901119894
= 1199041205972Π119862
120597119901119877119894120597119901119895
= minus1199041205731205972Π119862
1205971199012
119877119894
= minus2
1205972Π119862
120597119901119877119894120597119901119877119895
= 2120574
1205972Π119862
120597119901119877119895
120597119901119894
= minus1199041205731205972Π119862
120597119901119877119895
120597119901119895
= 1199041205972Π119862
120597119901119877119895
120597119901119877119894
= 2120574
1205972Π119862
1205971199012
119877119895
= minus2
(9)
Then the Hessian matrix is
|119867| =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus2 2120573 119904 minus119904120573
2120573 minus2 minus119904120573 119904
119904 minus119904120573 minus2 2120574
minus119904120573 119904 2120574 minus2
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(10)
By using the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt
1 we can obtain that |1198631| = minus2 lt 0 |119863
2| = 4(1 minus 120573) gt 0
|1198633| = minus2(1 minus 120573
2)(4 minus 119904
2) lt 0 and |1198634| = 16 + 16(120574
2minus 1205732(1 minus
1205742))+8119904
2(1205732+120574120573(1minus120573
2)minus1)+119904
4(1minus1205732)2gt 0 So119867 is negative
definiteΠ119862 is jointly concavewith respect to119901119862
1198941199011198623minus119894
119901119862119895 and
119901119862
3minus119895 thus the optimal retailing and repurchase prices can be
obtained by the first-order conditions as follows
120597Π119862
120597119901119894
= Φ119894minus 2119901119894+ 2120573119901
119895+ 119904119901119877119894
minus 119904120573119901119877119895
+ (1 minus 120573) (119888119898
minus 119904Δ)
= 0
(11)
120597Π119862
120597119901119895
= Φ119895 minus 2119901119895 + 2120573119901119894 + 119904119901119877119894 minus 119904120573119901119877119895 + (1 minus 120573) (119888119898 minus 119904Δ)
= 0
(12)
120597Π119862
120597119901119877119894
= minus 119904Φ119894+ 119904119901119894minus 119904120573119901
119895minus 2119901119877119894
+ 2120574119901119877119895
+ (1 minus 120574) Δ
= 0
(13)
120597Π119862
120597119901119877119895
= minus 119904Φ119895+ 119904119901119895minus 119904120573119901
119894minus 2119901119877119895
+ 2120574119901119877119894
+ (1 minus 120574) Δ
= 0
(14)
By combining (11) (12) (13) and (14) we can obtainthe optimal retail prices 119901
lowast
119894and 119901
lowast
119895 and optimal repurchase
prices 119901lowast
119877119894and 119901
lowast
119877119895
Corollary 2 In the case of coordinated collection the retailprices 119901
119862
119894and 119901
119862
119895are strictly monotonically decreasing with
respect to the recycling substitution effect
Proof According to (7) the first-order derivative of 119901119862119894with
respect to 120574 is
d119901119862119894
d120574= (1199042[1198832
1(Φ119894 minus Φ3minus119894)
minus1198832
2(Φ119894+ Φ3minus119894
+ (1 minus 120573) (Δ119904 minus 2119888119898))])
times (1198832
11198832
2)minus1
(15)
where1198831= 1199042(1minus120573)minus4(1minus120574) and119883
2= minus4(1minus120574)minus119904
2(1+120573)
It follows from the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt
120573 lt 1 that d119901119862119894d120574 lt 0 thus the retail prices 119901
119862
119894and 119901
119862
119895are
strictly decreasing with respect to the recycling substitutioneffect
Abstract and Applied Analysis 5
Corollary 3 In the case of coordinated collection the repur-chase prices 119901
119862
119877119894and 119901
119862
119877119895are strictly increasing with respect to
the recycling substitution effect
Proof According to (8) the first-order derivative of 119901119862119877119894is
d119901119862119877119894
d120574= 2119904 (119883
2
2[Φ119894+ Φ3minus119894
minus (2119888119898
minus Δ119904) (1 minus 120573)]
minus1198832
1(Φ119894minus Φ3minus119894
)) times (1198832
11198832
2)minus1
(16)
where1198831 = 1199042(1minus120573)minus4(1minus120574) and1198832 = minus4(1minus120574)minus119904
2(1+120573)
By assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1 wecan obtain that d119901119862119862
119895d120574 gt 0 thus the repurchase prices 119901
119862
119877119894
and 119901119862
119877119895are strictly increasing with respect to the recycling
substitution effect
Corollary 2 implies how the recycling substitution effectimpacts the profit of forward channel Specially themanufac-turer can obtain more profit from forward channel by induc-ing recycling substitution effect (select the lower retail priceto induce more recycling substitution effect) Corollary 3implies how the recycling substitution effect impacts theprofit of reverse channel themanufacturer can select optimalrepurchase price by inducing recycling substitution effect(inducing more recycling substitution effect to select thehigher retail price) in order to obtain more profit of reversechannel Corollaries 2 and 3 illustrate that the recyclingsubstitution effectrsquos impact on the profit of forward andreverse is converse so themanufacturer can induce a suitablerecycling substitution effect between the retailers to obtainthe maximum profit in forward and reverse supply chain
Proposition 4 In the case of coordinated collection the profitof forward channel is strictly increasing with respect to theretailing substitution effect and decreasing with respect to therecycling substitution effect While the profit of reverse channelis strictly decreasing with respect to the retailing substitutioneffect and increasing with respect to the recycling substitutioneffect
Proof First for the analysis of the profit in forward channelfor any given 119901119895 according to (1) the demand119863119894 is invariablefor the durable product and Φ119894 is constant for retailer 119894the retail price is increasing with respect to the retailingsubstitution effect Then it follows from (3) that the profitof forward channel sum2
119894=1(119901119894minus 119888119898)119863119894(119901119894 119901119895) is increasing with
respect to 119901119894and 119901
119895 Furthermore by Corollary 2 the retail
prices 119901119894and 119901
119895are both strictly decreasing with respect to
the recycling substitution effect Hence the profit of forwardchannel function sum
2
119894=1(119901119894minus 119888119898)119863119894(119901119894 119901119895) is decreasing with
respect to the recycling substitution effectSecond for the analysis of the profit in reverse channel
for any given 119901119877119894 since the recycling function119876
119894is invariable
for the durable product and 120601119894= 119904119863119894and 119901
119877119894are invariable
for retailer 119894 by (2) the repurchase price is increasing withrespect to recycling substitution effect Then the profit ofreverse channelsum2
119894=1(Δ minus 119901
119877119894)119876119894(119901119877119894 119901119877119895
) in (3) is decreasing
with respect to 119901119877119894
and 119901119877119895 According to Corollary 3 the
repurchase prices 119901119877119894
and 119901119877119895
are strictly increasing withrespect to the recycling substitution effect Therefore theprofit of reverse channel function sum
2
119894=1(Δ minus 119901
119877119894)119876119894(119901119877119894 119901119877119895
)
is decreasing with the recycling substitution effect
From what is discussed above in the case of coordi-nated collection we can obtain some managerial insights inpractice It is illustrated that the variation tendency of theforward channel profit is decided by the difference betweenthe increase of forward channel profit from the retailingsubstitution effect and the decrease of forward channelrsquos profitstemmed from the recycling substitution effect Similarly thevariation of reverse channel profit is decided by the differencebetween the increase of reverse channel profit for the retail-ing substitution effect and the decrease of reverse channelprofit for the recycling substitution effect Additionally thevariation of the total profit of forward and reverse channel isuncertain Furthermore the manufacturer can induce lowerretailing substitution effect and higher recovery substitutioneffect to collect more obsolete products and increase theprofit of reverse channel In other words the manufacturercan easily achieve the goal of remanufacturing strategy byrecovering the residual value of used products
32 Decentralized Collection In the case of decentralizedcollection the manufacturer decides on the wholesale price119908 and reimburses each retailer a fixed fee 119887 per unit returnedand the retailers compete with each other and decide on theretail price of the product as well as the repurchase price ofcollection used product
We can obtain retailer 119894rsquos reaction function by solving thefollowing optimization problem
max119901119894 119901119895
Π119863
119894= (119901119894 minus 119908)119863119894 (119901119894 119901119895) + (119887 minus 119901119877119895)119876119877119894 (119901119877119894 119901119877119895)
(17)
Proposition 5 In the case of decentralized collection theretailerrsquos optimal reaction retail price 119901
lowast119863
119894and the optimal
reaction repurchase price 119901lowast119863
119877119894satisfy
119901lowast119863
119894=
119887119904 (1 minus 120574) minus 119908 (2 minus 120574)
11988411
+(Φ119894+ Φ3minus119894
) 1198841
211988411
+(Φ119894minus Φ3minus119894
) 1198851
211988511
(18)
119901lowast119863
119877119894=
119904 (Φ119894+ Φ3minus119894
)
211988411
minus119904 (Φ119894minus Φ3minus119894
)
211988511
+ 119887[1199042(1 minus 120574)
119884111988411
+
(1 minus 1199042)1198851
119879]
minus 119908119904 [1198851
119879+ (2 minus 120574)]
(19)
where 1198841
= 1199042+ 120574 minus 2 119884
11= 1199042minus 2(2 minus 120574) + 120573(2 minus 119904
2minus 120574)
1198851 = 2 + 120574 minus 119904
2 11988511 = 2(2 + 120574) minus 120573(2 + 1199042minus 120574) minus 119904
2 and119879 = 120574
2minus 1199044minus 4(1 minus 119904
2)
6 Abstract and Applied Analysis
Proof The second-order partial derivatives of retailerrsquos profitΠ119863
119894are
1205972Π119863
119894
1205971199012
119894
= minus21205972Π119863
119894
120597119901119894120597119901119895
= 119904
1205972Π119863
119894
120597119901119895120597119901119894
= 1199041205972Π119863
119894
1205971199012
119895
= minus119904120573
(20)
Then the Hessian matrix is
|119867| =
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1205972Π119863
119894
1205971199012
119894
1205972Π119863
119894
120597119901119894120597119901119895
1205972Π119863
119894
120597119901119895120597119901119894
1205972Π119863
119894
1205971199012
119895
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(21)
By using the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt
120573 lt 1 we can obtain that |1198631| = 120597
2Π119863
1198941205971199012
119894= minus2 lt 0
and |1198632| = 2119904(120573 minus 119904) gt 0 So 119867 is negative definite Π119863 is
jointly concave with respect to 119901119894and 119901
119895 Thus the optimal
retailing and repurchase price can be obtained by the first-order condition as follows
According to (17) for retailer 1 the first-order conditionsof Π119863119894are
120597Π119863
119894
120597119901119894
= Φ119894 minus 2119901119894 + 120573119901119895 + 119908 minus 119904 (119887 minus 119901119877119894) = 0 (22)
120597Π119863
119894
120597119901119877119894
= minus119904 (Φ119894 minus 119901119894 + 120573119901119895) minus 119901119877119894 + 120574119901119877119895 + 119887 minus 119901119877119894 = 0
(23)
Similarly for retailer 2 the first-order partial conditions ofΠ119863
119895are
120597Π119863
119895
120597119901119895
= Φ119895minus 2119901119895+ 120573119901119894+ 119908 minus 119904 (119887 minus 119901
119877119895) = 0 (24)
120597Π119863
119895
120597119901119877119895
= minus119904 (Φ119895 minus 119901119895+ 120573119901119894) minus 119901
119877119895+ 120574119901119877119894
+ 119887 minus 119901119877119895
= 0
(25)
By combining (22) (23) (24) and (25) we can obtain theoptimal retail prices 119901119894 and 119901119895 and optimal repurchase prices119901119877119894 and 119901119877119895
Corollary 6 The retail price is increasing with respect to thewholesale price 119908 and decreasing with respect to the buy-backpayment 119887 But the repurchase price is decreasing with respectto the wholesale price119908 and increasing with respect to the buy-back payment 119887
Proof By the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1we can easily obtain119884
1= 1199042+120574minus2 lt 0119884
11= 1199042minus2(2minus120574)+120573(2minus
1199042minus120574) lt 0119885
1= 2+120574minus119904
2gt 0 and 119879 = 120574
2minus1199044minus4(1minus119904
2) lt 0
Thus according to (18) we can obtain that the retail prices119901119894and 119901
119895are increasing with respect to the wholesale price
119908 while decreasing with respect to the buy-back payment 119887And it follows from (19) that the repurchase prices 119901
119877119894and
119901119877119895are decreasingwith respect to thewholesale price119908 while
increasing with respect to the buy-back payment 119887
Substituting (18) and (19) into (17) yields
max119908119887
Π119863
119898= (119908 minus 119888119898)
2
sum
119894=1
119863119894 (119901lowast119863
119894 119901lowast119863
119895)
+ (Δ minus 119887)
2
sum
119894=1
119876119894(119901lowast119863
119877119894 119901lowast119863
119877119895)
(26)
Proposition 7 In the case of decentralized collection themanufacturerrsquos optimal wholesale price 119908
lowast and optimal buy-back payment 119887lowast satisfy
119908lowast
=(Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701 [1198701 (120573 + 1) + 4 (1 minus 120574)]
+1198703(1 minus 120574)
1198701(2 minus 120574)
(27)
119887lowast
=2Δ1198731+ 1205741198732+ 120573120574 (2119888
119898119904 minus 4Δ)
4120573 (2 minus 1199042) + 2120574 (8 minus 119904
2) minus 2 (2 + 120573120574) (4 minus 119904
2) (28)
where 1198701 = 4(120574 minus 1) + 1199042(2 minus 120574) 1198702 = (2119888119898 minus Δ119904)(1 minus 120574)
1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873
1= 120573(2 minus 119904
2) minus (4 minus 119904
2) and
1198732= 8Δ + (Φ
1+ Φ2minus 2119888119898)119904
Proof The second-order partial derivatives of Π119863119898are
1205972Π119863
119898
1205971199082
=41199042(2 minus 120574) (1 minus 120573) (1 minus 120574)
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
1205972Π119863
119898
1205971198872
=41199042(2 minus 120574) (1 minus 120574)
2
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
1205972Π119863
119898
120597119908120597119887
=1205972Π119863
119898
120597119887120597119908
=41199042(2 minus 120574)
2[1 minus 120573 + 2119904 (1 minus 120574) minus (2 minus 120573) (2 minus 120574)]
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
10038161003816100381610038161198671198981003816100381610038161003816 =
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1205972Π119863
119898
1205971199082
1205972Π119863
119898
120597119908120597119887
1205972Π119863
119898
120597119887120597119908
1205972Π119863119862
119898
1205971198872
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(29)
By the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1 wecan easily obtain that119867
119898is negative definite soΠ
119863
119898is jointly
concave with respect to the wholesale price 119908 and buy-back
Abstract and Applied Analysis 7
payment 119887Thus the optimal wholesale price119908 and buy-backpayment 119887 can be obtained by the first-order condition
According to (26) the first-order conditions of Π119863119898with
respect to 119908 and 119887 are as follows
120597Π119863
119898
120597119908
=2 (2 minus 120574) (2119908 minus 119887119904 + 119888
119898) + 2Δ119904
1199042+ 120574 minus 2
+ ((120574 minus 2) [2119887119904 (1 minus 120574) + (Φ1+ Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)
+ 2 (119888119898
minus 119908) (2 minus 120574 + 2119904 (119887 minus Δ)) ])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(30)
120597Π119863
119898
120597119887
=2 (1 minus 120574) (2119887 minus Δ + 119888
119898119904) minus 2119908119904 (2 minus 120574)
1199042+ 120574 minus 2
+ (119904 [2119887119904 (1 minus 120574) + (Φ1 + Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)]
+ 2119904 (1 minus 120574) [(119888119898
minus 119908) (2 minus 120574) + 119904 (119887 minus Δ)])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(31)
By combining (30) and (31) we can obtain the optimal retailwholesale price119908lowast and buy-back payment 119887lowast that is (27) and(28) hold
Proposition 8 In the case of decentralized collection theoptimal retail 119901lowast119863
119894and optimal repurchase price 119901
lowast119863
119877119894of retailer
119894 satisfy
119901lowast119863
119894
=(Φ119894 minus Φ3minus119894)
2[
1198851
11988511
minus1198841
11988411
]
+119904 (1 minus 120574)
211988411
[2Δ1198721 + 1205741198722 + 2120573120574 (119888119898119904 minus 2Δ)
21198721+ 120574119872
2
]
minus(2 minus 120574)
11988411
[(Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(1 + 120573) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
]
119901lowast119863
119877119894
= ((1 minus 120574) 119904
2
119884111988411
minus
(1 minus 1199042)1198851
119879)
times (2Δ1198721 + 1205741198723 + 120573120574 (2119888119898119904 minus 4Δ)
21198721+ 120574119872
2
) minus 119904 [(2 minus 120574) +1198851
119879]
times ((Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(120573 + 1) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
) + [119904 (Φ119894 + Φ3minus119894)
2](
1
11988411
minus1
11988511
)
(32)
where 1198721
= 120573(2 minus 1199042) minus (4 minus 119904
2) 1198722
= (8 minus 1199042) minus 120573(4 minus 119904
2)
1198723
= 8Δ + (Φ1+ Φ2minus 2119888119898)119904 1198701
= 4(120574 minus 1) + 1199042(2 minus 120574)
1198702= (2119888119898
minus Δ119904)(1 minus 120574) 1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873
1=
120573(2minus1199042)minus(4minus119904
2)1198732= 8Δ+(Φ
1+Φ2minus2119888119898)1199041198841= 1199042+120574minus2
11988411
= 1199042minus 2(2 minus 120574) + 120573(2 minus 119904
2minus 120574) 119885
1= 2 + 120574 minus 119904
2 11988511
=
2(2 + 120574) minus 120573(2 + 1199042minus 120574) minus 119904
2 and 119879 = 1205742minus 1199044minus 4(1 minus 119904
2)
Proof By substituting (27) and (28) into (18) and (19) respec-tively we can obtain the optimal retail price and repurchaseprice of retailer 119894
Proposition 9 In the case of decentralized collection themanufacturerrsquos profit of forward channel is strictly increasingwith respect to the retailing substitution effect and decreasingwith respect to the recycling substitution effect while themanufacturerrsquos reverse channelrsquos profit is increasingwith respectto the retailing substitution effect and decreasing with respect tothe recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in forward channelAccording to (1) the demand of retailing 119863
119894is invariable
for the durable product and Φ119894and 119901
119895are invariable for
retailer 119894 thus the retail price is increasing with respectto the retailing substitution effect that is increasing theretailing substitution effect 120573 will induce the increase ofretail price 119901119894 and according to Corollary 6 it will inducethe increase of wholesale price 119908 According to (4) themanufacturerrsquos profit in forward channel (119908minus119888119898)(119863119894(119901119894 119901119895)+
119863119895(119901119895 119901119894)) is increasing with respect to retailing substitutioneffect Similarly we can easily obtain that the manufacturerrsquosprofit in forward channel (119908 minus 119888119898)(119863119894(119901119894 119901119895) + 119863119895(119901119895 119901119894)) isdecreasing with respect to recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in reverse channelAccording to (2) the recycling function 119876
119894is invariable for
the durable product and 120601119894
= 119904119863119894and 119901
119877119895are invariable
8 Abstract and Applied Analysis
for retailer 119894 thus the repurchase price is increasing withrespect to recycling substitution effect Then we can easilyobtain that the increase of retailing substitution effect 120573
will induce the increase of retail price 119901119894 and according
to Corollary 6 it will induce the decrease of buy-backpayment 119887 According to (4) the manufacturerrsquos reversechannel (Δ minus 119887)(119876119894(119901119877119894 119901119877119895) + 119876
119895(119901119877119895
119901119877119894)) is increasing
with respect to retailing substitution effect Similarly we caneasily get that the manufacturerrsquos reverse channel profit (Δ minus
119887)(119876119894(119901119877119894 119901119877119895) + 119876119895(119901119877119895 119901119877119894)) is decreasing with respect torecycling substitution effect
It is illustrated that in the case of decentralized collectionthe variation of manufacturerrsquos profit in forward or reversechannel is decided by the difference between the increase offorward or reverse channel profit for the retailing substitutioneffect and the decrease of forward and reverse channelprofit for the recycling substitution effect Additionally thechange of closed-loop supply chainrsquos profit with respect tothe retailing and recycling substitution effects is uncertainin the case of decentralized collection On the other handthe manufacturer can induce higher retailing substitutioneffect and lower recovery substitution effect via choosingsuitable wholesale price and buy-back payment to collectmore obsolete products and increase the profit of reversechannel It means that the manufacturers can achieve theirgoal of remanufacturing strategy by recovering the residualvalue of used products
Proposition 10 In the case of decentralized collection theretailerrsquos profit of forward channel is strictly increasing withrespect to the retailing substitution effect and decreasing withrespect to the recycling substitution effect while his reversechannel profit is decreasing with respect to the retailing substi-tution effect and recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in forward channel Accordingto (1) the demand of retailing119863
119894is invariable for the durable
product and Φ119894and 119901
119895are invariable for retailer 119894 thus
the retail price is increasing with respect to the retailingsubstitution effect that is the increase of 120573 will induce theincrease of retail price 119901
119894 According to (5) each retailerrsquos
profit in forward channel (119901119894minus 119908)119863
119894(119901119894 119901119895) is increasing
with respect to the retailing substitution effect similarly forretailer 119895 We can easily obtain that each retailerrsquos profit inreverse channel (119901119894 minus 119908)119863119894(119901119894 119901119895) is increasing with respectto recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in reverse channel Accordingto (2) the recycling function119876119877119894 is invariable for the durableproduct and 120601119894 = 119904119863119894 and 119901119877119895 are invariable for retailer119894 thus the repurchase price is increasing with respect torecycling substitution effect Then we can easily obtain thatthe repurchase price is increasing with respect to recyclingsubstitution effect So the increase of retailing substitutioneffect 120573 will induce the increase of retail price 119901
119894 induce
the increase of buy-back payment 119887 and then will inducethe decrease of 119901
119877119894 According to (5) the retailer 119894rsquos profit in
