research article a closed-loop supply chain problem with...

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


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


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)


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)


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)


|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


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)]


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


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)])


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



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


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


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


0 02 04 06 08 1minus20

minus15

minus10

minus5

0

5

10

15

20

25

30

120573

pi

120574 = 02

120574 = 04



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


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


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


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


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


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


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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


Manufacturer


Customer


Manufacturer


Customer

Competing











119863119894(119901119894 119901119895) = Φ

119894minus 119901119894+ 120573119901119895 0 le 120573 lt 1 (1)


119894is the



119876119894(119901119877119894 119901119877119895

) = 120601119894+ 119901119877119894

minus 120574119901119877119895

0 le 120574 lt 1 (2)






Π119862(119901119894 119901119895 119901119877119894 119901119877119895

)

=

2

sum

119894=1

((119901119894minus 119888119898)119863119894(119901119894 119901119895) + (Δ minus 119901

119877119894) 119876119894(119901119894 119901119877119895

))

(3)

wheresum2



(Δminus119901119877119894)119876119894(119901119877119894 119901119877119895



Π119863

119898(119901119894 119901119895 119901119877119894 119901119877119895)

=

2

sum

119894=1

((119908 minus 119888119898)119863119894 (119901119894 119901119895) + (Δ minus 119887)119876119894 (119901119877119894 119901119877119895))

(4)


2

119894=1119863119894(119901119894 119901119895) denotes the


119887)sum2

119894=1119876119894(119901119877119894 119901119877119895



Π119863

119894= (119901119894minus 119908)119863

119894(119901119894 119901119895) + (119887 minus 119901

119877119894) 119876119894(119901119877119894 119901119877119895

) (5)

where (119901119894minus 119908)119863








max119901119894119901119895119901119877119894119901119877119895

Π119862(119901119894 119901119895 119901119877119894 119901119877119895

)

=

2

sum

119894=1

((119901119894minus 119888119898)119863119894(119901119894 119901119895) + (Δ minus 119901

119877119894) 119876119894(119901119877119894 119901119877119895

))

(6)



119862

119895satisfy

119901lowast119862

119894=

(Φ119894+ Φ119895120573)

2 (1 minus 1205732)

+4 (Δ119904 minus 2119888

119898) (1 minus 120574) + 119904

2(Φ119894+ Φ3minus119894

)

41198831


41198832

(7)

119901lowast119862

119877119894=

Δ

2+ s[

(Φ119894+ Φ3minus119894

) minus (2119888119898


21198831


)

21198832

]

(8)


2(1+120573)


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)


|119867| =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus2 2120573 119904 minus119904120573

2120573 minus2 minus119904120573 119904

119904 minus119904120573 minus2 2120574

minus119904120573 119904 2120574 minus2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(10)



2| = 4(1 minus 120573) gt 0

|1198633| = minus2(1 minus 120573

2)(4 minus 119904

2) lt 0 and |1198634| = 16 + 16(120574


1205742))+8119904

2(1205732+120574120573(1minus120573

2)minus1)+119904



1198941199011198623minus119894

119901119862119895 and

119901119862



120597Π119862

120597119901119894

= Φ119894minus 2119901119894+ 2120573119901

119895+ 119904119901119877119894

minus 119904120573119901119877119895

+ (1 minus 120573) (119888119898

minus 119904Δ)

= 0

(11)

120597Π119862

120597119901119895


= 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)


lowast

119894and 119901

lowast


prices 119901lowast

119877119894and 119901

lowast

119877119895


119862

119894and 119901

119862





d119901119862119894

d120574= (1199042[1198832


minus1198832

2(Φ119894+ Φ3minus119894

+ (1 minus 120573) (Δ119904 minus 2119888119898))])

times (1198832

11198832

2)minus1

(15)



2(1+120573)



119862

119894and 119901

119862

119895are




119862

119877119894and 119901

119862




d119901119862119877119894

d120574= 2119904 (119883

2

2[Φ119894+ Φ3minus119894

minus (2119888119898

minus Δ119904) (1 minus 120573)]

minus1198832


)) times (1198832

11198832

2)minus1

(16)