reverse channel (119887minus119901119877119894)119876119894(119901119877119894 119901119877119895
) is decreasingwith respectto retailing substitution effect Similarly we can easily get thatthe retailer 119894rsquos profit in reverse channel (119887 minus119901
119877119894)119876119894(119901119877119894 119901119877119895
) isincreasing with respect to recycling substitution effect
Proposition 10 illustrates that in the case of decentralizedcollection the variation of retailerrsquos profit in forward channelis decided by the difference between the increase of forwardchannelrsquos profit for the retailing substitution effect and thedecrease of forward channel profit for the recycling substitu-tion effect While for the retailerrsquos profit in reverse channelit is decreasing with respect to retailing substitution effectand recycling substitution effect Additionally because thechange of forward channel profit is uncertain the variationof the closed-loop supply chainrsquos profit is also uncertain Inthis case the retailer can induce lower retailing substitutioneffect and recovery substitution effect via choosing suitablewholesale price and buy-back payment to collect moreobsolete products and increase the profit of reverse channel
In a word the variations of manufacturer and retailerrsquosrespective profit with the retailing substitution effect inforward channel are uniform but in the reverse channelthey are inconsistent For example the higher recyclingsubstitution effect will induce the manufacturer to get moreprofits in reverse channel (ie augments the scale of theman-ufacturerrsquos reverse supply chains) But the higher recyclingsubstitution effect will cause the retailer to obtain less profitsin reverse channel (ie cut down the scale of the retailerrsquosreverse supply chains) There is a contradiction betweenthe profits of the manufacturer and the retailer in reversechannel because the manufacturer has sufficient channelpower over the retailers to act as a Stackelberg leader thus themanufacturer can choose the appropriate wholesale price andbuy-back payment to achieve the remanufacturing strategyand balance the retailerrsquos profit And we make a comparisonwith the coordinated collection we can easily know thatthe coordinated collection is more effective to augment thescale of the whole reverse supply chains than decentralizedcollection
4 Numerical Examples
In this section numerical examples are provided to comparethe results obtained in the above two different collectionmodels and to analyze the impacts of retailing and recyclingsubstitution effects on the decisions and profits of chainmembers Some managerial insights can be derived throughthe numerical analysis The numerical analysis is performedfor Φ1= 100 Φ
2= 50 119888
119898= 20 Δ = 10 120574 = 04 or 120574 = 02
and 119904 = 01
41 Numerical Analysis of Model 119862 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofcoordinated collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect to retail-ing substitution effect that is the higher retailing substitution
Abstract and Applied Analysis 9
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
120574 = 02
120574 = 04
pi
(a) Impact on the retail price
0
5
10
15
20
25
30
35
40
45
50
pRi
0 02 04 06 08 1120573
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 2 Impact of the retailing and recycling substitution effects on price in model 119862
effect will induce the increase of retail price Moreover thehigher recycling substitution effect will induce the increaseof retail price the higher recycling substitution effect willinduce the decrease of repurchase price In addition thehigher retailing substitution effect will induce the decrease ofretail price The green and blue lines in Figures 2(a) and 2(b)give us a visual presentation
From the previous assumptions and analysis we knowthat the profit of forward channel is strictly increasing withrespect to retailing substitution effect and decreasing withrespect to recycling substitution effect that is the manu-facturer can induce higher retailing competition and lowerrecycling competition to obtain more profit from forwardchannel
Andwe known that the profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andincreasing with respect to recycling substitution effect thatis the manufacturer can induce lower recycling substitutioneffect to obtain more profit in the reverse channel In otherwords themanufacturer can induce higher retailing substitu-tion effect and lower recycling substitution effect could obtainmore profit in closed-loop supply chain The green and bluelines in Figures 3(a) and 3(b) give us a visual presentation
42 Numerical Analysis of Model 119863 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofdecentralized collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect toretailing substitution effect that is the higher retailing substi-tution effect will induce the increase of retail price And the
higher recycling substitution effect will induce the decreaseof retail price And we known that the repurchase priceis strictly decreasing with respect to recycling substitutioneffect that is the higher recycling substitution effect willinduce the increase of repurchase price while the higherretailing substitution effect will induce the decrease of retailprice The green and blue lines in Figures 4(a) and 4(b) giveus a visual presentation
According to the assumptions and the analysis weknow that the manufacturerrsquos profit of forward channel isstrictly increasing with respect to retailing substitution effectand decreasing with respect to recycling substitution effectAnd the manufacturerrsquos profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andrecycling substitution effect In other words in the case ofdecentralized collection the manufacturer can induce higherretailing competition and lower recycling competition toobtain more profit in the forward channel and induce lowerrecycling substitution effect to obtain more profit in thethe reverse channel So the manufacturer can select higherretailing substitution effect and lower recycling substitutioneffect to obtain more profit in their closed-loop supply chainMoreover the higher recycling competition will induce thelower profit of forward channel and induce lower profit ofreverse channel
For the forward channel profit of retailers is strictlyincreasing with respect to retailing substitution effect anddecreasing with respect to recycling substitution effect Andthe retailersrsquo profit of reverse channel is strictly decreasingwith respect to retailing substitution effect and increasingwith respect to recycling substitution effect In other wordsin the case of decentralized collection for the retailers thehigher retailing competition and lower recycling competition
10 Abstract and Applied Analysis
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(a) Impact on the profit in forward channel
0 02 04 06 08 10
200
400
600
800
1000
1200
1400
1600
1800
2000
120573
120574 = 02
120574 = 04120587
(b) Impact on the profit in reverse channel
Figure 3 Impact of the retailing and recycling substitution effects on channel profit in model 119862
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
pi
120574 = 02
120574 = 04
(a) Impact on the retail price
0 02 04 06 08 1minus20
minus15
minus10
minus5
0
5
10
15
20
25
30
120573
pi
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 4 Impact of the retailing and recycling substitution effects on price in model 119863
can make them obtain more profit in their forward channelbut the lower retailing substitution effect and higher recyclingsubstitution effect can make them obtain more profit in thereverse channel So the retailers tend to select suitable retailprice and repurchase price to induce retailing and recyclingcompetitions (not too high or too low the higher retailsubstitution effect will make the profit of reverse channel
negative or the lower recycling substitution effect will makethe profit of forward channel decrease) to obtain more profitin their closed-loop supply chain The lines in Figure 5 giveus a visual presentation
43 Comparison of Model119862 andModel119863 In this subsectionwe discuss the impact of retailing substitution effect on the
Abstract and Applied Analysis 11
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120587
(a) Impact on the manufacturerrsquos profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120587
(b) Impact on the manufacturerrsquos profit in reverse channel
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(c) Impact on the retailersrsquo profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120574 = 02
120574 = 04
120587
(d) Impact on the retailersrsquo profit in reverse channel
Figure 5 Impact of the retailing and recycling substitution effects on channel profit in model 119863119862
optimal prices and optimal channel profits and comparethem in the case of coordinated collection with that ofdecentralized collection
From the previous assumptions and analysis we knowthat the retail price in the case of coordinated collection islower than that in decentralized collection that is facingthe same circumstances the retail price in the case ofdecentralized collection is higher than that in coordinatedcollection And the repurchase price in the case of decentral-ized collection is lower than that in coordinated collectionIt implies that for the retailer the coordinated collectionhas more price advantages (lower retail price and higherrepurchase price) than the decentralized collection in the
closed-loop channel The lines in Figure 6 give us a visualpresentation
In what is discussed above we know that the channelprofit in model 119862 is higher than that in model 119863 Theprofit of reverse channel in model 119863 is very low even thevalue is negative In this case the manufacturer and retailerstend to select coordinated collection to obtain more profitsfor example HP Xerox and Apple select the coordinatedcollection to collect their used product The lines in Figure 6give us a visual presentation
44 Impact of Retailing and Recycling Substitution Effects onClosed Supply Chainrsquos Profit In this subsection we discuss
12 Abstract and Applied Analysis
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
Model CCModel DC
120573
pi
(a) Comparison of retail price
minus100
minus50
0
50
100
150
200
pRi
0 02 04 06 08 1
Model CCModel DC
120573
(b) Comparison of repurchase price
Figure 6 Pricing decisions of coordinated collection versus decentralized collection
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
Model CCModel DC
times104
120573
120587
(a) Comparison of forward channel profits
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
Model CCModel DC
120573
120587
(b) Comparison of reverse channel profits
Figure 7 The profit of coordinated collection versus decentralized collection
the impact of retailing and recycling substitution effects onthe optimal forward channel profits and optimal reversechannel profits in the cases of coordinated collection anddecentralized collection
From the previous assumptions and analysis weknow that the recycling substitution effectrsquos impact on theclosed-loop supply channel profit in the case of coordinated
collection is more obvious than that in decentralizedcollection In other words the manufacturer can obtainmore profit than in decentralized collection more easily bychanging the recycling substitution effect The reason maybe as follows in the case of coordinated collection the retailprice and repurchase price are made by the center planner(ie manufacturer) this collection model can adjust the
Abstract and Applied Analysis 13
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(a) Impact on the closed-loop supply channelrsquos profit in model 119862119862
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(b) Impact on the closed-loop supply channelrsquos profit in model119863119862
Figure 8 Impact of the retailing and recycling substitution effects on closed-Loop supply chainrsquos profit
retail and repurchase price fast and accurately But in caseof decentralized collection there is a pricing game betweenmanufacturer and retailers The manufacturer firstly selectsoptimal wholesale price and buy-back payment then theretailers select their optimal reaction retail price and repur-chase price that is the pricing game and pricing decision areunsynchronized between the manufacturer and retailer sothe manufacturer can obtain more profits in the case of coor-dinated collection than the case of decentralized collectionThe lines in Figures 7 and 8 give us a visual presentation
5 Conclusion
In this paper we investigated a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers Theresults demonstrate that more intense retailing competitioninduces the increase of the forward and reverse profitsof manufacturer and the forward channel of retailers anddecrease of the retailers profit in reverse channel while moreintense recycling competition induces the decrease of theprofits of the manufacturer and retailers in forward andreverse channels while the whole supply chainrsquos profit ofmodel 119862 is increasing with the respect to the free recyclingproportional coefficient but the effect on model 119863 is uncer-tain The results have important managerial insights for themanufacturer and retailers
This paper makes a contribution to the literature onreverse channel choice and coordination by drawing atten-tion to closed-loop supply chains with competition of retail-ing and recycling Our future research will be as follows ourmodel does not consider any fixed costs of retailingrecyclinga collection system If such costs were incurred a minimumnumber of retailingreturns would be required to justify thecost of operations Last but not least our cost formulation
does not include any operationalcapacity constraints inthe channel of products collection Operationalcapacityconstraint on the channel of collection can be incorporatedinto the model by defining an upper bound on the numbercollected via each channel Another possibility is to modelthe collection cost as a quadratic function of the quantityreturned
Notations
120574 Recycling substitution effect120573 Retailing substitution effectΦ Market size of the retailers in the forward
channel120601 Market size of the retailers by free recovering
in the reverse channel120575 Unit saving cost by remanufacturing119904 Recovering coefficient119888119903 Unit cost of remanufacturing a returned
product into a new one119888119898 Unit cost of manufacturing a new product
119908 New productrsquos wholesale price of themanufacturer
119887 Obsolete productrsquos repurchase price from theretailers to the manufacturer
119863119894(119901119894 119901119895) Demand function of retailer 119894
119876119894(119901119877119894 119901119877119895
) Recycling function of retailer 119894
119901119862
119894 119901119863
119895 Retailer 119894 and 119895rsquos respective retail price in
models 119862 and 119863
119901119862
119877119894 119901119863
119877119895 Retailer 119894 and 119895rsquos repurchase price in models
119862 and 119863
Π119870
119894 Profit function for channel member 119894 in
model 119896 Superscript 119896 takes the values of 119862and 119863 Subscript 119894 takes the values of 119898 119903and 119890 denoting the manufacturer theretailer and the 119890-tailer respectively
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
Corollary 3 In the case of coordinated collection the repur-chase prices 119901
119862
119877119894and 119901
119862
119877119895are strictly increasing with respect to
the recycling substitution effect
Proof According to (8) the first-order derivative of 119901119862119877119894is
d119901119862119877119894
d120574= 2119904 (119883
2
2[Φ119894+ Φ3minus119894
minus (2119888119898
minus Δ119904) (1 minus 120573)]
minus1198832
1(Φ119894minus Φ3minus119894
)) times (1198832
11198832
2)minus1
(16)
where1198831 = 1199042(1minus120573)minus4(1minus120574) and1198832 = minus4(1minus120574)minus119904
2(1+120573)
By assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1 wecan obtain that d119901119862119862
119895d120574 gt 0 thus the repurchase prices 119901
119862
119877119894
and 119901119862
119877119895are strictly increasing with respect to the recycling
substitution effect
Corollary 2 implies how the recycling substitution effectimpacts the profit of forward channel Specially themanufac-turer can obtain more profit from forward channel by induc-ing recycling substitution effect (select the lower retail priceto induce more recycling substitution effect) Corollary 3implies how the recycling substitution effect impacts theprofit of reverse channel themanufacturer can select optimalrepurchase price by inducing recycling substitution effect(inducing more recycling substitution effect to select thehigher retail price) in order to obtain more profit of reversechannel Corollaries 2 and 3 illustrate that the recyclingsubstitution effectrsquos impact on the profit of forward andreverse is converse so themanufacturer can induce a suitablerecycling substitution effect between the retailers to obtainthe maximum profit in forward and reverse supply chain
Proposition 4 In the case of coordinated collection the profitof forward channel is strictly increasing with respect to theretailing substitution effect and decreasing with respect to therecycling substitution effect While the profit of reverse channelis strictly decreasing with respect to the retailing substitutioneffect and increasing with respect to the recycling substitutioneffect
Proof First for the analysis of the profit in forward channelfor any given 119901119895 according to (1) the demand119863119894 is invariablefor the durable product and Φ119894 is constant for retailer 119894the retail price is increasing with respect to the retailingsubstitution effect Then it follows from (3) that the profitof forward channel sum2
119894=1(119901119894minus 119888119898)119863119894(119901119894 119901119895) is increasing with
respect to 119901119894and 119901
119895 Furthermore by Corollary 2 the retail
prices 119901119894and 119901
119895are both strictly decreasing with respect to
the recycling substitution effect Hence the profit of forwardchannel function sum
2
119894=1(119901119894minus 119888119898)119863119894(119901119894 119901119895) is decreasing with
respect to the recycling substitution effectSecond for the analysis of the profit in reverse channel
for any given 119901119877119894 since the recycling function119876
119894is invariable
for the durable product and 120601119894= 119904119863119894and 119901
119877119894are invariable
for retailer 119894 by (2) the repurchase price is increasing withrespect to recycling substitution effect Then the profit ofreverse channelsum2
119894=1(Δ minus 119901
119877119894)119876119894(119901119877119894 119901119877119895
) in (3) is decreasing
with respect to 119901119877119894
and 119901119877119895 According to Corollary 3 the
repurchase prices 119901119877119894
and 119901119877119895
are strictly increasing withrespect to the recycling substitution effect Therefore theprofit of reverse channel function sum
2
119894=1(Δ minus 119901
119877119894)119876119894(119901119877119894 119901119877119895
)
is decreasing with the recycling substitution effect
From what is discussed above in the case of coordi-nated collection we can obtain some managerial insights inpractice It is illustrated that the variation tendency of theforward channel profit is decided by the difference betweenthe increase of forward channel profit from the retailingsubstitution effect and the decrease of forward channelrsquos profitstemmed from the recycling substitution effect Similarly thevariation of reverse channel profit is decided by the differencebetween the increase of reverse channel profit for the retail-ing substitution effect and the decrease of reverse channelprofit for the recycling substitution effect Additionally thevariation of the total profit of forward and reverse channel isuncertain Furthermore the manufacturer can induce lowerretailing substitution effect and higher recovery substitutioneffect to collect more obsolete products and increase theprofit of reverse channel In other words the manufacturercan easily achieve the goal of remanufacturing strategy byrecovering the residual value of used products
32 Decentralized Collection In the case of decentralizedcollection the manufacturer decides on the wholesale price119908 and reimburses each retailer a fixed fee 119887 per unit returnedand the retailers compete with each other and decide on theretail price of the product as well as the repurchase price ofcollection used product
We can obtain retailer 119894rsquos reaction function by solving thefollowing optimization problem
max119901119894 119901119895
Π119863
119894= (119901119894 minus 119908)119863119894 (119901119894 119901119895) + (119887 minus 119901119877119895)119876119877119894 (119901119877119894 119901119877119895)
(17)
Proposition 5 In the case of decentralized collection theretailerrsquos optimal reaction retail price 119901
lowast119863
119894and the optimal
reaction repurchase price 119901lowast119863
119877119894satisfy
119901lowast119863
119894=
119887119904 (1 minus 120574) minus 119908 (2 minus 120574)
11988411
+(Φ119894+ Φ3minus119894
) 1198841
211988411
+(Φ119894minus Φ3minus119894
) 1198851
211988511
(18)
119901lowast119863
119877119894=
119904 (Φ119894+ Φ3minus119894
)
211988411
minus119904 (Φ119894minus Φ3minus119894
)
211988511
+ 119887[1199042(1 minus 120574)
119884111988411
+
(1 minus 1199042)1198851
119879]
minus 119908119904 [1198851
119879+ (2 minus 120574)]
(19)
where 1198841
= 1199042+ 120574 minus 2 119884
11= 1199042minus 2(2 minus 120574) + 120573(2 minus 119904
2minus 120574)
1198851 = 2 + 120574 minus 119904
2 11988511 = 2(2 + 120574) minus 120573(2 + 1199042minus 120574) minus 119904
2 and119879 = 120574
2minus 1199044minus 4(1 minus 119904
2)
6 Abstract and Applied Analysis
Proof The second-order partial derivatives of retailerrsquos profitΠ119863
119894are
1205972Π119863
119894
1205971199012
119894
= minus21205972Π119863
119894
120597119901119894120597119901119895
= 119904
1205972Π119863
119894
120597119901119895120597119901119894
= 1199041205972Π119863
119894
1205971199012
119895
= minus119904120573
(20)
Then the Hessian matrix is
|119867| =
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1205972Π119863
119894
1205971199012
119894
1205972Π119863
119894
120597119901119894120597119901119895
1205972Π119863
119894
120597119901119895120597119901119894
1205972Π119863
119894
1205971199012
119895
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(21)
By using the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt
120573 lt 1 we can obtain that |1198631| = 120597
2Π119863
1198941205971199012
119894= minus2 lt 0
and |1198632| = 2119904(120573 minus 119904) gt 0 So 119867 is negative definite Π119863 is
jointly concave with respect to 119901119894and 119901
119895 Thus the optimal
retailing and repurchase price can be obtained by the first-order condition as follows
According to (17) for retailer 1 the first-order conditionsof Π119863119894are
120597Π119863
119894
120597119901119894
= Φ119894 minus 2119901119894 + 120573119901119895 + 119908 minus 119904 (119887 minus 119901119877119894) = 0 (22)
120597Π119863
119894
120597119901119877119894
= minus119904 (Φ119894 minus 119901119894 + 120573119901119895) minus 119901119877119894 + 120574119901119877119895 + 119887 minus 119901119877119894 = 0
(23)
Similarly for retailer 2 the first-order partial conditions ofΠ119863
119895are
120597Π119863
119895
120597119901119895
= Φ119895minus 2119901119895+ 120573119901119894+ 119908 minus 119904 (119887 minus 119901
119877119895) = 0 (24)
120597Π119863
119895
120597119901119877119895
= minus119904 (Φ119895 minus 119901119895+ 120573119901119894) minus 119901
119877119895+ 120574119901119877119894
+ 119887 minus 119901119877119895
= 0
(25)
By combining (22) (23) (24) and (25) we can obtain theoptimal retail prices 119901119894 and 119901119895 and optimal repurchase prices119901119877119894 and 119901119877119895
Corollary 6 The retail price is increasing with respect to thewholesale price 119908 and decreasing with respect to the buy-backpayment 119887 But the repurchase price is decreasing with respectto the wholesale price119908 and increasing with respect to the buy-back payment 119887
Proof By the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1we can easily obtain119884
1= 1199042+120574minus2 lt 0119884
11= 1199042minus2(2minus120574)+120573(2minus
1199042minus120574) lt 0119885
1= 2+120574minus119904
2gt 0 and 119879 = 120574
2minus1199044minus4(1minus119904
2) lt 0
Thus according to (18) we can obtain that the retail prices119901119894and 119901
119895are increasing with respect to the wholesale price
119908 while decreasing with respect to the buy-back payment 119887And it follows from (19) that the repurchase prices 119901
119877119894and
119901119877119895are decreasingwith respect to thewholesale price119908 while
increasing with respect to the buy-back payment 119887
Substituting (18) and (19) into (17) yields
max119908119887
Π119863
119898= (119908 minus 119888119898)
2
sum
119894=1
119863119894 (119901lowast119863
119894 119901lowast119863
119895)
+ (Δ minus 119887)
2
sum
119894=1
119876119894(119901lowast119863
119877119894 119901lowast119863
119877119895)
(26)
Proposition 7 In the case of decentralized collection themanufacturerrsquos optimal wholesale price 119908
lowast and optimal buy-back payment 119887lowast satisfy
119908lowast
=(Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701 [1198701 (120573 + 1) + 4 (1 minus 120574)]
+1198703(1 minus 120574)
1198701(2 minus 120574)
(27)
119887lowast
=2Δ1198731+ 1205741198732+ 120573120574 (2119888
119898119904 minus 4Δ)
4120573 (2 minus 1199042) + 2120574 (8 minus 119904
2) minus 2 (2 + 120573120574) (4 minus 119904
2) (28)
where 1198701 = 4(120574 minus 1) + 1199042(2 minus 120574) 1198702 = (2119888119898 minus Δ119904)(1 minus 120574)
1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873
1= 120573(2 minus 119904
2) minus (4 minus 119904
2) and
1198732= 8Δ + (Φ
1+ Φ2minus 2119888119898)119904
Proof The second-order partial derivatives of Π119863119898are
1205972Π119863
119898
1205971199082
=41199042(2 minus 120574) (1 minus 120573) (1 minus 120574)
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
1205972Π119863
119898
1205971198872
=41199042(2 minus 120574) (1 minus 120574)
2
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
1205972Π119863
119898
120597119908120597119887
=1205972Π119863
119898
120597119887120597119908
=41199042(2 minus 120574)
2[1 minus 120573 + 2119904 (1 minus 120574) minus (2 minus 120573) (2 minus 120574)]
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
10038161003816100381610038161198671198981003816100381610038161003816 =
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1205972Π119863
119898
1205971199082
1205972Π119863
119898
120597119908120597119887
1205972Π119863
119898
120597119887120597119908
1205972Π119863119862
119898
1205971198872
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(29)
By the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1 wecan easily obtain that119867
119898is negative definite soΠ
119863
119898is jointly
concave with respect to the wholesale price 119908 and buy-back
Abstract and Applied Analysis 7
payment 119887Thus the optimal wholesale price119908 and buy-backpayment 119887 can be obtained by the first-order condition
According to (26) the first-order conditions of Π119863119898with
respect to 119908 and 119887 are as follows
120597Π119863
119898
120597119908
=2 (2 minus 120574) (2119908 minus 119887119904 + 119888
119898) + 2Δ119904
1199042+ 120574 minus 2
+ ((120574 minus 2) [2119887119904 (1 minus 120574) + (Φ1+ Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)
+ 2 (119888119898
minus 119908) (2 minus 120574 + 2119904 (119887 minus Δ)) ])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(30)
120597Π119863
119898
120597119887
=2 (1 minus 120574) (2119887 minus Δ + 119888
119898119904) minus 2119908119904 (2 minus 120574)
1199042+ 120574 minus 2
+ (119904 [2119887119904 (1 minus 120574) + (Φ1 + Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)]
+ 2119904 (1 minus 120574) [(119888119898
minus 119908) (2 minus 120574) + 119904 (119887 minus Δ)])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(31)
By combining (30) and (31) we can obtain the optimal retailwholesale price119908lowast and buy-back payment 119887lowast that is (27) and(28) hold
Proposition 8 In the case of decentralized collection theoptimal retail 119901lowast119863
119894and optimal repurchase price 119901
lowast119863
119877119894of retailer
119894 satisfy
119901lowast119863
119894
=(Φ119894 minus Φ3minus119894)
2[
1198851
11988511
minus1198841
11988411
]
+119904 (1 minus 120574)
211988411
[2Δ1198721 + 1205741198722 + 2120573120574 (119888119898119904 minus 2Δ)
21198721+ 120574119872
2
]
minus(2 minus 120574)
11988411
[(Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(1 + 120573) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
]
119901lowast119863
119877119894
= ((1 minus 120574) 119904
2
119884111988411
minus
(1 minus 1199042)1198851
119879)
times (2Δ1198721 + 1205741198723 + 120573120574 (2119888119898119904 minus 4Δ)
21198721+ 120574119872
2
) minus 119904 [(2 minus 120574) +1198851
119879]
times ((Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(120573 + 1) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
) + [119904 (Φ119894 + Φ3minus119894)
2](
1
11988411
minus1
11988511
)
(32)
where 1198721
= 120573(2 minus 1199042) minus (4 minus 119904
2) 1198722
= (8 minus 1199042) minus 120573(4 minus 119904
2)
1198723
= 8Δ + (Φ1+ Φ2minus 2119888119898)119904 1198701
= 4(120574 minus 1) + 1199042(2 minus 120574)
1198702= (2119888119898
minus Δ119904)(1 minus 120574) 1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873
1=
120573(2minus1199042)minus(4minus119904
2)1198732= 8Δ+(Φ
1+Φ2minus2119888119898)1199041198841= 1199042+120574minus2
11988411
= 1199042minus 2(2 minus 120574) + 120573(2 minus 119904
2minus 120574) 119885
1= 2 + 120574 minus 119904
2 11988511
=
2(2 + 120574) minus 120573(2 + 1199042minus 120574) minus 119904
2 and 119879 = 1205742minus 1199044minus 4(1 minus 119904
2)
Proof By substituting (27) and (28) into (18) and (19) respec-tively we can obtain the optimal retail price and repurchaseprice of retailer 119894
Proposition 9 In the case of decentralized collection themanufacturerrsquos profit of forward channel is strictly increasingwith respect to the retailing substitution effect and decreasingwith respect to the recycling substitution effect while themanufacturerrsquos reverse channelrsquos profit is increasingwith respectto the retailing substitution effect and decreasing with respect tothe recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in forward channelAccording to (1) the demand of retailing 119863
119894is invariable
for the durable product and Φ119894and 119901
119895are invariable for
retailer 119894 thus the retail price is increasing with respectto the retailing substitution effect that is increasing theretailing substitution effect 120573 will induce the increase ofretail price 119901119894 and according to Corollary 6 it will inducethe increase of wholesale price 119908 According to (4) themanufacturerrsquos profit in forward channel (119908minus119888119898)(119863119894(119901119894 119901119895)+
119863119895(119901119895 119901119894)) is increasing with respect to retailing substitutioneffect Similarly we can easily obtain that the manufacturerrsquosprofit in forward channel (119908 minus 119888119898)(119863119894(119901119894 119901119895) + 119863119895(119901119895 119901119894)) isdecreasing with respect to recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in reverse channelAccording to (2) the recycling function 119876
119894is invariable for
the durable product and 120601119894
= 119904119863119894and 119901
119877119895are invariable
8 Abstract and Applied Analysis
for retailer 119894 thus the repurchase price is increasing withrespect to recycling substitution effect Then we can easilyobtain that the increase of retailing substitution effect 120573
will induce the increase of retail price 119901119894 and according
to Corollary 6 it will induce the decrease of buy-backpayment 119887 According to (4) the manufacturerrsquos reversechannel (Δ minus 119887)(119876119894(119901119877119894 119901119877119895) + 119876
119895(119901119877119895
119901119877119894)) is increasing
with respect to retailing substitution effect Similarly we caneasily get that the manufacturerrsquos reverse channel profit (Δ minus
119887)(119876119894(119901119877119894 119901119877119895) + 119876119895(119901119877119895 119901119877119894)) is decreasing with respect torecycling substitution effect
It is illustrated that in the case of decentralized collectionthe variation of manufacturerrsquos profit in forward or reversechannel is decided by the difference between the increase offorward or reverse channel profit for the retailing substitutioneffect and the decrease of forward and reverse channelprofit for the recycling substitution effect Additionally thechange of closed-loop supply chainrsquos profit with respect tothe retailing and recycling substitution effects is uncertainin the case of decentralized collection On the other handthe manufacturer can induce higher retailing substitutioneffect and lower recovery substitution effect via choosingsuitable wholesale price and buy-back payment to collectmore obsolete products and increase the profit of reversechannel It means that the manufacturers can achieve theirgoal of remanufacturing strategy by recovering the residualvalue of used products
Proposition 10 In the case of decentralized collection theretailerrsquos profit of forward channel is strictly increasing withrespect to the retailing substitution effect and decreasing withrespect to the recycling substitution effect while his reversechannel profit is decreasing with respect to the retailing substi-tution effect and recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in forward channel Accordingto (1) the demand of retailing119863
119894is invariable for the durable
product and Φ119894and 119901
119895are invariable for retailer 119894 thus
the retail price is increasing with respect to the retailingsubstitution effect that is the increase of 120573 will induce theincrease of retail price 119901
119894 According to (5) each retailerrsquos
profit in forward channel (119901119894minus 119908)119863
119894(119901119894 119901119895) is increasing
with respect to the retailing substitution effect similarly forretailer 119895 We can easily obtain that each retailerrsquos profit inreverse channel (119901119894 minus 119908)119863119894(119901119894 119901119895) is increasing with respectto recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in reverse channel Accordingto (2) the recycling function119876119877119894 is invariable for the durableproduct and 120601119894 = 119904119863119894 and 119901119877119895 are invariable for retailer119894 thus the repurchase price is increasing with respect torecycling substitution effect Then we can easily obtain thatthe repurchase price is increasing with respect to recyclingsubstitution effect So the increase of retailing substitutioneffect 120573 will induce the increase of retail price 119901
119894 induce
the increase of buy-back payment 119887 and then will inducethe decrease of 119901
119877119894 According to (5) the retailer 119894rsquos profit in
reverse channel (119887minus119901119877119894)119876119894(119901119877119894 119901119877119895
) is decreasingwith respectto retailing substitution effect Similarly we can easily get thatthe retailer 119894rsquos profit in reverse channel (119887 minus119901
119877119894)119876119894(119901119877119894 119901119877119895
) isincreasing with respect to recycling substitution effect
Proposition 10 illustrates that in the case of decentralizedcollection the variation of retailerrsquos profit in forward channelis decided by the difference between the increase of forwardchannelrsquos profit for the retailing substitution effect and thedecrease of forward channel profit for the recycling substitu-tion effect While for the retailerrsquos profit in reverse channelit is decreasing with respect to retailing substitution effectand recycling substitution effect Additionally because thechange of forward channel profit is uncertain the variationof the closed-loop supply chainrsquos profit is also uncertain Inthis case the retailer can induce lower retailing substitutioneffect and recovery substitution effect via choosing suitablewholesale price and buy-back payment to collect moreobsolete products and increase the profit of reverse channel
In a word the variations of manufacturer and retailerrsquosrespective profit with the retailing substitution effect inforward channel are uniform but in the reverse channelthey are inconsistent For example the higher recyclingsubstitution effect will induce the manufacturer to get moreprofits in reverse channel (ie augments the scale of theman-ufacturerrsquos reverse supply chains) But the higher recyclingsubstitution effect will cause the retailer to obtain less profitsin reverse channel (ie cut down the scale of the retailerrsquosreverse supply chains) There is a contradiction betweenthe profits of the manufacturer and the retailer in reversechannel because the manufacturer has sufficient channelpower over the retailers to act as a Stackelberg leader thus themanufacturer can choose the appropriate wholesale price andbuy-back payment to achieve the remanufacturing strategyand balance the retailerrsquos profit And we make a comparisonwith the coordinated collection we can easily know thatthe coordinated collection is more effective to augment thescale of the whole reverse supply chains than decentralizedcollection
4 Numerical Examples
In this section numerical examples are provided to comparethe results obtained in the above two different collectionmodels and to analyze the impacts of retailing and recyclingsubstitution effects on the decisions and profits of chainmembers Some managerial insights can be derived throughthe numerical analysis The numerical analysis is performedfor Φ1= 100 Φ
2= 50 119888
119898= 20 Δ = 10 120574 = 04 or 120574 = 02
and 119904 = 01
41 Numerical Analysis of Model 119862 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofcoordinated collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect to retail-ing substitution effect that is the higher retailing substitution
Abstract and Applied Analysis 9
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
120574 = 02
120574 = 04
pi
(a) Impact on the retail price
0
5
10
15
20
25
30
35
40
45
50
pRi
0 02 04 06 08 1120573
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 2 Impact of the retailing and recycling substitution effects on price in model 119862
effect will induce the increase of retail price Moreover thehigher recycling substitution effect will induce the increaseof retail price the higher recycling substitution effect willinduce the decrease of repurchase price In addition thehigher retailing substitution effect will induce the decrease ofretail price The green and blue lines in Figures 2(a) and 2(b)give us a visual presentation
From the previous assumptions and analysis we knowthat the profit of forward channel is strictly increasing withrespect to retailing substitution effect and decreasing withrespect to recycling substitution effect that is the manu-facturer can induce higher retailing competition and lowerrecycling competition to obtain more profit from forwardchannel
Andwe known that the profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andincreasing with respect to recycling substitution effect thatis the manufacturer can induce lower recycling substitutioneffect to obtain more profit in the reverse channel In otherwords themanufacturer can induce higher retailing substitu-tion effect and lower recycling substitution effect could obtainmore profit in closed-loop supply chain The green and bluelines in Figures 3(a) and 3(b) give us a visual presentation
42 Numerical Analysis of Model 119863 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofdecentralized collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect toretailing substitution effect that is the higher retailing substi-tution effect will induce the increase of retail price And the
higher recycling substitution effect will induce the decreaseof retail price And we known that the repurchase priceis strictly decreasing with respect to recycling substitutioneffect that is the higher recycling substitution effect willinduce the increase of repurchase price while the higherretailing substitution effect will induce the decrease of retailprice The green and blue lines in Figures 4(a) and 4(b) giveus a visual presentation
According to the assumptions and the analysis weknow that the manufacturerrsquos profit of forward channel isstrictly increasing with respect to retailing substitution effectand decreasing with respect to recycling substitution effectAnd the manufacturerrsquos profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andrecycling substitution effect In other words in the case ofdecentralized collection the manufacturer can induce higherretailing competition and lower recycling competition toobtain more profit in the forward channel and induce lowerrecycling substitution effect to obtain more profit in thethe reverse channel So the manufacturer can select higherretailing substitution effect and lower recycling substitutioneffect to obtain more profit in their closed-loop supply chainMoreover the higher recycling competition will induce thelower profit of forward channel and induce lower profit ofreverse channel
For the forward channel profit of retailers is strictlyincreasing with respect to retailing substitution effect anddecreasing with respect to recycling substitution effect Andthe retailersrsquo profit of reverse channel is strictly decreasingwith respect to retailing substitution effect and increasingwith respect to recycling substitution effect In other wordsin the case of decentralized collection for the retailers thehigher retailing competition and lower recycling competition
10 Abstract and Applied Analysis
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(a) Impact on the profit in forward channel
0 02 04 06 08 10
200
400
600
800
1000
1200
1400
1600
1800
2000
120573
120574 = 02
120574 = 04120587
(b) Impact on the profit in reverse channel
Figure 3 Impact of the retailing and recycling substitution effects on channel profit in model 119862
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
pi
120574 = 02
120574 = 04
(a) Impact on the retail price
0 02 04 06 08 1minus20
minus15
minus10
minus5
0
5
10
15
20
25
30
120573
pi
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 4 Impact of the retailing and recycling substitution effects on price in model 119863
can make them obtain more profit in their forward channelbut the lower retailing substitution effect and higher recyclingsubstitution effect can make them obtain more profit in thereverse channel So the retailers tend to select suitable retailprice and repurchase price to induce retailing and recyclingcompetitions (not too high or too low the higher retailsubstitution effect will make the profit of reverse channel
negative or the lower recycling substitution effect will makethe profit of forward channel decrease) to obtain more profitin their closed-loop supply chain The lines in Figure 5 giveus a visual presentation
43 Comparison of Model119862 andModel119863 In this subsectionwe discuss the impact of retailing substitution effect on the
Abstract and Applied Analysis 11
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120587
(a) Impact on the manufacturerrsquos profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120587
(b) Impact on the manufacturerrsquos profit in reverse channel
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(c) Impact on the retailersrsquo profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120574 = 02
120574 = 04
120587
(d) Impact on the retailersrsquo profit in reverse channel
Figure 5 Impact of the retailing and recycling substitution effects on channel profit in model 119863119862
optimal prices and optimal channel profits and comparethem in the case of coordinated collection with that ofdecentralized collection
From the previous assumptions and analysis we knowthat the retail price in the case of coordinated collection islower than that in decentralized collection that is facingthe same circumstances the retail price in the case ofdecentralized collection is higher than that in coordinatedcollection And the repurchase price in the case of decentral-ized collection is lower than that in coordinated collectionIt implies that for the retailer the coordinated collectionhas more price advantages (lower retail price and higherrepurchase price) than the decentralized collection in the
closed-loop channel The lines in Figure 6 give us a visualpresentation
In what is discussed above we know that the channelprofit in model 119862 is higher than that in model 119863 Theprofit of reverse channel in model 119863 is very low even thevalue is negative In this case the manufacturer and retailerstend to select coordinated collection to obtain more profitsfor example HP Xerox and Apple select the coordinatedcollection to collect their used product The lines in Figure 6give us a visual presentation
44 Impact of Retailing and Recycling Substitution Effects onClosed Supply Chainrsquos Profit In this subsection we discuss
12 Abstract and Applied Analysis
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
Model CCModel DC
120573
pi
(a) Comparison of retail price
minus100
minus50
0
50
100
150
200
pRi
0 02 04 06 08 1
Model CCModel DC
120573
(b) Comparison of repurchase price
Figure 6 Pricing decisions of coordinated collection versus decentralized collection
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
Model CCModel DC
times104
120573
120587
(a) Comparison of forward channel profits
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
Model CCModel DC
120573
120587
(b) Comparison of reverse channel profits
Figure 7 The profit of coordinated collection versus decentralized collection
the impact of retailing and recycling substitution effects onthe optimal forward channel profits and optimal reversechannel profits in the cases of coordinated collection anddecentralized collection
From the previous assumptions and analysis weknow that the recycling substitution effectrsquos impact on theclosed-loop supply channel profit in the case of coordinated
collection is more obvious than that in decentralizedcollection In other words the manufacturer can obtainmore profit than in decentralized collection more easily bychanging the recycling substitution effect The reason maybe as follows in the case of coordinated collection the retailprice and repurchase price are made by the center planner(ie manufacturer) this collection model can adjust the