2(1+120573)



119862

119877119894

and 119901119862


substitution effect





respect to 119901119894and 119901


prices 119901119894and 119901



2




119894is invariable




119894=1(Δ minus 119901

119877119894)119876119894(119901119877119894 119901119877119895


with respect to 119901119877119894



and 119901119877119895


2

119894=1(Δ minus 119901

119877119894)119876119894(119901119877119894 119901119877119895

)





max119901119894 119901119895

Π119863

119894= (119901119894 minus 119908)119863119894 (119901119894 119901119895) + (119887 minus 119901119877119895)119876119877119894 (119901119877119894 119901119877119895)

(17)


lowast119863



119877119894satisfy

119901lowast119863

119894=


11988411

+(Φ119894+ Φ3minus119894

) 1198841

211988411


) 1198851

211988511

(18)

119901lowast119863

119877119894=

119904 (Φ119894+ Φ3minus119894

)

211988411


)

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


2minus 120574)

1198851 = 2 + 120574 minus 119904


2 and119879 = 120574


2)



119894are

1205972Π119863

119894

1205971199012

119894

= minus21205972Π119863

119894

120597119901119894120597119901119895

= 119904

1205972Π119863

119894

120597119901119895120597119901119894

= 1199041205972Π119863

119894

1205971199012

119895

= minus119904120573

(20)


|119867| =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972Π119863

119894

1205971199012

119894

1205972Π119863

119894

120597119901119894120597119901119895

1205972Π119863

119894

120597119901119895120597119901119894

1205972Π119863

119894

1205971199012

119895

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(21)



2Π119863

1198941205971199012

119894= minus2 lt 0






120597Π119863

119894

120597119901119894


120597Π119863

119894

120597119901119877119894


(23)


119895are

120597Π119863

119895

120597119901119895


119877119895) = 0 (24)

120597Π119863

119895

120597119901119877119895


119877119895+ 120574119901119877119894

+ 119887 minus 119901119877119895

= 0

(25)




1= 1199042+120574minus2 lt 0119884


1199042minus120574) lt 0119885

1= 2+120574minus119904

2gt 0 and 119879 = 120574


2) lt 0




119877119894and




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)



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)


1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873

1= 120573(2 minus 119904


2) and

1198732= 8Δ + (Φ

1+ Φ2minus 2119888119898)119904


1205972Π119863

119898

1205971199082



1205972Π119863

119898

1205971198872

=41199042(2 minus 120574) (1 minus 120574)

2


1205972Π119863

119898

120597119908120597119887

=1205972Π119863

119898

120597119887120597119908

=41199042(2 minus 120574)



10038161003816100381610038161198671198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972Π119863

119898

1205971199082

1205972Π119863

119898

120597119908120597119887

1205972Π119863

119898

120597119887120597119908

1205972Π119863119862

119898

1205971198872

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(29)



119863

119898is jointly






120597Π119863

119898

120597119908

=2 (2 minus 120574) (2119908 minus 119887119904 + 119888

119898) + 2Δ119904

1199042+ 120574 minus 2


+ 2119908 (120574 minus 2)

+ 2 (119888119898



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



2+ 120574 minus 2) + 119904

2minus 4])minus1

= 0

(31)




lowast119863


119894 satisfy

119901lowast119863

119894


2[

1198851

11988511

minus1198841

11988411

]

+119904 (1 minus 120574)

211988411

[2Δ1198721 + 1205741198722 + 2120573120574 (119888119898119904 minus 2Δ)

21198721+ 120574119872

2

]


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


2) 1198722


2)

1198723

= 8Δ + (Φ1+ Φ2minus 2119888119898)119904 1198701

= 4(120574 minus 1) + 1199042(2 minus 120574)

1198702= (2119888119898


1=


2)1198732= 8Δ+(Φ

1+Φ2minus2119888119898)1199041198841= 1199042+120574minus2

11988411


2minus 120574) 119885

1= 2 + 120574 minus 119904

2 11988511

=



2)