Abstract and Applied Analysis 13
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(a) Impact on the closed-loop supply channelrsquos profit in model 119862119862
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(b) Impact on the closed-loop supply channelrsquos profit in model119863119862
Figure 8 Impact of the retailing and recycling substitution effects on closed-Loop supply chainrsquos profit
retail and repurchase price fast and accurately But in caseof decentralized collection there is a pricing game betweenmanufacturer and retailers The manufacturer firstly selectsoptimal wholesale price and buy-back payment then theretailers select their optimal reaction retail price and repur-chase price that is the pricing game and pricing decision areunsynchronized between the manufacturer and retailer sothe manufacturer can obtain more profits in the case of coor-dinated collection than the case of decentralized collectionThe lines in Figures 7 and 8 give us a visual presentation
5 Conclusion
In this paper we investigated a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers Theresults demonstrate that more intense retailing competitioninduces the increase of the forward and reverse profitsof manufacturer and the forward channel of retailers anddecrease of the retailers profit in reverse channel while moreintense recycling competition induces the decrease of theprofits of the manufacturer and retailers in forward andreverse channels while the whole supply chainrsquos profit ofmodel 119862 is increasing with the respect to the free recyclingproportional coefficient but the effect on model 119863 is uncer-tain The results have important managerial insights for themanufacturer and retailers
This paper makes a contribution to the literature onreverse channel choice and coordination by drawing atten-tion to closed-loop supply chains with competition of retail-ing and recycling Our future research will be as follows ourmodel does not consider any fixed costs of retailingrecyclinga collection system If such costs were incurred a minimumnumber of retailingreturns would be required to justify thecost of operations Last but not least our cost formulation
does not include any operationalcapacity constraints inthe channel of products collection Operationalcapacityconstraint on the channel of collection can be incorporatedinto the model by defining an upper bound on the numbercollected via each channel Another possibility is to modelthe collection cost as a quadratic function of the quantityreturned
Notations
120574 Recycling substitution effect120573 Retailing substitution effectΦ Market size of the retailers in the forward
channel120601 Market size of the retailers by free recovering
in the reverse channel120575 Unit saving cost by remanufacturing119904 Recovering coefficient119888119903 Unit cost of remanufacturing a returned
product into a new one119888119898 Unit cost of manufacturing a new product
119908 New productrsquos wholesale price of themanufacturer
119887 Obsolete productrsquos repurchase price from theretailers to the manufacturer
119863119894(119901119894 119901119895) Demand function of retailer 119894
119876119894(119901119877119894 119901119877119895
) Recycling function of retailer 119894
119901119862
119894 119901119863
119895 Retailer 119894 and 119895rsquos respective retail price in
models 119862 and 119863
119901119862
119877119894 119901119863
119877119895 Retailer 119894 and 119895rsquos repurchase price in models
119862 and 119863
Π119870
119894 Profit function for channel member 119894 in
model 119896 Superscript 119896 takes the values of 119862and 119863 Subscript 119894 takes the values of 119898 119903and 119890 denoting the manufacturer theretailer and the 119890-tailer respectively
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
Proof The second-order partial derivatives of retailerrsquos profitΠ119863
119894are
1205972Π119863
119894
1205971199012
119894
= minus21205972Π119863
119894
120597119901119894120597119901119895
= 119904
1205972Π119863
119894
120597119901119895120597119901119894
= 1199041205972Π119863
119894
1205971199012
119895
= minus119904120573
(20)
Then the Hessian matrix is
|119867| =
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1205972Π119863
119894
1205971199012
119894
1205972Π119863
119894
120597119901119894120597119901119895
1205972Π119863
119894
120597119901119895120597119901119894
1205972Π119863
119894
1205971199012
119895
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(21)
By using the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt
120573 lt 1 we can obtain that |1198631| = 120597
2Π119863
1198941205971199012
119894= minus2 lt 0
and |1198632| = 2119904(120573 minus 119904) gt 0 So 119867 is negative definite Π119863 is
jointly concave with respect to 119901119894and 119901
119895 Thus the optimal
retailing and repurchase price can be obtained by the first-order condition as follows
According to (17) for retailer 1 the first-order conditionsof Π119863119894are
120597Π119863
119894
120597119901119894
= Φ119894 minus 2119901119894 + 120573119901119895 + 119908 minus 119904 (119887 minus 119901119877119894) = 0 (22)
120597Π119863
119894
120597119901119877119894
= minus119904 (Φ119894 minus 119901119894 + 120573119901119895) minus 119901119877119894 + 120574119901119877119895 + 119887 minus 119901119877119894 = 0
(23)
Similarly for retailer 2 the first-order partial conditions ofΠ119863
119895are
120597Π119863
119895
120597119901119895
= Φ119895minus 2119901119895+ 120573119901119894+ 119908 minus 119904 (119887 minus 119901
119877119895) = 0 (24)
120597Π119863
119895
120597119901119877119895
= minus119904 (Φ119895 minus 119901119895+ 120573119901119894) minus 119901
119877119895+ 120574119901119877119894
+ 119887 minus 119901119877119895
= 0
(25)
By combining (22) (23) (24) and (25) we can obtain theoptimal retail prices 119901119894 and 119901119895 and optimal repurchase prices119901119877119894 and 119901119877119895
Corollary 6 The retail price is increasing with respect to thewholesale price 119908 and decreasing with respect to the buy-backpayment 119887 But the repurchase price is decreasing with respectto the wholesale price119908 and increasing with respect to the buy-back payment 119887
Proof By the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1we can easily obtain119884
1= 1199042+120574minus2 lt 0119884
11= 1199042minus2(2minus120574)+120573(2minus
1199042minus120574) lt 0119885
1= 2+120574minus119904
2gt 0 and 119879 = 120574
2minus1199044minus4(1minus119904
2) lt 0
Thus according to (18) we can obtain that the retail prices119901119894and 119901
119895are increasing with respect to the wholesale price
119908 while decreasing with respect to the buy-back payment 119887And it follows from (19) that the repurchase prices 119901
119877119894and
119901119877119895are decreasingwith respect to thewholesale price119908 while
increasing with respect to the buy-back payment 119887
Substituting (18) and (19) into (17) yields
max119908119887
Π119863
119898= (119908 minus 119888119898)
2
sum
119894=1
119863119894 (119901lowast119863
119894 119901lowast119863
119895)
+ (Δ minus 119887)
2
sum
119894=1
119876119894(119901lowast119863
119877119894 119901lowast119863
119877119895)
(26)
Proposition 7 In the case of decentralized collection themanufacturerrsquos optimal wholesale price 119908
lowast and optimal buy-back payment 119887lowast satisfy
119908lowast
=(Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701 [1198701 (120573 + 1) + 4 (1 minus 120574)]
+1198703(1 minus 120574)
1198701(2 minus 120574)
(27)
119887lowast
=2Δ1198731+ 1205741198732+ 120573120574 (2119888
119898119904 minus 4Δ)
4120573 (2 minus 1199042) + 2120574 (8 minus 119904
2) minus 2 (2 + 120573120574) (4 minus 119904
2) (28)
where 1198701 = 4(120574 minus 1) + 1199042(2 minus 120574) 1198702 = (2119888119898 minus Δ119904)(1 minus 120574)
1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873
1= 120573(2 minus 119904
2) minus (4 minus 119904
2) and
1198732= 8Δ + (Φ
1+ Φ2minus 2119888119898)119904
Proof The second-order partial derivatives of Π119863119898are
1205972Π119863
119898
1205971199082
=41199042(2 minus 120574) (1 minus 120573) (1 minus 120574)
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
1205972Π119863
119898
1205971198872
=41199042(2 minus 120574) (1 minus 120574)
2
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
1205972Π119863
119898
120597119908120597119887
=1205972Π119863
119898
120597119887120597119908
=41199042(2 minus 120574)
2[1 minus 120573 + 2119904 (1 minus 120574) minus (2 minus 120573) (2 minus 120574)]
(2 minus 120574 minus 1199042) [1199042(1 minus 120573) minus (2 minus 120573) (2 minus 120574)]
10038161003816100381610038161198671198981003816100381610038161003816 =
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1205972Π119863
119898
1205971199082
1205972Π119863
119898
120597119908120597119887
1205972Π119863
119898
120597119887120597119908
1205972Π119863119862
119898
1205971198872
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(29)
By the assumptions 0 le 120574 lt 120573 lt 1 and 0 le 119904 lt 120573 lt 1 wecan easily obtain that119867
119898is negative definite soΠ
119863
119898is jointly
concave with respect to the wholesale price 119908 and buy-back
Abstract and Applied Analysis 7
payment 119887Thus the optimal wholesale price119908 and buy-backpayment 119887 can be obtained by the first-order condition
According to (26) the first-order conditions of Π119863119898with
respect to 119908 and 119887 are as follows
120597Π119863
119898
120597119908
=2 (2 minus 120574) (2119908 minus 119887119904 + 119888
119898) + 2Δ119904
1199042+ 120574 minus 2
+ ((120574 minus 2) [2119887119904 (1 minus 120574) + (Φ1+ Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)
+ 2 (119888119898
minus 119908) (2 minus 120574 + 2119904 (119887 minus Δ)) ])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(30)
120597Π119863
119898
120597119887
=2 (1 minus 120574) (2119887 minus Δ + 119888
119898119904) minus 2119908119904 (2 minus 120574)
1199042+ 120574 minus 2
+ (119904 [2119887119904 (1 minus 120574) + (Φ1 + Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)]
+ 2119904 (1 minus 120574) [(119888119898
minus 119908) (2 minus 120574) + 119904 (119887 minus Δ)])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(31)
By combining (30) and (31) we can obtain the optimal retailwholesale price119908lowast and buy-back payment 119887lowast that is (27) and(28) hold
Proposition 8 In the case of decentralized collection theoptimal retail 119901lowast119863
119894and optimal repurchase price 119901
lowast119863
119877119894of retailer
119894 satisfy
119901lowast119863
119894
=(Φ119894 minus Φ3minus119894)
2[
1198851
11988511
minus1198841
11988411
]
+119904 (1 minus 120574)
211988411
[2Δ1198721 + 1205741198722 + 2120573120574 (119888119898119904 minus 2Δ)
21198721+ 120574119872
2
]
minus(2 minus 120574)
11988411
[(Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(1 + 120573) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
]
119901lowast119863
119877119894
= ((1 minus 120574) 119904
2
119884111988411
minus
(1 minus 1199042)1198851
119879)
times (2Δ1198721 + 1205741198723 + 120573120574 (2119888119898119904 minus 4Δ)
21198721+ 120574119872
2
) minus 119904 [(2 minus 120574) +1198851
119879]
times ((Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(120573 + 1) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
) + [119904 (Φ119894 + Φ3minus119894)
2](
1
11988411
minus1
11988511
)
(32)
where 1198721
= 120573(2 minus 1199042) minus (4 minus 119904
2) 1198722
= (8 minus 1199042) minus 120573(4 minus 119904
2)
1198723
= 8Δ + (Φ1+ Φ2minus 2119888119898)119904 1198701
= 4(120574 minus 1) + 1199042(2 minus 120574)
1198702= (2119888119898
minus Δ119904)(1 minus 120574) 1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873
1=
120573(2minus1199042)minus(4minus119904
2)1198732= 8Δ+(Φ
1+Φ2minus2119888119898)1199041198841= 1199042+120574minus2
11988411
= 1199042minus 2(2 minus 120574) + 120573(2 minus 119904
2minus 120574) 119885
1= 2 + 120574 minus 119904
2 11988511
=
2(2 + 120574) minus 120573(2 + 1199042minus 120574) minus 119904
2 and 119879 = 1205742minus 1199044minus 4(1 minus 119904
2)
Proof By substituting (27) and (28) into (18) and (19) respec-tively we can obtain the optimal retail price and repurchaseprice of retailer 119894
Proposition 9 In the case of decentralized collection themanufacturerrsquos profit of forward channel is strictly increasingwith respect to the retailing substitution effect and decreasingwith respect to the recycling substitution effect while themanufacturerrsquos reverse channelrsquos profit is increasingwith respectto the retailing substitution effect and decreasing with respect tothe recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in forward channelAccording to (1) the demand of retailing 119863
119894is invariable
for the durable product and Φ119894and 119901
119895are invariable for
retailer 119894 thus the retail price is increasing with respectto the retailing substitution effect that is increasing theretailing substitution effect 120573 will induce the increase ofretail price 119901119894 and according to Corollary 6 it will inducethe increase of wholesale price 119908 According to (4) themanufacturerrsquos profit in forward channel (119908minus119888119898)(119863119894(119901119894 119901119895)+
119863119895(119901119895 119901119894)) is increasing with respect to retailing substitutioneffect Similarly we can easily obtain that the manufacturerrsquosprofit in forward channel (119908 minus 119888119898)(119863119894(119901119894 119901119895) + 119863119895(119901119895 119901119894)) isdecreasing with respect to recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in reverse channelAccording to (2) the recycling function 119876
119894is invariable for
the durable product and 120601119894
= 119904119863119894and 119901
119877119895are invariable
8 Abstract and Applied Analysis
for retailer 119894 thus the repurchase price is increasing withrespect to recycling substitution effect Then we can easilyobtain that the increase of retailing substitution effect 120573
will induce the increase of retail price 119901119894 and according
to Corollary 6 it will induce the decrease of buy-backpayment 119887 According to (4) the manufacturerrsquos reversechannel (Δ minus 119887)(119876119894(119901119877119894 119901119877119895) + 119876
119895(119901119877119895
119901119877119894)) is increasing
with respect to retailing substitution effect Similarly we caneasily get that the manufacturerrsquos reverse channel profit (Δ minus
119887)(119876119894(119901119877119894 119901119877119895) + 119876119895(119901119877119895 119901119877119894)) is decreasing with respect torecycling substitution effect
It is illustrated that in the case of decentralized collectionthe variation of manufacturerrsquos profit in forward or reversechannel is decided by the difference between the increase offorward or reverse channel profit for the retailing substitutioneffect and the decrease of forward and reverse channelprofit for the recycling substitution effect Additionally thechange of closed-loop supply chainrsquos profit with respect tothe retailing and recycling substitution effects is uncertainin the case of decentralized collection On the other handthe manufacturer can induce higher retailing substitutioneffect and lower recovery substitution effect via choosingsuitable wholesale price and buy-back payment to collectmore obsolete products and increase the profit of reversechannel It means that the manufacturers can achieve theirgoal of remanufacturing strategy by recovering the residualvalue of used products
Proposition 10 In the case of decentralized collection theretailerrsquos profit of forward channel is strictly increasing withrespect to the retailing substitution effect and decreasing withrespect to the recycling substitution effect while his reversechannel profit is decreasing with respect to the retailing substi-tution effect and recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in forward channel Accordingto (1) the demand of retailing119863
119894is invariable for the durable
product and Φ119894and 119901
119895are invariable for retailer 119894 thus
the retail price is increasing with respect to the retailingsubstitution effect that is the increase of 120573 will induce theincrease of retail price 119901
119894 According to (5) each retailerrsquos
profit in forward channel (119901119894minus 119908)119863
119894(119901119894 119901119895) is increasing
with respect to the retailing substitution effect similarly forretailer 119895 We can easily obtain that each retailerrsquos profit inreverse channel (119901119894 minus 119908)119863119894(119901119894 119901119895) is increasing with respectto recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in reverse channel Accordingto (2) the recycling function119876119877119894 is invariable for the durableproduct and 120601119894 = 119904119863119894 and 119901119877119895 are invariable for retailer119894 thus the repurchase price is increasing with respect torecycling substitution effect Then we can easily obtain thatthe repurchase price is increasing with respect to recyclingsubstitution effect So the increase of retailing substitutioneffect 120573 will induce the increase of retail price 119901
119894 induce
the increase of buy-back payment 119887 and then will inducethe decrease of 119901
119877119894 According to (5) the retailer 119894rsquos profit in
reverse channel (119887minus119901119877119894)119876119894(119901119877119894 119901119877119895
) is decreasingwith respectto retailing substitution effect Similarly we can easily get thatthe retailer 119894rsquos profit in reverse channel (119887 minus119901
119877119894)119876119894(119901119877119894 119901119877119895
) isincreasing with respect to recycling substitution effect
Proposition 10 illustrates that in the case of decentralizedcollection the variation of retailerrsquos profit in forward channelis decided by the difference between the increase of forwardchannelrsquos profit for the retailing substitution effect and thedecrease of forward channel profit for the recycling substitu-tion effect While for the retailerrsquos profit in reverse channelit is decreasing with respect to retailing substitution effectand recycling substitution effect Additionally because thechange of forward channel profit is uncertain the variationof the closed-loop supply chainrsquos profit is also uncertain Inthis case the retailer can induce lower retailing substitutioneffect and recovery substitution effect via choosing suitablewholesale price and buy-back payment to collect moreobsolete products and increase the profit of reverse channel
In a word the variations of manufacturer and retailerrsquosrespective profit with the retailing substitution effect inforward channel are uniform but in the reverse channelthey are inconsistent For example the higher recyclingsubstitution effect will induce the manufacturer to get moreprofits in reverse channel (ie augments the scale of theman-ufacturerrsquos reverse supply chains) But the higher recyclingsubstitution effect will cause the retailer to obtain less profitsin reverse channel (ie cut down the scale of the retailerrsquosreverse supply chains) There is a contradiction betweenthe profits of the manufacturer and the retailer in reversechannel because the manufacturer has sufficient channelpower over the retailers to act as a Stackelberg leader thus themanufacturer can choose the appropriate wholesale price andbuy-back payment to achieve the remanufacturing strategyand balance the retailerrsquos profit And we make a comparisonwith the coordinated collection we can easily know thatthe coordinated collection is more effective to augment thescale of the whole reverse supply chains than decentralizedcollection
4 Numerical Examples
In this section numerical examples are provided to comparethe results obtained in the above two different collectionmodels and to analyze the impacts of retailing and recyclingsubstitution effects on the decisions and profits of chainmembers Some managerial insights can be derived throughthe numerical analysis The numerical analysis is performedfor Φ1= 100 Φ
2= 50 119888
119898= 20 Δ = 10 120574 = 04 or 120574 = 02
and 119904 = 01
41 Numerical Analysis of Model 119862 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofcoordinated collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect to retail-ing substitution effect that is the higher retailing substitution
Abstract and Applied Analysis 9
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
120574 = 02
120574 = 04
pi
(a) Impact on the retail price
0
5
10
15
20
25
30
35
40
45
50
pRi
0 02 04 06 08 1120573
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 2 Impact of the retailing and recycling substitution effects on price in model 119862
effect will induce the increase of retail price Moreover thehigher recycling substitution effect will induce the increaseof retail price the higher recycling substitution effect willinduce the decrease of repurchase price In addition thehigher retailing substitution effect will induce the decrease ofretail price The green and blue lines in Figures 2(a) and 2(b)give us a visual presentation
From the previous assumptions and analysis we knowthat the profit of forward channel is strictly increasing withrespect to retailing substitution effect and decreasing withrespect to recycling substitution effect that is the manu-facturer can induce higher retailing competition and lowerrecycling competition to obtain more profit from forwardchannel
Andwe known that the profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andincreasing with respect to recycling substitution effect thatis the manufacturer can induce lower recycling substitutioneffect to obtain more profit in the reverse channel In otherwords themanufacturer can induce higher retailing substitu-tion effect and lower recycling substitution effect could obtainmore profit in closed-loop supply chain The green and bluelines in Figures 3(a) and 3(b) give us a visual presentation
42 Numerical Analysis of Model 119863 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofdecentralized collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect toretailing substitution effect that is the higher retailing substi-tution effect will induce the increase of retail price And the
higher recycling substitution effect will induce the decreaseof retail price And we known that the repurchase priceis strictly decreasing with respect to recycling substitutioneffect that is the higher recycling substitution effect willinduce the increase of repurchase price while the higherretailing substitution effect will induce the decrease of retailprice The green and blue lines in Figures 4(a) and 4(b) giveus a visual presentation
According to the assumptions and the analysis weknow that the manufacturerrsquos profit of forward channel isstrictly increasing with respect to retailing substitution effectand decreasing with respect to recycling substitution effectAnd the manufacturerrsquos profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andrecycling substitution effect In other words in the case ofdecentralized collection the manufacturer can induce higherretailing competition and lower recycling competition toobtain more profit in the forward channel and induce lowerrecycling substitution effect to obtain more profit in thethe reverse channel So the manufacturer can select higherretailing substitution effect and lower recycling substitutioneffect to obtain more profit in their closed-loop supply chainMoreover the higher recycling competition will induce thelower profit of forward channel and induce lower profit ofreverse channel
For the forward channel profit of retailers is strictlyincreasing with respect to retailing substitution effect anddecreasing with respect to recycling substitution effect Andthe retailersrsquo profit of reverse channel is strictly decreasingwith respect to retailing substitution effect and increasingwith respect to recycling substitution effect In other wordsin the case of decentralized collection for the retailers thehigher retailing competition and lower recycling competition
10 Abstract and Applied Analysis
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(a) Impact on the profit in forward channel
0 02 04 06 08 10
200
400
600
800
1000
1200
1400
1600
1800
2000
120573
120574 = 02
120574 = 04120587
(b) Impact on the profit in reverse channel
Figure 3 Impact of the retailing and recycling substitution effects on channel profit in model 119862
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
pi
120574 = 02
120574 = 04
(a) Impact on the retail price
0 02 04 06 08 1minus20
minus15
minus10
minus5
0
5
10
15
20
25
30
120573
pi
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 4 Impact of the retailing and recycling substitution effects on price in model 119863
can make them obtain more profit in their forward channelbut the lower retailing substitution effect and higher recyclingsubstitution effect can make them obtain more profit in thereverse channel So the retailers tend to select suitable retailprice and repurchase price to induce retailing and recyclingcompetitions (not too high or too low the higher retailsubstitution effect will make the profit of reverse channel
negative or the lower recycling substitution effect will makethe profit of forward channel decrease) to obtain more profitin their closed-loop supply chain The lines in Figure 5 giveus a visual presentation
43 Comparison of Model119862 andModel119863 In this subsectionwe discuss the impact of retailing substitution effect on the
Abstract and Applied Analysis 11
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120587
(a) Impact on the manufacturerrsquos profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120587
(b) Impact on the manufacturerrsquos profit in reverse channel
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(c) Impact on the retailersrsquo profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120574 = 02
120574 = 04
120587
(d) Impact on the retailersrsquo profit in reverse channel
Figure 5 Impact of the retailing and recycling substitution effects on channel profit in model 119863119862
optimal prices and optimal channel profits and comparethem in the case of coordinated collection with that ofdecentralized collection
From the previous assumptions and analysis we knowthat the retail price in the case of coordinated collection islower than that in decentralized collection that is facingthe same circumstances the retail price in the case ofdecentralized collection is higher than that in coordinatedcollection And the repurchase price in the case of decentral-ized collection is lower than that in coordinated collectionIt implies that for the retailer the coordinated collectionhas more price advantages (lower retail price and higherrepurchase price) than the decentralized collection in the
closed-loop channel The lines in Figure 6 give us a visualpresentation
In what is discussed above we know that the channelprofit in model 119862 is higher than that in model 119863 Theprofit of reverse channel in model 119863 is very low even thevalue is negative In this case the manufacturer and retailerstend to select coordinated collection to obtain more profitsfor example HP Xerox and Apple select the coordinatedcollection to collect their used product The lines in Figure 6give us a visual presentation
44 Impact of Retailing and Recycling Substitution Effects onClosed Supply Chainrsquos Profit In this subsection we discuss
12 Abstract and Applied Analysis
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
Model CCModel DC
120573
pi
(a) Comparison of retail price
minus100
minus50
0
50
100
150
200
pRi
0 02 04 06 08 1
Model CCModel DC
120573
(b) Comparison of repurchase price
Figure 6 Pricing decisions of coordinated collection versus decentralized collection
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
Model CCModel DC
times104
120573
120587
(a) Comparison of forward channel profits
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
Model CCModel DC
120573
120587
(b) Comparison of reverse channel profits
Figure 7 The profit of coordinated collection versus decentralized collection
the impact of retailing and recycling substitution effects onthe optimal forward channel profits and optimal reversechannel profits in the cases of coordinated collection anddecentralized collection
From the previous assumptions and analysis weknow that the recycling substitution effectrsquos impact on theclosed-loop supply channel profit in the case of coordinated
collection is more obvious than that in decentralizedcollection In other words the manufacturer can obtainmore profit than in decentralized collection more easily bychanging the recycling substitution effect The reason maybe as follows in the case of coordinated collection the retailprice and repurchase price are made by the center planner(ie manufacturer) this collection model can adjust the
Abstract and Applied Analysis 13
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(a) Impact on the closed-loop supply channelrsquos profit in model 119862119862
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(b) Impact on the closed-loop supply channelrsquos profit in model119863119862
Figure 8 Impact of the retailing and recycling substitution effects on closed-Loop supply chainrsquos profit
retail and repurchase price fast and accurately But in caseof decentralized collection there is a pricing game betweenmanufacturer and retailers The manufacturer firstly selectsoptimal wholesale price and buy-back payment then theretailers select their optimal reaction retail price and repur-chase price that is the pricing game and pricing decision areunsynchronized between the manufacturer and retailer sothe manufacturer can obtain more profits in the case of coor-dinated collection than the case of decentralized collectionThe lines in Figures 7 and 8 give us a visual presentation
5 Conclusion
In this paper we investigated a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers Theresults demonstrate that more intense retailing competitioninduces the increase of the forward and reverse profitsof manufacturer and the forward channel of retailers anddecrease of the retailers profit in reverse channel while moreintense recycling competition induces the decrease of theprofits of the manufacturer and retailers in forward andreverse channels while the whole supply chainrsquos profit ofmodel 119862 is increasing with the respect to the free recyclingproportional coefficient but the effect on model 119863 is uncer-tain The results have important managerial insights for themanufacturer and retailers
This paper makes a contribution to the literature onreverse channel choice and coordination by drawing atten-tion to closed-loop supply chains with competition of retail-ing and recycling Our future research will be as follows ourmodel does not consider any fixed costs of retailingrecyclinga collection system If such costs were incurred a minimumnumber of retailingreturns would be required to justify thecost of operations Last but not least our cost formulation
does not include any operationalcapacity constraints inthe channel of products collection Operationalcapacityconstraint on the channel of collection can be incorporatedinto the model by defining an upper bound on the numbercollected via each channel Another possibility is to modelthe collection cost as a quadratic function of the quantityreturned
Notations
120574 Recycling substitution effect120573 Retailing substitution effectΦ Market size of the retailers in the forward
channel120601 Market size of the retailers by free recovering
in the reverse channel120575 Unit saving cost by remanufacturing119904 Recovering coefficient119888119903 Unit cost of remanufacturing a returned
product into a new one119888119898 Unit cost of manufacturing a new product
119908 New productrsquos wholesale price of themanufacturer
119887 Obsolete productrsquos repurchase price from theretailers to the manufacturer
119863119894(119901119894 119901119895) Demand function of retailer 119894
119876119894(119901119877119894 119901119877119895
) Recycling function of retailer 119894
119901119862
119894 119901119863
119895 Retailer 119894 and 119895rsquos respective retail price in
models 119862 and 119863
119901119862
119877119894 119901119863
119877119895 Retailer 119894 and 119895rsquos repurchase price in models
119862 and 119863
Π119870
119894 Profit function for channel member 119894 in
model 119896 Superscript 119896 takes the values of 119862and 119863 Subscript 119894 takes the values of 119898 119903and 119890 denoting the manufacturer theretailer and the 119890-tailer respectively
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
payment 119887Thus the optimal wholesale price119908 and buy-backpayment 119887 can be obtained by the first-order condition
According to (26) the first-order conditions of Π119863119898with
respect to 119908 and 119887 are as follows
120597Π119863
119898
120597119908
=2 (2 minus 120574) (2119908 minus 119887119904 + 119888
119898) + 2Δ119904
1199042+ 120574 minus 2
+ ((120574 minus 2) [2119887119904 (1 minus 120574) + (Φ1+ Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)
+ 2 (119888119898
minus 119908) (2 minus 120574 + 2119904 (119887 minus Δ)) ])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(30)
120597Π119863
119898
120597119887
=2 (1 minus 120574) (2119887 minus Δ + 119888
119898119904) minus 2119908119904 (2 minus 120574)
1199042+ 120574 minus 2
+ (119904 [2119887119904 (1 minus 120574) + (Φ1 + Φ2) (1199042+ 120574 minus 2)
+ 2119908 (120574 minus 2)]
+ 2119904 (1 minus 120574) [(119888119898
minus 119908) (2 minus 120574) + 119904 (119887 minus Δ)])
times ((1199042+ 120574 minus 2)[2120574 minus 120573(119904
2+ 120574 minus 2) + 119904
2minus 4])minus1
= 0
(31)
By combining (30) and (31) we can obtain the optimal retailwholesale price119908lowast and buy-back payment 119887lowast that is (27) and(28) hold
Proposition 8 In the case of decentralized collection theoptimal retail 119901lowast119863
119894and optimal repurchase price 119901
lowast119863
119877119894of retailer
119894 satisfy
119901lowast119863
119894
=(Φ119894 minus Φ3minus119894)
2[
1198851
11988511
minus1198841
11988411
]
+119904 (1 minus 120574)
211988411
[2Δ1198721 + 1205741198722 + 2120573120574 (119888119898119904 minus 2Δ)
21198721+ 120574119872
2
]
minus(2 minus 120574)
11988411
[(Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(1 + 120573) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
]
119901lowast119863
119877119894
= ((1 minus 120574) 119904
2
119884111988411
minus
(1 minus 1199042)1198851
119879)
times (2Δ1198721 + 1205741198723 + 120573120574 (2119888119898119904 minus 4Δ)
21198721+ 120574119872
2
) minus 119904 [(2 minus 120574) +1198851
119879]
times ((Φ1+ Φ2)
4 (1 minus 120573)+
1205741199042[41198702minus 1198701(Φ1+ Φ2)]
41198701[1198701(120573 + 1) + 4 (1 minus 120574)]
+1198703 (1 minus 120574)
1198701(2 minus 120574)
) + [119904 (Φ119894 + Φ3minus119894)
2](
1
11988411
minus1
11988511
)
(32)
where 1198721
= 120573(2 minus 1199042) minus (4 minus 119904
2) 1198722
= (8 minus 1199042) minus 120573(4 minus 119904
2)
1198723
= 8Δ + (Φ1+ Φ2minus 2119888119898)119904 1198701
= 4(120574 minus 1) + 1199042(2 minus 120574)
1198702= (2119888119898
minus Δ119904)(1 minus 120574) 1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873
1=
120573(2minus1199042)minus(4minus119904
2)1198732= 8Δ+(Φ
1+Φ2minus2119888119898)1199041198841= 1199042+120574minus2
11988411
= 1199042minus 2(2 minus 120574) + 120573(2 minus 119904
2minus 120574) 119885
1= 2 + 120574 minus 119904
2 11988511
=
2(2 + 120574) minus 120573(2 + 1199042minus 120574) minus 119904
2 and 119879 = 1205742minus 1199044minus 4(1 minus 119904
2)
Proof By substituting (27) and (28) into (18) and (19) respec-tively we can obtain the optimal retail price and repurchaseprice of retailer 119894
Proposition 9 In the case of decentralized collection themanufacturerrsquos profit of forward channel is strictly increasingwith respect to the retailing substitution effect and decreasingwith respect to the recycling substitution effect while themanufacturerrsquos reverse channelrsquos profit is increasingwith respectto the retailing substitution effect and decreasing with respect tothe recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in forward channelAccording to (1) the demand of retailing 119863
119894is invariable
for the durable product and Φ119894and 119901
119895are invariable for
retailer 119894 thus the retail price is increasing with respectto the retailing substitution effect that is increasing theretailing substitution effect 120573 will induce the increase ofretail price 119901119894 and according to Corollary 6 it will inducethe increase of wholesale price 119908 According to (4) themanufacturerrsquos profit in forward channel (119908minus119888119898)(119863119894(119901119894 119901119895)+
119863119895(119901119895 119901119894)) is increasing with respect to retailing substitutioneffect Similarly we can easily obtain that the manufacturerrsquosprofit in forward channel (119908 minus 119888119898)(119863119894(119901119894 119901119895) + 119863119895(119901119895 119901119894)) isdecreasing with respect to recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the manufacturerrsquos profit in reverse channelAccording to (2) the recycling function 119876
119894is invariable for
the durable product and 120601119894
= 119904119863119894and 119901
119877119895are invariable
8 Abstract and Applied Analysis
for retailer 119894 thus the repurchase price is increasing withrespect to recycling substitution effect Then we can easilyobtain that the increase of retailing substitution effect 120573
will induce the increase of retail price 119901119894 and according
to Corollary 6 it will induce the decrease of buy-backpayment 119887 According to (4) the manufacturerrsquos reversechannel (Δ minus 119887)(119876119894(119901119877119894 119901119877119895) + 119876
119895(119901119877119895
119901119877119894)) is increasing
with respect to retailing substitution effect Similarly we caneasily get that the manufacturerrsquos reverse channel profit (Δ minus
119887)(119876119894(119901119877119894 119901119877119895) + 119876119895(119901119877119895 119901119877119894)) is decreasing with respect torecycling substitution effect
It is illustrated that in the case of decentralized collectionthe variation of manufacturerrsquos profit in forward or reversechannel is decided by the difference between the increase offorward or reverse channel profit for the retailing substitutioneffect and the decrease of forward and reverse channelprofit for the recycling substitution effect Additionally thechange of closed-loop supply chainrsquos profit with respect tothe retailing and recycling substitution effects is uncertainin the case of decentralized collection On the other handthe manufacturer can induce higher retailing substitutioneffect and lower recovery substitution effect via choosingsuitable wholesale price and buy-back payment to collectmore obsolete products and increase the profit of reversechannel It means that the manufacturers can achieve theirgoal of remanufacturing strategy by recovering the residualvalue of used products
Proposition 10 In the case of decentralized collection theretailerrsquos profit of forward channel is strictly increasing withrespect to the retailing substitution effect and decreasing withrespect to the recycling substitution effect while his reversechannel profit is decreasing with respect to the retailing substi-tution effect and recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in forward channel Accordingto (1) the demand of retailing119863
119894is invariable for the durable
product and Φ119894and 119901
119895are invariable for retailer 119894 thus
the retail price is increasing with respect to the retailingsubstitution effect that is the increase of 120573 will induce theincrease of retail price 119901
119894 According to (5) each retailerrsquos
profit in forward channel (119901119894minus 119908)119863
119894(119901119894 119901119895) is increasing
with respect to the retailing substitution effect similarly forretailer 119895 We can easily obtain that each retailerrsquos profit inreverse channel (119901119894 minus 119908)119863119894(119901119894 119901119895) is increasing with respectto recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in reverse channel Accordingto (2) the recycling function119876119877119894 is invariable for the durableproduct and 120601119894 = 119904119863119894 and 119901119877119895 are invariable for retailer119894 thus the repurchase price is increasing with respect torecycling substitution effect Then we can easily obtain thatthe repurchase price is increasing with respect to recyclingsubstitution effect So the increase of retailing substitutioneffect 120573 will induce the increase of retail price 119901
119894 induce
the increase of buy-back payment 119887 and then will inducethe decrease of 119901
119877119894 According to (5) the retailer 119894rsquos profit in
reverse channel (119887minus119901119877119894)119876119894(119901119877119894 119901119877119895
) is decreasingwith respectto retailing substitution effect Similarly we can easily get thatthe retailer 119894rsquos profit in reverse channel (119887 minus119901
119877119894)119876119894(119901119877119894 119901119877119895
) isincreasing with respect to recycling substitution effect
Proposition 10 illustrates that in the case of decentralizedcollection the variation of retailerrsquos profit in forward channelis decided by the difference between the increase of forwardchannelrsquos profit for the retailing substitution effect and thedecrease of forward channel profit for the recycling substitu-tion effect While for the retailerrsquos profit in reverse channelit is decreasing with respect to retailing substitution effectand recycling substitution effect Additionally because thechange of forward channel profit is uncertain the variationof the closed-loop supply chainrsquos profit is also uncertain Inthis case the retailer can induce lower retailing substitutioneffect and recovery substitution effect via choosing suitablewholesale price and buy-back payment to collect moreobsolete products and increase the profit of reverse channel
In a word the variations of manufacturer and retailerrsquosrespective profit with the retailing substitution effect inforward channel are uniform but in the reverse channelthey are inconsistent For example the higher recyclingsubstitution effect will induce the manufacturer to get moreprofits in reverse channel (ie augments the scale of theman-ufacturerrsquos reverse supply chains) But the higher recyclingsubstitution effect will cause the retailer to obtain less profitsin reverse channel (ie cut down the scale of the retailerrsquosreverse supply chains) There is a contradiction betweenthe profits of the manufacturer and the retailer in reversechannel because the manufacturer has sufficient channelpower over the retailers to act as a Stackelberg leader thus themanufacturer can choose the appropriate wholesale price andbuy-back payment to achieve the remanufacturing strategyand balance the retailerrsquos profit And we make a comparisonwith the coordinated collection we can easily know thatthe coordinated collection is more effective to augment thescale of the whole reverse supply chains than decentralizedcollection
4 Numerical Examples
In this section numerical examples are provided to comparethe results obtained in the above two different collectionmodels and to analyze the impacts of retailing and recyclingsubstitution effects on the decisions and profits of chainmembers Some managerial insights can be derived throughthe numerical analysis The numerical analysis is performedfor Φ1= 100 Φ
2= 50 119888
119898= 20 Δ = 10 120574 = 04 or 120574 = 02
and 119904 = 01
41 Numerical Analysis of Model 119862 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofcoordinated collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect to retail-ing substitution effect that is the higher retailing substitution
Abstract and Applied Analysis 9
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
120574 = 02
120574 = 04
pi
(a) Impact on the retail price
0
5
10
15
20
25
30
35
40
45
50
pRi
0 02 04 06 08 1120573
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 2 Impact of the retailing and recycling substitution effects on price in model 119862
effect will induce the increase of retail price Moreover thehigher recycling substitution effect will induce the increaseof retail price the higher recycling substitution effect willinduce the decrease of repurchase price In addition thehigher retailing substitution effect will induce the decrease ofretail price The green and blue lines in Figures 2(a) and 2(b)give us a visual presentation
From the previous assumptions and analysis we knowthat the profit of forward channel is strictly increasing withrespect to retailing substitution effect and decreasing withrespect to recycling substitution effect that is the manu-facturer can induce higher retailing competition and lowerrecycling competition to obtain more profit from forwardchannel
Andwe known that the profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andincreasing with respect to recycling substitution effect thatis the manufacturer can induce lower recycling substitutioneffect to obtain more profit in the reverse channel In otherwords themanufacturer can induce higher retailing substitu-tion effect and lower recycling substitution effect could obtainmore profit in closed-loop supply chain The green and bluelines in Figures 3(a) and 3(b) give us a visual presentation
42 Numerical Analysis of Model 119863 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofdecentralized collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect toretailing substitution effect that is the higher retailing substi-tution effect will induce the increase of retail price And the
higher recycling substitution effect will induce the decreaseof retail price And we known that the repurchase priceis strictly decreasing with respect to recycling substitutioneffect that is the higher recycling substitution effect willinduce the increase of repurchase price while the higherretailing substitution effect will induce the decrease of retailprice The green and blue lines in Figures 4(a) and 4(b) giveus a visual presentation
According to the assumptions and the analysis weknow that the manufacturerrsquos profit of forward channel isstrictly increasing with respect to retailing substitution effectand decreasing with respect to recycling substitution effectAnd the manufacturerrsquos profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andrecycling substitution effect In other words in the case ofdecentralized collection the manufacturer can induce higherretailing competition and lower recycling competition toobtain more profit in the forward channel and induce lowerrecycling substitution effect to obtain more profit in thethe reverse channel So the manufacturer can select higherretailing substitution effect and lower recycling substitutioneffect to obtain more profit in their closed-loop supply chainMoreover the higher recycling competition will induce thelower profit of forward channel and induce lower profit ofreverse channel