119894is invariable








= 119904119863119894and 119901






119895(119901119877119895

119901119877119894)) is increasing












119894(119901119894 119901119895) is increasing



119894 induce





119877119894)119876119894(119901119877119894 119901119877119895






2= 50 119888

119898= 20 Δ = 10 120574 = 04 or 120574 = 02

and 119904 = 01




0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

120574 = 02

120574 = 04

pi


0

5

10

15

20

25

30

35

40

45

50

pRi

0 02 04 06 08 1120573

120574 = 02

120574 = 04












0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 10

200

400

600

800

1000

1200

1400

1600

1800

2000

120573

120574 = 02

120574 = 04120587



0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

pi

120574 = 02

120574 = 04


0 02 04 06 08 1minus20

minus15

minus10

minus5

0

5

10

15

20

25

30

120573

pi

120574 = 02

120574 = 04







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120574 = 02

120574 = 04

120587









0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

Model CCModel DC

120573

pi


minus100

minus50

0

50

100

150

200

pRi

0 02 04 06 08 1

Model CCModel DC

120573



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

Model CCModel DC

times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

Model CCModel DC

120573

120587







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587




5 Conclusion




Notations








119876119894(119901119877119894 119901119877119895


119901119862

119894 119901119863


models 119862 and 119863

119901119862

119877119894 119901119863


119862 and 119863

Π119870






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Manufacturer


Customer


Manufacturer


Customer

Competing











119863119894(119901119894 119901119895) = Φ

119894minus 119901119894+ 120573119901119895 0 le 120573 lt 1 (1)


119894is the



119876119894(119901119877119894 119901119877119895

) = 120601119894+ 119901119877119894

minus 120574119901119877119895

0 le 120574 lt 1 (2)






Π119862(119901119894 119901119895 119901119877119894 119901119877119895

)

=

2

sum

119894=1

((119901119894minus 119888119898)119863119894(119901119894 119901119895) + (Δ minus 119901

119877119894) 119876119894(119901119894 119901119877119895

))

(3)

wheresum2



(Δminus119901119877119894)119876119894(119901119877119894 119901119877119895



Π119863

119898(119901119894 119901119895 119901119877119894 119901119877119895)

=

2

sum

119894=1

((119908 minus 119888119898)119863119894 (119901119894 119901119895) + (Δ minus 119887)119876119894 (119901119877119894 119901119877119895))

(4)


2

119894=1119863119894(119901119894 119901119895) denotes the


119887)sum2

119894=1119876119894(119901119877119894 119901119877119895



Π119863

119894= (119901119894minus 119908)119863

119894(119901119894 119901119895) + (119887 minus 119901

119877119894) 119876119894(119901119877119894 119901119877119895

) (5)

where (119901119894minus 119908)119863








max119901119894119901119895119901119877119894119901119877119895

Π119862(119901119894 119901119895 119901119877119894 119901119877119895

)

=

2

sum

119894=1

((119901119894minus 119888119898)119863119894(119901119894 119901119895) + (Δ minus 119901

119877119894) 119876119894(119901119877119894 119901119877119895

))

(6)



119862

119895satisfy

119901lowast119862

119894=

(Φ119894+ Φ119895120573)

2 (1 minus 1205732)

+4 (Δ119904 minus 2119888

119898) (1 minus 120574) + 119904

2(Φ119894+ Φ3minus119894

)

41198831


41198832

(7)

119901lowast119862

119877119894=

Δ

2+ s[

(Φ119894+ Φ3minus119894

) minus (2119888119898


21198831


)

21198832

]

(8)


2(1+120573)


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)


|119867| =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus2 2120573 119904 minus119904120573

2120573 minus2 minus119904120573 119904

119904 minus119904120573 minus2 2120574

minus119904120573 119904 2120574 minus2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(10)