For the forward channel profit of retailers is strictlyincreasing with respect to retailing substitution effect anddecreasing with respect to recycling substitution effect Andthe retailersrsquo profit of reverse channel is strictly decreasingwith respect to retailing substitution effect and increasingwith respect to recycling substitution effect In other wordsin the case of decentralized collection for the retailers thehigher retailing competition and lower recycling competition
10 Abstract and Applied Analysis
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(a) Impact on the profit in forward channel
0 02 04 06 08 10
200
400
600
800
1000
1200
1400
1600
1800
2000
120573
120574 = 02
120574 = 04120587
(b) Impact on the profit in reverse channel
Figure 3 Impact of the retailing and recycling substitution effects on channel profit in model 119862
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
pi
120574 = 02
120574 = 04
(a) Impact on the retail price
0 02 04 06 08 1minus20
minus15
minus10
minus5
0
5
10
15
20
25
30
120573
pi
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 4 Impact of the retailing and recycling substitution effects on price in model 119863
can make them obtain more profit in their forward channelbut the lower retailing substitution effect and higher recyclingsubstitution effect can make them obtain more profit in thereverse channel So the retailers tend to select suitable retailprice and repurchase price to induce retailing and recyclingcompetitions (not too high or too low the higher retailsubstitution effect will make the profit of reverse channel
negative or the lower recycling substitution effect will makethe profit of forward channel decrease) to obtain more profitin their closed-loop supply chain The lines in Figure 5 giveus a visual presentation
43 Comparison of Model119862 andModel119863 In this subsectionwe discuss the impact of retailing substitution effect on the
Abstract and Applied Analysis 11
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120587
(a) Impact on the manufacturerrsquos profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120587
(b) Impact on the manufacturerrsquos profit in reverse channel
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(c) Impact on the retailersrsquo profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120574 = 02
120574 = 04
120587
(d) Impact on the retailersrsquo profit in reverse channel
Figure 5 Impact of the retailing and recycling substitution effects on channel profit in model 119863119862
optimal prices and optimal channel profits and comparethem in the case of coordinated collection with that ofdecentralized collection
From the previous assumptions and analysis we knowthat the retail price in the case of coordinated collection islower than that in decentralized collection that is facingthe same circumstances the retail price in the case ofdecentralized collection is higher than that in coordinatedcollection And the repurchase price in the case of decentral-ized collection is lower than that in coordinated collectionIt implies that for the retailer the coordinated collectionhas more price advantages (lower retail price and higherrepurchase price) than the decentralized collection in the
closed-loop channel The lines in Figure 6 give us a visualpresentation
In what is discussed above we know that the channelprofit in model 119862 is higher than that in model 119863 Theprofit of reverse channel in model 119863 is very low even thevalue is negative In this case the manufacturer and retailerstend to select coordinated collection to obtain more profitsfor example HP Xerox and Apple select the coordinatedcollection to collect their used product The lines in Figure 6give us a visual presentation
44 Impact of Retailing and Recycling Substitution Effects onClosed Supply Chainrsquos Profit In this subsection we discuss
12 Abstract and Applied Analysis
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
Model CCModel DC
120573
pi
(a) Comparison of retail price
minus100
minus50
0
50
100
150
200
pRi
0 02 04 06 08 1
Model CCModel DC
120573
(b) Comparison of repurchase price
Figure 6 Pricing decisions of coordinated collection versus decentralized collection
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
Model CCModel DC
times104
120573
120587
(a) Comparison of forward channel profits
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
Model CCModel DC
120573
120587
(b) Comparison of reverse channel profits
Figure 7 The profit of coordinated collection versus decentralized collection
the impact of retailing and recycling substitution effects onthe optimal forward channel profits and optimal reversechannel profits in the cases of coordinated collection anddecentralized collection
From the previous assumptions and analysis weknow that the recycling substitution effectrsquos impact on theclosed-loop supply channel profit in the case of coordinated
collection is more obvious than that in decentralizedcollection In other words the manufacturer can obtainmore profit than in decentralized collection more easily bychanging the recycling substitution effect The reason maybe as follows in the case of coordinated collection the retailprice and repurchase price are made by the center planner(ie manufacturer) this collection model can adjust the
Abstract and Applied Analysis 13
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(a) Impact on the closed-loop supply channelrsquos profit in model 119862119862
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(b) Impact on the closed-loop supply channelrsquos profit in model119863119862
Figure 8 Impact of the retailing and recycling substitution effects on closed-Loop supply chainrsquos profit
retail and repurchase price fast and accurately But in caseof decentralized collection there is a pricing game betweenmanufacturer and retailers The manufacturer firstly selectsoptimal wholesale price and buy-back payment then theretailers select their optimal reaction retail price and repur-chase price that is the pricing game and pricing decision areunsynchronized between the manufacturer and retailer sothe manufacturer can obtain more profits in the case of coor-dinated collection than the case of decentralized collectionThe lines in Figures 7 and 8 give us a visual presentation
5 Conclusion
In this paper we investigated a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers Theresults demonstrate that more intense retailing competitioninduces the increase of the forward and reverse profitsof manufacturer and the forward channel of retailers anddecrease of the retailers profit in reverse channel while moreintense recycling competition induces the decrease of theprofits of the manufacturer and retailers in forward andreverse channels while the whole supply chainrsquos profit ofmodel 119862 is increasing with the respect to the free recyclingproportional coefficient but the effect on model 119863 is uncer-tain The results have important managerial insights for themanufacturer and retailers
This paper makes a contribution to the literature onreverse channel choice and coordination by drawing atten-tion to closed-loop supply chains with competition of retail-ing and recycling Our future research will be as follows ourmodel does not consider any fixed costs of retailingrecyclinga collection system If such costs were incurred a minimumnumber of retailingreturns would be required to justify thecost of operations Last but not least our cost formulation
does not include any operationalcapacity constraints inthe channel of products collection Operationalcapacityconstraint on the channel of collection can be incorporatedinto the model by defining an upper bound on the numbercollected via each channel Another possibility is to modelthe collection cost as a quadratic function of the quantityreturned
Notations
120574 Recycling substitution effect120573 Retailing substitution effectΦ Market size of the retailers in the forward
channel120601 Market size of the retailers by free recovering
in the reverse channel120575 Unit saving cost by remanufacturing119904 Recovering coefficient119888119903 Unit cost of remanufacturing a returned
product into a new one119888119898 Unit cost of manufacturing a new product
119908 New productrsquos wholesale price of themanufacturer
119887 Obsolete productrsquos repurchase price from theretailers to the manufacturer
119863119894(119901119894 119901119895) Demand function of retailer 119894
119876119894(119901119877119894 119901119877119895
) Recycling function of retailer 119894
119901119862
119894 119901119863
119895 Retailer 119894 and 119895rsquos respective retail price in
models 119862 and 119863
119901119862
119877119894 119901119863
119877119895 Retailer 119894 and 119895rsquos repurchase price in models
119862 and 119863
Π119870
119894 Profit function for channel member 119894 in
model 119896 Superscript 119896 takes the values of 119862and 119863 Subscript 119894 takes the values of 119898 119903and 119890 denoting the manufacturer theretailer and the 119890-tailer respectively
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
for retailer 119894 thus the repurchase price is increasing withrespect to recycling substitution effect Then we can easilyobtain that the increase of retailing substitution effect 120573
will induce the increase of retail price 119901119894 and according
to Corollary 6 it will induce the decrease of buy-backpayment 119887 According to (4) the manufacturerrsquos reversechannel (Δ minus 119887)(119876119894(119901119877119894 119901119877119895) + 119876
119895(119901119877119895
119901119877119894)) is increasing
with respect to retailing substitution effect Similarly we caneasily get that the manufacturerrsquos reverse channel profit (Δ minus
119887)(119876119894(119901119877119894 119901119877119895) + 119876119895(119901119877119895 119901119877119894)) is decreasing with respect torecycling substitution effect
It is illustrated that in the case of decentralized collectionthe variation of manufacturerrsquos profit in forward or reversechannel is decided by the difference between the increase offorward or reverse channel profit for the retailing substitutioneffect and the decrease of forward and reverse channelprofit for the recycling substitution effect Additionally thechange of closed-loop supply chainrsquos profit with respect tothe retailing and recycling substitution effects is uncertainin the case of decentralized collection On the other handthe manufacturer can induce higher retailing substitutioneffect and lower recovery substitution effect via choosingsuitable wholesale price and buy-back payment to collectmore obsolete products and increase the profit of reversechannel It means that the manufacturers can achieve theirgoal of remanufacturing strategy by recovering the residualvalue of used products
Proposition 10 In the case of decentralized collection theretailerrsquos profit of forward channel is strictly increasing withrespect to the retailing substitution effect and decreasing withrespect to the recycling substitution effect while his reversechannel profit is decreasing with respect to the retailing substi-tution effect and recycling substitution effect
Proof First we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in forward channel Accordingto (1) the demand of retailing119863
119894is invariable for the durable
product and Φ119894and 119901
119895are invariable for retailer 119894 thus
the retail price is increasing with respect to the retailingsubstitution effect that is the increase of 120573 will induce theincrease of retail price 119901
119894 According to (5) each retailerrsquos
profit in forward channel (119901119894minus 119908)119863
119894(119901119894 119901119895) is increasing
with respect to the retailing substitution effect similarly forretailer 119895 We can easily obtain that each retailerrsquos profit inreverse channel (119901119894 minus 119908)119863119894(119901119894 119901119895) is increasing with respectto recycling substitution effect
Second we analyze the retailing substitution effectrsquosimpact on the retailerrsquos profit in reverse channel Accordingto (2) the recycling function119876119877119894 is invariable for the durableproduct and 120601119894 = 119904119863119894 and 119901119877119895 are invariable for retailer119894 thus the repurchase price is increasing with respect torecycling substitution effect Then we can easily obtain thatthe repurchase price is increasing with respect to recyclingsubstitution effect So the increase of retailing substitutioneffect 120573 will induce the increase of retail price 119901
119894 induce
the increase of buy-back payment 119887 and then will inducethe decrease of 119901
119877119894 According to (5) the retailer 119894rsquos profit in
reverse channel (119887minus119901119877119894)119876119894(119901119877119894 119901119877119895
) is decreasingwith respectto retailing substitution effect Similarly we can easily get thatthe retailer 119894rsquos profit in reverse channel (119887 minus119901
119877119894)119876119894(119901119877119894 119901119877119895
) isincreasing with respect to recycling substitution effect
Proposition 10 illustrates that in the case of decentralizedcollection the variation of retailerrsquos profit in forward channelis decided by the difference between the increase of forwardchannelrsquos profit for the retailing substitution effect and thedecrease of forward channel profit for the recycling substitu-tion effect While for the retailerrsquos profit in reverse channelit is decreasing with respect to retailing substitution effectand recycling substitution effect Additionally because thechange of forward channel profit is uncertain the variationof the closed-loop supply chainrsquos profit is also uncertain Inthis case the retailer can induce lower retailing substitutioneffect and recovery substitution effect via choosing suitablewholesale price and buy-back payment to collect moreobsolete products and increase the profit of reverse channel
In a word the variations of manufacturer and retailerrsquosrespective profit with the retailing substitution effect inforward channel are uniform but in the reverse channelthey are inconsistent For example the higher recyclingsubstitution effect will induce the manufacturer to get moreprofits in reverse channel (ie augments the scale of theman-ufacturerrsquos reverse supply chains) But the higher recyclingsubstitution effect will cause the retailer to obtain less profitsin reverse channel (ie cut down the scale of the retailerrsquosreverse supply chains) There is a contradiction betweenthe profits of the manufacturer and the retailer in reversechannel because the manufacturer has sufficient channelpower over the retailers to act as a Stackelberg leader thus themanufacturer can choose the appropriate wholesale price andbuy-back payment to achieve the remanufacturing strategyand balance the retailerrsquos profit And we make a comparisonwith the coordinated collection we can easily know thatthe coordinated collection is more effective to augment thescale of the whole reverse supply chains than decentralizedcollection
4 Numerical Examples
In this section numerical examples are provided to comparethe results obtained in the above two different collectionmodels and to analyze the impacts of retailing and recyclingsubstitution effects on the decisions and profits of chainmembers Some managerial insights can be derived throughthe numerical analysis The numerical analysis is performedfor Φ1= 100 Φ
2= 50 119888
119898= 20 Δ = 10 120574 = 04 or 120574 = 02
and 119904 = 01
41 Numerical Analysis of Model 119862 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofcoordinated collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect to retail-ing substitution effect that is the higher retailing substitution
Abstract and Applied Analysis 9
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
120574 = 02
120574 = 04
pi
(a) Impact on the retail price
0
5
10
15
20
25
30
35
40
45
50
pRi
0 02 04 06 08 1120573
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 2 Impact of the retailing and recycling substitution effects on price in model 119862
effect will induce the increase of retail price Moreover thehigher recycling substitution effect will induce the increaseof retail price the higher recycling substitution effect willinduce the decrease of repurchase price In addition thehigher retailing substitution effect will induce the decrease ofretail price The green and blue lines in Figures 2(a) and 2(b)give us a visual presentation
From the previous assumptions and analysis we knowthat the profit of forward channel is strictly increasing withrespect to retailing substitution effect and decreasing withrespect to recycling substitution effect that is the manu-facturer can induce higher retailing competition and lowerrecycling competition to obtain more profit from forwardchannel
Andwe known that the profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andincreasing with respect to recycling substitution effect thatis the manufacturer can induce lower recycling substitutioneffect to obtain more profit in the reverse channel In otherwords themanufacturer can induce higher retailing substitu-tion effect and lower recycling substitution effect could obtainmore profit in closed-loop supply chain The green and bluelines in Figures 3(a) and 3(b) give us a visual presentation
42 Numerical Analysis of Model 119863 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofdecentralized collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect toretailing substitution effect that is the higher retailing substi-tution effect will induce the increase of retail price And the
higher recycling substitution effect will induce the decreaseof retail price And we known that the repurchase priceis strictly decreasing with respect to recycling substitutioneffect that is the higher recycling substitution effect willinduce the increase of repurchase price while the higherretailing substitution effect will induce the decrease of retailprice The green and blue lines in Figures 4(a) and 4(b) giveus a visual presentation
According to the assumptions and the analysis weknow that the manufacturerrsquos profit of forward channel isstrictly increasing with respect to retailing substitution effectand decreasing with respect to recycling substitution effectAnd the manufacturerrsquos profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andrecycling substitution effect In other words in the case ofdecentralized collection the manufacturer can induce higherretailing competition and lower recycling competition toobtain more profit in the forward channel and induce lowerrecycling substitution effect to obtain more profit in thethe reverse channel So the manufacturer can select higherretailing substitution effect and lower recycling substitutioneffect to obtain more profit in their closed-loop supply chainMoreover the higher recycling competition will induce thelower profit of forward channel and induce lower profit ofreverse channel
For the forward channel profit of retailers is strictlyincreasing with respect to retailing substitution effect anddecreasing with respect to recycling substitution effect Andthe retailersrsquo profit of reverse channel is strictly decreasingwith respect to retailing substitution effect and increasingwith respect to recycling substitution effect In other wordsin the case of decentralized collection for the retailers thehigher retailing competition and lower recycling competition
10 Abstract and Applied Analysis
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(a) Impact on the profit in forward channel
0 02 04 06 08 10
200
400
600
800
1000
1200
1400
1600
1800
2000
120573
120574 = 02
120574 = 04120587
(b) Impact on the profit in reverse channel
Figure 3 Impact of the retailing and recycling substitution effects on channel profit in model 119862
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
pi
120574 = 02
120574 = 04
(a) Impact on the retail price
0 02 04 06 08 1minus20
minus15
minus10
minus5
0
5
10
15
20
25
30
120573
pi
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 4 Impact of the retailing and recycling substitution effects on price in model 119863
can make them obtain more profit in their forward channelbut the lower retailing substitution effect and higher recyclingsubstitution effect can make them obtain more profit in thereverse channel So the retailers tend to select suitable retailprice and repurchase price to induce retailing and recyclingcompetitions (not too high or too low the higher retailsubstitution effect will make the profit of reverse channel
negative or the lower recycling substitution effect will makethe profit of forward channel decrease) to obtain more profitin their closed-loop supply chain The lines in Figure 5 giveus a visual presentation
43 Comparison of Model119862 andModel119863 In this subsectionwe discuss the impact of retailing substitution effect on the
Abstract and Applied Analysis 11
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120587
(a) Impact on the manufacturerrsquos profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120587
(b) Impact on the manufacturerrsquos profit in reverse channel
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(c) Impact on the retailersrsquo profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120574 = 02
120574 = 04
120587
(d) Impact on the retailersrsquo profit in reverse channel
Figure 5 Impact of the retailing and recycling substitution effects on channel profit in model 119863119862
optimal prices and optimal channel profits and comparethem in the case of coordinated collection with that ofdecentralized collection
From the previous assumptions and analysis we knowthat the retail price in the case of coordinated collection islower than that in decentralized collection that is facingthe same circumstances the retail price in the case ofdecentralized collection is higher than that in coordinatedcollection And the repurchase price in the case of decentral-ized collection is lower than that in coordinated collectionIt implies that for the retailer the coordinated collectionhas more price advantages (lower retail price and higherrepurchase price) than the decentralized collection in the
closed-loop channel The lines in Figure 6 give us a visualpresentation
In what is discussed above we know that the channelprofit in model 119862 is higher than that in model 119863 Theprofit of reverse channel in model 119863 is very low even thevalue is negative In this case the manufacturer and retailerstend to select coordinated collection to obtain more profitsfor example HP Xerox and Apple select the coordinatedcollection to collect their used product The lines in Figure 6give us a visual presentation
44 Impact of Retailing and Recycling Substitution Effects onClosed Supply Chainrsquos Profit In this subsection we discuss
12 Abstract and Applied Analysis
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
Model CCModel DC
120573
pi
(a) Comparison of retail price
minus100
minus50
0
50
100
150
200
pRi
0 02 04 06 08 1
Model CCModel DC
120573
(b) Comparison of repurchase price
Figure 6 Pricing decisions of coordinated collection versus decentralized collection
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
Model CCModel DC
times104
120573
120587
(a) Comparison of forward channel profits
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
Model CCModel DC
120573
120587
(b) Comparison of reverse channel profits
Figure 7 The profit of coordinated collection versus decentralized collection
the impact of retailing and recycling substitution effects onthe optimal forward channel profits and optimal reversechannel profits in the cases of coordinated collection anddecentralized collection
From the previous assumptions and analysis weknow that the recycling substitution effectrsquos impact on theclosed-loop supply channel profit in the case of coordinated
collection is more obvious than that in decentralizedcollection In other words the manufacturer can obtainmore profit than in decentralized collection more easily bychanging the recycling substitution effect The reason maybe as follows in the case of coordinated collection the retailprice and repurchase price are made by the center planner(ie manufacturer) this collection model can adjust the
Abstract and Applied Analysis 13
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(a) Impact on the closed-loop supply channelrsquos profit in model 119862119862
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(b) Impact on the closed-loop supply channelrsquos profit in model119863119862
Figure 8 Impact of the retailing and recycling substitution effects on closed-Loop supply chainrsquos profit
retail and repurchase price fast and accurately But in caseof decentralized collection there is a pricing game betweenmanufacturer and retailers The manufacturer firstly selectsoptimal wholesale price and buy-back payment then theretailers select their optimal reaction retail price and repur-chase price that is the pricing game and pricing decision areunsynchronized between the manufacturer and retailer sothe manufacturer can obtain more profits in the case of coor-dinated collection than the case of decentralized collectionThe lines in Figures 7 and 8 give us a visual presentation
5 Conclusion
In this paper we investigated a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers Theresults demonstrate that more intense retailing competitioninduces the increase of the forward and reverse profitsof manufacturer and the forward channel of retailers anddecrease of the retailers profit in reverse channel while moreintense recycling competition induces the decrease of theprofits of the manufacturer and retailers in forward andreverse channels while the whole supply chainrsquos profit ofmodel 119862 is increasing with the respect to the free recyclingproportional coefficient but the effect on model 119863 is uncer-tain The results have important managerial insights for themanufacturer and retailers
This paper makes a contribution to the literature onreverse channel choice and coordination by drawing atten-tion to closed-loop supply chains with competition of retail-ing and recycling Our future research will be as follows ourmodel does not consider any fixed costs of retailingrecyclinga collection system If such costs were incurred a minimumnumber of retailingreturns would be required to justify thecost of operations Last but not least our cost formulation
does not include any operationalcapacity constraints inthe channel of products collection Operationalcapacityconstraint on the channel of collection can be incorporatedinto the model by defining an upper bound on the numbercollected via each channel Another possibility is to modelthe collection cost as a quadratic function of the quantityreturned
Notations
120574 Recycling substitution effect120573 Retailing substitution effectΦ Market size of the retailers in the forward
channel120601 Market size of the retailers by free recovering
in the reverse channel120575 Unit saving cost by remanufacturing119904 Recovering coefficient119888119903 Unit cost of remanufacturing a returned
product into a new one119888119898 Unit cost of manufacturing a new product
119908 New productrsquos wholesale price of themanufacturer
119887 Obsolete productrsquos repurchase price from theretailers to the manufacturer
119863119894(119901119894 119901119895) Demand function of retailer 119894
119876119894(119901119877119894 119901119877119895
) Recycling function of retailer 119894
119901119862
119894 119901119863
119895 Retailer 119894 and 119895rsquos respective retail price in
models 119862 and 119863
119901119862
119877119894 119901119863
119877119895 Retailer 119894 and 119895rsquos repurchase price in models
119862 and 119863
Π119870
119894 Profit function for channel member 119894 in
model 119896 Superscript 119896 takes the values of 119862and 119863 Subscript 119894 takes the values of 119898 119903and 119890 denoting the manufacturer theretailer and the 119890-tailer respectively
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
120574 = 02
120574 = 04
pi
(a) Impact on the retail price
0
5
10
15
20
25
30
35
40
45
50
pRi
0 02 04 06 08 1120573
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 2 Impact of the retailing and recycling substitution effects on price in model 119862
effect will induce the increase of retail price Moreover thehigher recycling substitution effect will induce the increaseof retail price the higher recycling substitution effect willinduce the decrease of repurchase price In addition thehigher retailing substitution effect will induce the decrease ofretail price The green and blue lines in Figures 2(a) and 2(b)give us a visual presentation
From the previous assumptions and analysis we knowthat the profit of forward channel is strictly increasing withrespect to retailing substitution effect and decreasing withrespect to recycling substitution effect that is the manu-facturer can induce higher retailing competition and lowerrecycling competition to obtain more profit from forwardchannel
Andwe known that the profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andincreasing with respect to recycling substitution effect thatis the manufacturer can induce lower recycling substitutioneffect to obtain more profit in the reverse channel In otherwords themanufacturer can induce higher retailing substitu-tion effect and lower recycling substitution effect could obtainmore profit in closed-loop supply chain The green and bluelines in Figures 3(a) and 3(b) give us a visual presentation
42 Numerical Analysis of Model 119863 In this subsection wediscuss the impacts of retailing and recycling substitutioneffects on the retail price repurchase price the forwardchannel profit and reverse channel profit in the case ofdecentralized collection respectively
From the previous assumptions and analysis we knowthat the retail price is strictly increasing with respect toretailing substitution effect that is the higher retailing substi-tution effect will induce the increase of retail price And the
higher recycling substitution effect will induce the decreaseof retail price And we known that the repurchase priceis strictly decreasing with respect to recycling substitutioneffect that is the higher recycling substitution effect willinduce the increase of repurchase price while the higherretailing substitution effect will induce the decrease of retailprice The green and blue lines in Figures 4(a) and 4(b) giveus a visual presentation
According to the assumptions and the analysis weknow that the manufacturerrsquos profit of forward channel isstrictly increasing with respect to retailing substitution effectand decreasing with respect to recycling substitution effectAnd the manufacturerrsquos profit of reverse channel is strictlydecreasing with respect to retailing substitution effect andrecycling substitution effect In other words in the case ofdecentralized collection the manufacturer can induce higherretailing competition and lower recycling competition toobtain more profit in the forward channel and induce lowerrecycling substitution effect to obtain more profit in thethe reverse channel So the manufacturer can select higherretailing substitution effect and lower recycling substitutioneffect to obtain more profit in their closed-loop supply chainMoreover the higher recycling competition will induce thelower profit of forward channel and induce lower profit ofreverse channel
For the forward channel profit of retailers is strictlyincreasing with respect to retailing substitution effect anddecreasing with respect to recycling substitution effect Andthe retailersrsquo profit of reverse channel is strictly decreasingwith respect to retailing substitution effect and increasingwith respect to recycling substitution effect In other wordsin the case of decentralized collection for the retailers thehigher retailing competition and lower recycling competition
10 Abstract and Applied Analysis
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(a) Impact on the profit in forward channel
0 02 04 06 08 10
200
400
600
800
1000
1200
1400
1600
1800
2000
120573
120574 = 02
120574 = 04120587
(b) Impact on the profit in reverse channel
Figure 3 Impact of the retailing and recycling substitution effects on channel profit in model 119862
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
pi
120574 = 02
120574 = 04
(a) Impact on the retail price
0 02 04 06 08 1minus20
minus15
minus10
minus5
0
5
10
15
20
25
30
120573
pi
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 4 Impact of the retailing and recycling substitution effects on price in model 119863
can make them obtain more profit in their forward channelbut the lower retailing substitution effect and higher recyclingsubstitution effect can make them obtain more profit in thereverse channel So the retailers tend to select suitable retailprice and repurchase price to induce retailing and recyclingcompetitions (not too high or too low the higher retailsubstitution effect will make the profit of reverse channel
negative or the lower recycling substitution effect will makethe profit of forward channel decrease) to obtain more profitin their closed-loop supply chain The lines in Figure 5 giveus a visual presentation
43 Comparison of Model119862 andModel119863 In this subsectionwe discuss the impact of retailing substitution effect on the
Abstract and Applied Analysis 11
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120587
(a) Impact on the manufacturerrsquos profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120587
(b) Impact on the manufacturerrsquos profit in reverse channel
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(c) Impact on the retailersrsquo profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120574 = 02
120574 = 04
120587
(d) Impact on the retailersrsquo profit in reverse channel
Figure 5 Impact of the retailing and recycling substitution effects on channel profit in model 119863119862
optimal prices and optimal channel profits and comparethem in the case of coordinated collection with that ofdecentralized collection
From the previous assumptions and analysis we knowthat the retail price in the case of coordinated collection islower than that in decentralized collection that is facingthe same circumstances the retail price in the case ofdecentralized collection is higher than that in coordinatedcollection And the repurchase price in the case of decentral-ized collection is lower than that in coordinated collectionIt implies that for the retailer the coordinated collectionhas more price advantages (lower retail price and higherrepurchase price) than the decentralized collection in the
closed-loop channel The lines in Figure 6 give us a visualpresentation
In what is discussed above we know that the channelprofit in model 119862 is higher than that in model 119863 Theprofit of reverse channel in model 119863 is very low even thevalue is negative In this case the manufacturer and retailerstend to select coordinated collection to obtain more profitsfor example HP Xerox and Apple select the coordinatedcollection to collect their used product The lines in Figure 6give us a visual presentation
44 Impact of Retailing and Recycling Substitution Effects onClosed Supply Chainrsquos Profit In this subsection we discuss
12 Abstract and Applied Analysis
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
Model CCModel DC
120573
pi
(a) Comparison of retail price
minus100
minus50
0
50
100
150
200
pRi
0 02 04 06 08 1
Model CCModel DC
120573
(b) Comparison of repurchase price
Figure 6 Pricing decisions of coordinated collection versus decentralized collection
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
Model CCModel DC
times104
120573
120587
(a) Comparison of forward channel profits
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
Model CCModel DC
120573
120587
(b) Comparison of reverse channel profits
Figure 7 The profit of coordinated collection versus decentralized collection
the impact of retailing and recycling substitution effects onthe optimal forward channel profits and optimal reversechannel profits in the cases of coordinated collection anddecentralized collection
From the previous assumptions and analysis weknow that the recycling substitution effectrsquos impact on theclosed-loop supply channel profit in the case of coordinated
collection is more obvious than that in decentralizedcollection In other words the manufacturer can obtainmore profit than in decentralized collection more easily bychanging the recycling substitution effect The reason maybe as follows in the case of coordinated collection the retailprice and repurchase price are made by the center planner(ie manufacturer) this collection model can adjust the
Abstract and Applied Analysis 13
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(a) Impact on the closed-loop supply channelrsquos profit in model 119862119862
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(b) Impact on the closed-loop supply channelrsquos profit in model119863119862
Figure 8 Impact of the retailing and recycling substitution effects on closed-Loop supply chainrsquos profit
retail and repurchase price fast and accurately But in caseof decentralized collection there is a pricing game betweenmanufacturer and retailers The manufacturer firstly selectsoptimal wholesale price and buy-back payment then theretailers select their optimal reaction retail price and repur-chase price that is the pricing game and pricing decision areunsynchronized between the manufacturer and retailer sothe manufacturer can obtain more profits in the case of coor-dinated collection than the case of decentralized collectionThe lines in Figures 7 and 8 give us a visual presentation
5 Conclusion
In this paper we investigated a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers Theresults demonstrate that more intense retailing competitioninduces the increase of the forward and reverse profitsof manufacturer and the forward channel of retailers anddecrease of the retailers profit in reverse channel while moreintense recycling competition induces the decrease of theprofits of the manufacturer and retailers in forward andreverse channels while the whole supply chainrsquos profit ofmodel 119862 is increasing with the respect to the free recyclingproportional coefficient but the effect on model 119863 is uncer-tain The results have important managerial insights for themanufacturer and retailers
This paper makes a contribution to the literature onreverse channel choice and coordination by drawing atten-tion to closed-loop supply chains with competition of retail-ing and recycling Our future research will be as follows ourmodel does not consider any fixed costs of retailingrecyclinga collection system If such costs were incurred a minimumnumber of retailingreturns would be required to justify thecost of operations Last but not least our cost formulation
does not include any operationalcapacity constraints inthe channel of products collection Operationalcapacityconstraint on the channel of collection can be incorporatedinto the model by defining an upper bound on the numbercollected via each channel Another possibility is to modelthe collection cost as a quadratic function of the quantityreturned
Notations
120574 Recycling substitution effect120573 Retailing substitution effectΦ Market size of the retailers in the forward
channel120601 Market size of the retailers by free recovering
in the reverse channel120575 Unit saving cost by remanufacturing119904 Recovering coefficient119888119903 Unit cost of remanufacturing a returned
product into a new one119888119898 Unit cost of manufacturing a new product
119908 New productrsquos wholesale price of themanufacturer
119887 Obsolete productrsquos repurchase price from theretailers to the manufacturer
119863119894(119901119894 119901119895) Demand function of retailer 119894
119876119894(119901119877119894 119901119877119895
) Recycling function of retailer 119894
119901119862
119894 119901119863
119895 Retailer 119894 and 119895rsquos respective retail price in
models 119862 and 119863
119901119862
119877119894 119901119863
119877119895 Retailer 119894 and 119895rsquos repurchase price in models
119862 and 119863
Π119870
119894 Profit function for channel member 119894 in
model 119896 Superscript 119896 takes the values of 119862and 119863 Subscript 119894 takes the values of 119898 119903and 119890 denoting the manufacturer theretailer and the 119890-tailer respectively
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Abstract and Applied Analysis
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(a) Impact on the profit in forward channel
0 02 04 06 08 10
200
400
600
800
1000
1200
1400
1600
1800
2000
120573
120574 = 02
120574 = 04120587
(b) Impact on the profit in reverse channel
Figure 3 Impact of the retailing and recycling substitution effects on channel profit in model 119862
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
120573
pi
120574 = 02
120574 = 04
(a) Impact on the retail price
0 02 04 06 08 1minus20
minus15
minus10
minus5
0
5
10
15
20
25
30
120573
pi
120574 = 02
120574 = 04
(b) Impact on the repurchase price
Figure 4 Impact of the retailing and recycling substitution effects on price in model 119863
can make them obtain more profit in their forward channelbut the lower retailing substitution effect and higher recyclingsubstitution effect can make them obtain more profit in thereverse channel So the retailers tend to select suitable retailprice and repurchase price to induce retailing and recyclingcompetitions (not too high or too low the higher retailsubstitution effect will make the profit of reverse channel
negative or the lower recycling substitution effect will makethe profit of forward channel decrease) to obtain more profitin their closed-loop supply chain The lines in Figure 5 giveus a visual presentation
43 Comparison of Model119862 andModel119863 In this subsectionwe discuss the impact of retailing substitution effect on the
Abstract and Applied Analysis 11
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120587
(a) Impact on the manufacturerrsquos profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120587
(b) Impact on the manufacturerrsquos profit in reverse channel
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(c) Impact on the retailersrsquo profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120574 = 02
120574 = 04
120587
(d) Impact on the retailersrsquo profit in reverse channel
Figure 5 Impact of the retailing and recycling substitution effects on channel profit in model 119863119862
optimal prices and optimal channel profits and comparethem in the case of coordinated collection with that ofdecentralized collection
From the previous assumptions and analysis we knowthat the retail price in the case of coordinated collection islower than that in decentralized collection that is facingthe same circumstances the retail price in the case ofdecentralized collection is higher than that in coordinatedcollection And the repurchase price in the case of decentral-ized collection is lower than that in coordinated collectionIt implies that for the retailer the coordinated collectionhas more price advantages (lower retail price and higherrepurchase price) than the decentralized collection in the
closed-loop channel The lines in Figure 6 give us a visualpresentation
In what is discussed above we know that the channelprofit in model 119862 is higher than that in model 119863 Theprofit of reverse channel in model 119863 is very low even thevalue is negative In this case the manufacturer and retailerstend to select coordinated collection to obtain more profitsfor example HP Xerox and Apple select the coordinatedcollection to collect their used product The lines in Figure 6give us a visual presentation
44 Impact of Retailing and Recycling Substitution Effects onClosed Supply Chainrsquos Profit In this subsection we discuss
12 Abstract and Applied Analysis
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
Model CCModel DC
120573
pi
(a) Comparison of retail price
minus100
minus50
0
50
100
150
200
pRi
0 02 04 06 08 1
Model CCModel DC
120573
(b) Comparison of repurchase price
Figure 6 Pricing decisions of coordinated collection versus decentralized collection
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
Model CCModel DC
times104
120573
120587
(a) Comparison of forward channel profits
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
Model CCModel DC
120573
120587
(b) Comparison of reverse channel profits
Figure 7 The profit of coordinated collection versus decentralized collection
the impact of retailing and recycling substitution effects onthe optimal forward channel profits and optimal reversechannel profits in the cases of coordinated collection anddecentralized collection
From the previous assumptions and analysis weknow that the recycling substitution effectrsquos impact on theclosed-loop supply channel profit in the case of coordinated
collection is more obvious than that in decentralizedcollection In other words the manufacturer can obtainmore profit than in decentralized collection more easily bychanging the recycling substitution effect The reason maybe as follows in the case of coordinated collection the retailprice and repurchase price are made by the center planner(ie manufacturer) this collection model can adjust the
Abstract and Applied Analysis 13
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(a) Impact on the closed-loop supply channelrsquos profit in model 119862119862
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(b) Impact on the closed-loop supply channelrsquos profit in model119863119862
Figure 8 Impact of the retailing and recycling substitution effects on closed-Loop supply chainrsquos profit
retail and repurchase price fast and accurately But in caseof decentralized collection there is a pricing game betweenmanufacturer and retailers The manufacturer firstly selectsoptimal wholesale price and buy-back payment then theretailers select their optimal reaction retail price and repur-chase price that is the pricing game and pricing decision areunsynchronized between the manufacturer and retailer sothe manufacturer can obtain more profits in the case of coor-dinated collection than the case of decentralized collectionThe lines in Figures 7 and 8 give us a visual presentation
5 Conclusion
In this paper we investigated a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers Theresults demonstrate that more intense retailing competitioninduces the increase of the forward and reverse profitsof manufacturer and the forward channel of retailers anddecrease of the retailers profit in reverse channel while moreintense recycling competition induces the decrease of theprofits of the manufacturer and retailers in forward andreverse channels while the whole supply chainrsquos profit ofmodel 119862 is increasing with the respect to the free recyclingproportional coefficient but the effect on model 119863 is uncer-tain The results have important managerial insights for themanufacturer and retailers
This paper makes a contribution to the literature onreverse channel choice and coordination by drawing atten-tion to closed-loop supply chains with competition of retail-ing and recycling Our future research will be as follows ourmodel does not consider any fixed costs of retailingrecyclinga collection system If such costs were incurred a minimumnumber of retailingreturns would be required to justify thecost of operations Last but not least our cost formulation
does not include any operationalcapacity constraints inthe channel of products collection Operationalcapacityconstraint on the channel of collection can be incorporatedinto the model by defining an upper bound on the numbercollected via each channel Another possibility is to modelthe collection cost as a quadratic function of the quantityreturned
Notations
120574 Recycling substitution effect120573 Retailing substitution effectΦ Market size of the retailers in the forward
channel120601 Market size of the retailers by free recovering
in the reverse channel120575 Unit saving cost by remanufacturing119904 Recovering coefficient119888119903 Unit cost of remanufacturing a returned
product into a new one119888119898 Unit cost of manufacturing a new product
119908 New productrsquos wholesale price of themanufacturer
119887 Obsolete productrsquos repurchase price from theretailers to the manufacturer
119863119894(119901119894 119901119895) Demand function of retailer 119894
119876119894(119901119877119894 119901119877119895
) Recycling function of retailer 119894
119901119862
119894 119901119863
119895 Retailer 119894 and 119895rsquos respective retail price in
models 119862 and 119863
119901119862
119877119894 119901119863
119877119895 Retailer 119894 and 119895rsquos repurchase price in models
119862 and 119863
Π119870
119894 Profit function for channel member 119894 in
model 119896 Superscript 119896 takes the values of 119862and 119863 Subscript 119894 takes the values of 119898 119903and 119890 denoting the manufacturer theretailer and the 119890-tailer respectively
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 11
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120587
(a) Impact on the manufacturerrsquos profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120587
(b) Impact on the manufacturerrsquos profit in reverse channel
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
120573
120574 = 02
120574 = 04
120587
times104
(c) Impact on the retailersrsquo profit in forward channel
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
120573
120574 = 02
120574 = 04
120587
(d) Impact on the retailersrsquo profit in reverse channel
Figure 5 Impact of the retailing and recycling substitution effects on channel profit in model 119863119862
optimal prices and optimal channel profits and comparethem in the case of coordinated collection with that ofdecentralized collection
From the previous assumptions and analysis we knowthat the retail price in the case of coordinated collection islower than that in decentralized collection that is facingthe same circumstances the retail price in the case ofdecentralized collection is higher than that in coordinatedcollection And the repurchase price in the case of decentral-ized collection is lower than that in coordinated collectionIt implies that for the retailer the coordinated collectionhas more price advantages (lower retail price and higherrepurchase price) than the decentralized collection in the
closed-loop channel The lines in Figure 6 give us a visualpresentation
In what is discussed above we know that the channelprofit in model 119862 is higher than that in model 119863 Theprofit of reverse channel in model 119863 is very low even thevalue is negative In this case the manufacturer and retailerstend to select coordinated collection to obtain more profitsfor example HP Xerox and Apple select the coordinatedcollection to collect their used product The lines in Figure 6give us a visual presentation
44 Impact of Retailing and Recycling Substitution Effects onClosed Supply Chainrsquos Profit In this subsection we discuss
12 Abstract and Applied Analysis
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
Model CCModel DC
120573
pi
(a) Comparison of retail price
minus100
minus50
0
50
100
150
200
pRi
0 02 04 06 08 1
Model CCModel DC
120573
(b) Comparison of repurchase price
Figure 6 Pricing decisions of coordinated collection versus decentralized collection
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
Model CCModel DC
times104
120573
120587
(a) Comparison of forward channel profits
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
Model CCModel DC
120573
120587
(b) Comparison of reverse channel profits
Figure 7 The profit of coordinated collection versus decentralized collection
the impact of retailing and recycling substitution effects onthe optimal forward channel profits and optimal reversechannel profits in the cases of coordinated collection anddecentralized collection
From the previous assumptions and analysis weknow that the recycling substitution effectrsquos impact on theclosed-loop supply channel profit in the case of coordinated
collection is more obvious than that in decentralizedcollection In other words the manufacturer can obtainmore profit than in decentralized collection more easily bychanging the recycling substitution effect The reason maybe as follows in the case of coordinated collection the retailprice and repurchase price are made by the center planner(ie manufacturer) this collection model can adjust the
Abstract and Applied Analysis 13
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(a) Impact on the closed-loop supply channelrsquos profit in model 119862119862
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(b) Impact on the closed-loop supply channelrsquos profit in model119863119862
Figure 8 Impact of the retailing and recycling substitution effects on closed-Loop supply chainrsquos profit
retail and repurchase price fast and accurately But in caseof decentralized collection there is a pricing game betweenmanufacturer and retailers The manufacturer firstly selectsoptimal wholesale price and buy-back payment then theretailers select their optimal reaction retail price and repur-chase price that is the pricing game and pricing decision areunsynchronized between the manufacturer and retailer sothe manufacturer can obtain more profits in the case of coor-dinated collection than the case of decentralized collectionThe lines in Figures 7 and 8 give us a visual presentation
5 Conclusion
In this paper we investigated a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers Theresults demonstrate that more intense retailing competitioninduces the increase of the forward and reverse profitsof manufacturer and the forward channel of retailers anddecrease of the retailers profit in reverse channel while moreintense recycling competition induces the decrease of theprofits of the manufacturer and retailers in forward andreverse channels while the whole supply chainrsquos profit ofmodel 119862 is increasing with the respect to the free recyclingproportional coefficient but the effect on model 119863 is uncer-tain The results have important managerial insights for themanufacturer and retailers
This paper makes a contribution to the literature onreverse channel choice and coordination by drawing atten-tion to closed-loop supply chains with competition of retail-ing and recycling Our future research will be as follows ourmodel does not consider any fixed costs of retailingrecyclinga collection system If such costs were incurred a minimumnumber of retailingreturns would be required to justify thecost of operations Last but not least our cost formulation
does not include any operationalcapacity constraints inthe channel of products collection Operationalcapacityconstraint on the channel of collection can be incorporatedinto the model by defining an upper bound on the numbercollected via each channel Another possibility is to modelthe collection cost as a quadratic function of the quantityreturned
Notations
120574 Recycling substitution effect120573 Retailing substitution effectΦ Market size of the retailers in the forward
channel120601 Market size of the retailers by free recovering
in the reverse channel120575 Unit saving cost by remanufacturing119904 Recovering coefficient119888119903 Unit cost of remanufacturing a returned
product into a new one119888119898 Unit cost of manufacturing a new product
119908 New productrsquos wholesale price of themanufacturer
119887 Obsolete productrsquos repurchase price from theretailers to the manufacturer
119863119894(119901119894 119901119895) Demand function of retailer 119894
119876119894(119901119877119894 119901119877119895
) Recycling function of retailer 119894
119901119862
119894 119901119863
119895 Retailer 119894 and 119895rsquos respective retail price in
models 119862 and 119863
119901119862
119877119894 119901119863
119877119895 Retailer 119894 and 119895rsquos repurchase price in models
119862 and 119863
Π119870
119894 Profit function for channel member 119894 in
model 119896 Superscript 119896 takes the values of 119862and 119863 Subscript 119894 takes the values of 119898 119903and 119890 denoting the manufacturer theretailer and the 119890-tailer respectively
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Abstract and Applied Analysis
0 02 04 06 08 10
50
100
150
200
250
300
350
400
450
500
Model CCModel DC
120573
pi
(a) Comparison of retail price
minus100
minus50
0
50
100
150
200
pRi
0 02 04 06 08 1
Model CCModel DC
120573
(b) Comparison of repurchase price
Figure 6 Pricing decisions of coordinated collection versus decentralized collection
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2
Model CCModel DC
times104
120573
120587
(a) Comparison of forward channel profits
0 02 04 06 08 1minus1000
minus500
0
500
1000
1500
2000
Model CCModel DC
120573
120587
(b) Comparison of reverse channel profits
Figure 7 The profit of coordinated collection versus decentralized collection
the impact of retailing and recycling substitution effects onthe optimal forward channel profits and optimal reversechannel profits in the cases of coordinated collection anddecentralized collection
From the previous assumptions and analysis weknow that the recycling substitution effectrsquos impact on theclosed-loop supply channel profit in the case of coordinated
collection is more obvious than that in decentralizedcollection In other words the manufacturer can obtainmore profit than in decentralized collection more easily bychanging the recycling substitution effect The reason maybe as follows in the case of coordinated collection the retailprice and repurchase price are made by the center planner(ie manufacturer) this collection model can adjust the
Abstract and Applied Analysis 13
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(a) Impact on the closed-loop supply channelrsquos profit in model 119862119862
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(b) Impact on the closed-loop supply channelrsquos profit in model119863119862
Figure 8 Impact of the retailing and recycling substitution effects on closed-Loop supply chainrsquos profit
retail and repurchase price fast and accurately But in caseof decentralized collection there is a pricing game betweenmanufacturer and retailers The manufacturer firstly selectsoptimal wholesale price and buy-back payment then theretailers select their optimal reaction retail price and repur-chase price that is the pricing game and pricing decision areunsynchronized between the manufacturer and retailer sothe manufacturer can obtain more profits in the case of coor-dinated collection than the case of decentralized collectionThe lines in Figures 7 and 8 give us a visual presentation
5 Conclusion
In this paper we investigated a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers Theresults demonstrate that more intense retailing competitioninduces the increase of the forward and reverse profitsof manufacturer and the forward channel of retailers anddecrease of the retailers profit in reverse channel while moreintense recycling competition induces the decrease of theprofits of the manufacturer and retailers in forward andreverse channels while the whole supply chainrsquos profit ofmodel 119862 is increasing with the respect to the free recyclingproportional coefficient but the effect on model 119863 is uncer-tain The results have important managerial insights for themanufacturer and retailers
This paper makes a contribution to the literature onreverse channel choice and coordination by drawing atten-tion to closed-loop supply chains with competition of retail-ing and recycling Our future research will be as follows ourmodel does not consider any fixed costs of retailingrecyclinga collection system If such costs were incurred a minimumnumber of retailingreturns would be required to justify thecost of operations Last but not least our cost formulation
does not include any operationalcapacity constraints inthe channel of products collection Operationalcapacityconstraint on the channel of collection can be incorporatedinto the model by defining an upper bound on the numbercollected via each channel Another possibility is to modelthe collection cost as a quadratic function of the quantityreturned
Notations
120574 Recycling substitution effect120573 Retailing substitution effectΦ Market size of the retailers in the forward
channel120601 Market size of the retailers by free recovering
in the reverse channel120575 Unit saving cost by remanufacturing119904 Recovering coefficient119888119903 Unit cost of remanufacturing a returned
product into a new one119888119898 Unit cost of manufacturing a new product
119908 New productrsquos wholesale price of themanufacturer
119887 Obsolete productrsquos repurchase price from theretailers to the manufacturer
119863119894(119901119894 119901119895) Demand function of retailer 119894
119876119894(119901119877119894 119901119877119895
) Recycling function of retailer 119894
119901119862
119894 119901119863
119895 Retailer 119894 and 119895rsquos respective retail price in
models 119862 and 119863
119901119862
119877119894 119901119863
119877119895 Retailer 119894 and 119895rsquos repurchase price in models
119862 and 119863
Π119870
119894 Profit function for channel member 119894 in
model 119896 Superscript 119896 takes the values of 119862and 119863 Subscript 119894 takes the values of 119898 119903and 119890 denoting the manufacturer theretailer and the 119890-tailer respectively
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 13
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(a) Impact on the closed-loop supply channelrsquos profit in model 119862119862
0 02 04 06 08 10
02
04
06
08
1
12
14
16
18
2times104
120573
120574 = 02
120574 = 04
120587
(b) Impact on the closed-loop supply channelrsquos profit in model119863119862
Figure 8 Impact of the retailing and recycling substitution effects on closed-Loop supply chainrsquos profit
retail and repurchase price fast and accurately But in caseof decentralized collection there is a pricing game betweenmanufacturer and retailers The manufacturer firstly selectsoptimal wholesale price and buy-back payment then theretailers select their optimal reaction retail price and repur-chase price that is the pricing game and pricing decision areunsynchronized between the manufacturer and retailer sothe manufacturer can obtain more profits in the case of coor-dinated collection than the case of decentralized collectionThe lines in Figures 7 and 8 give us a visual presentation
5 Conclusion
In this paper we investigated a durable product retailing andrecycling problem in a closed-loop supply chain consistingof a single manufacturer and two competitive retailers Theresults demonstrate that more intense retailing competitioninduces the increase of the forward and reverse profitsof manufacturer and the forward channel of retailers anddecrease of the retailers profit in reverse channel while moreintense recycling competition induces the decrease of theprofits of the manufacturer and retailers in forward andreverse channels while the whole supply chainrsquos profit ofmodel 119862 is increasing with the respect to the free recyclingproportional coefficient but the effect on model 119863 is uncer-tain The results have important managerial insights for themanufacturer and retailers
This paper makes a contribution to the literature onreverse channel choice and coordination by drawing atten-tion to closed-loop supply chains with competition of retail-ing and recycling Our future research will be as follows ourmodel does not consider any fixed costs of retailingrecyclinga collection system If such costs were incurred a minimumnumber of retailingreturns would be required to justify thecost of operations Last but not least our cost formulation
does not include any operationalcapacity constraints inthe channel of products collection Operationalcapacityconstraint on the channel of collection can be incorporatedinto the model by defining an upper bound on the numbercollected via each channel Another possibility is to modelthe collection cost as a quadratic function of the quantityreturned
Notations
120574 Recycling substitution effect120573 Retailing substitution effectΦ Market size of the retailers in the forward
channel120601 Market size of the retailers by free recovering
in the reverse channel120575 Unit saving cost by remanufacturing119904 Recovering coefficient119888119903 Unit cost of remanufacturing a returned
product into a new one119888119898 Unit cost of manufacturing a new product
119908 New productrsquos wholesale price of themanufacturer
119887 Obsolete productrsquos repurchase price from theretailers to the manufacturer
119863119894(119901119894 119901119895) Demand function of retailer 119894
119876119894(119901119877119894 119901119877119895
) Recycling function of retailer 119894
119901119862
119894 119901119863
119895 Retailer 119894 and 119895rsquos respective retail price in
models 119862 and 119863
119901119862
119877119894 119901119863
119877119895 Retailer 119894 and 119895rsquos repurchase price in models
119862 and 119863
Π119870
119894 Profit function for channel member 119894 in
model 119896 Superscript 119896 takes the values of 119862and 119863 Subscript 119894 takes the values of 119898 119903and 119890 denoting the manufacturer theretailer and the 119890-tailer respectively
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Abstract and Applied Analysis
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the referees for their usefulcomments and suggestions which improved the presentationof the paper This work is supported by the Humanityand Social Science Foundation of Ministry of EducationChina no 12YJAZH052 and the National Natural ScienceFoundation of China under Grant no 71301114
References
[1] R C Savaskan S Bhattacharya and L N Van WassenhoveldquoClosed-loop supply chain models with product remanufactur-ingrdquoManagement Science vol 50 no 2 pp 239ndash252 2004
[2] Pconlinecom website 2012 httpofficepconlinecomcnhua-nbao12062812375html
[3] L Debo L Toktay and L van Wassenhove ldquoMarket segmen-tation and product technology selection for remanufacturableproductsrdquo Management Science vol 51 no 8 pp 1193ndash12052002
[4] V D R Guide Jr R H Teunter and L N van WassenhoveldquoMatching demand and supply to maximiz e profits fromremanufacturingrdquoManufacturing and Service Operations Man-agement vol 5 no 4 pp 303ndash316 2003
[5] J D Shulman and A T Coughlan ldquoUsed goods not used badsprofitable secondary market sales for a durable goods channelrdquoQuantitative Marketing and Economics vol 5 no 2 pp 191ndash2102007
[6] S Yin S Ray H Gurnani and A Animesh ldquoDurable productswith multiple used goods markets product upgrade and retailpricing implicationsrdquoMarketing Science vol 29 no 3 pp 540ndash560 2010
[7] T Choi D Li and H Yan ldquoOptimal returns policy for supplychain with e-marketplacerdquo International Journal of ProductionEconomics vol 88 no 2 pp 205ndash227 2004
[8] B Yalabik N C Petruzzi and D Chhajed ldquoAn integrated prod-uct returns model with logistics and marketing coordinationrdquoEuropean Journal of Operational Research vol 161 no 1 pp 162ndash182 2005
[9] Y Liang S Pokharel and G H Lim ldquoPricing used products forremanufacturingrdquo European Journal of Operational Researchvol 193 no 2 pp 390ndash395 2009
[10] K Inderfurth ldquoOptimal policies in hybrid manufacturingremanufacturing systems with product substitutionrdquo Interna-tional Journal of Production Economics vol 90 no 3 pp 325ndash343 2004
[11] R C Savaskan and L N van Wassenhove ldquoReverse channeldesign the case of competing retailersrdquo Management Sciencevol 52 no 1 pp 1ndash14 2006
[12] E Ofek Z Katona and M Sarvary ldquolsquoBricks and clicksrsquo theimpact of product returns on the strategies of multichannelretailersrdquoMarketing Science vol 30 no 1 pp 42ndash60 2011
[13] X Hong Z Wang D Wang and H Zhang ldquoDecision modelsof closed-loop supply chainwith remanufacturing under hybriddual-channel collectionrdquo International Journal of AdvancedManufacturing Technology vol 68 no 5ndash8 pp 1851ndash1865 2013
[14] T Choi Y Li and L Xu ldquoChannel leadership performanceand coordination in closed loop supply chainsrdquo InternationalJournal of Production Economics vol 146 no 1 pp 371ndash3802013
[15] S Webster and S Mitra ldquoCompetitive strategy in remanufac-turing and the impact of take-back lawsrdquo Journal of OperationsManagement vol 25 no 6 pp 1123ndash1140 2007
[16] M E Ferguson and L B Toktay ldquoThe effect of competition onrecovery strategiesrdquo Production and Operations Managementvol 15 no 3 pp 351ndash368 2006
[17] C Wu ldquoOEM product design in a price competition withremanufactured productrdquo Omega vol 41 no 2 pp 287ndash2982013
[18] S Tayur and R Ganeshan Quantitative Models for SupplyChain Management Kluwer Academic Publishers BostonMass USA 1998
[19] P Majumder and H Groenevelt ldquoCompetition in remanufac-turingrdquo Production and Operations Management vol 10 no 2pp 125ndash141 2001
[20] K Hickey ldquoHere and thererdquo Traffic World Magazine vol 265no 40 p 21 2001
[21] K S Jung and H Hwang ldquoCompetition and cooperation in aremanufacturing system with take-back requirementrdquo Journalof Intelligent Manufacturing vol 22 no 3 pp 427ndash433 2011
[22] C-C Chern S-T Lei and K-L Huang ldquoSolving a multi-objective master planning problem with substitution and arecycling process for a capacitated multi-commodity supplychain networkrdquo Journal of Intelligent Manufacturing vol 25 no1 pp 1ndash25 2014
[23] L van Wassenhove A Atasu and L Toktay ldquoHow collectioncost structure drives a manufacturerrsquos reverse channel choicerdquoProduction andOperationsManagement vol 22 no 5 pp 1089ndash1102 2013
[24] E Lee and R Staelin ldquoVertical strategic interaction implica-tions for channel pricing strategyrdquoMarketing Science vol 16 no3 pp 185ndash207 1997
[25] I Dobos and K Richter ldquoA productionrecycling model withquality considerationrdquo International Journal of Production Eco-nomics vol 104 no 2 pp 571ndash579 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of