2| = 4(1 minus 120573) gt 0

|1198633| = minus2(1 minus 120573

2)(4 minus 119904

2) lt 0 and |1198634| = 16 + 16(120574


1205742))+8119904

2(1205732+120574120573(1minus120573

2)minus1)+119904



1198941199011198623minus119894

119901119862119895 and

119901119862



120597Π119862

120597119901119894

= Φ119894minus 2119901119894+ 2120573119901

119895+ 119904119901119877119894

minus 119904120573119901119877119895

+ (1 minus 120573) (119888119898

minus 119904Δ)

= 0

(11)

120597Π119862

120597119901119895


= 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)


lowast

119894and 119901

lowast


prices 119901lowast

119877119894and 119901

lowast

119877119895


119862

119894and 119901

119862





d119901119862119894

d120574= (1199042[1198832


minus1198832

2(Φ119894+ Φ3minus119894

+ (1 minus 120573) (Δ119904 minus 2119888119898))])

times (1198832

11198832

2)minus1

(15)



2(1+120573)



119862

119894and 119901

119862

119895are




119862

119877119894and 119901

119862




d119901119862119877119894

d120574= 2119904 (119883

2

2[Φ119894+ Φ3minus119894

minus (2119888119898

minus Δ119904) (1 minus 120573)]

minus1198832


)) times (1198832

11198832

2)minus1

(16)


2(1+120573)



119862

119877119894

and 119901119862


substitution effect





respect to 119901119894and 119901


prices 119901119894and 119901



2




119894is invariable




119894=1(Δ minus 119901

119877119894)119876119894(119901119877119894 119901119877119895


with respect to 119901119877119894



and 119901119877119895


2

119894=1(Δ minus 119901

119877119894)119876119894(119901119877119894 119901119877119895

)





max119901119894 119901119895

Π119863

119894= (119901119894 minus 119908)119863119894 (119901119894 119901119895) + (119887 minus 119901119877119895)119876119877119894 (119901119877119894 119901119877119895)

(17)


lowast119863



119877119894satisfy

119901lowast119863

119894=


11988411

+(Φ119894+ Φ3minus119894

) 1198841

211988411


) 1198851

211988511

(18)

119901lowast119863

119877119894=

119904 (Φ119894+ Φ3minus119894

)

211988411


)

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


2minus 120574)

1198851 = 2 + 120574 minus 119904


2 and119879 = 120574


2)



119894are

1205972Π119863

119894

1205971199012

119894

= minus21205972Π119863

119894

120597119901119894120597119901119895

= 119904

1205972Π119863

119894

120597119901119895120597119901119894

= 1199041205972Π119863

119894

1205971199012

119895

= minus119904120573

(20)


|119867| =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972Π119863

119894

1205971199012

119894

1205972Π119863

119894

120597119901119894120597119901119895

1205972Π119863

119894

120597119901119895120597119901119894

1205972Π119863

119894

1205971199012

119895

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(21)



2Π119863

1198941205971199012

119894= minus2 lt 0






120597Π119863

119894

120597119901119894


120597Π119863

119894

120597119901119877119894


(23)


119895are

120597Π119863

119895

120597119901119895


119877119895) = 0 (24)

120597Π119863

119895

120597119901119877119895


119877119895+ 120574119901119877119894

+ 119887 minus 119901119877119895

= 0

(25)




1= 1199042+120574minus2 lt 0119884


1199042minus120574) lt 0119885

1= 2+120574minus119904

2gt 0 and 119879 = 120574


2) lt 0




119877119894and




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)



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)


1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873

1= 120573(2 minus 119904


2) and

1198732= 8Δ + (Φ

1+ Φ2minus 2119888119898)119904


1205972Π119863

119898

1205971199082



1205972Π119863

119898

1205971198872

=41199042(2 minus 120574) (1 minus 120574)

2


1205972Π119863

119898

120597119908120597119887

=1205972Π119863

119898

120597119887120597119908

=41199042(2 minus 120574)



10038161003816100381610038161198671198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972Π119863

119898

1205971199082

1205972Π119863

119898

120597119908120597119887

1205972Π119863

119898

120597119887120597119908

1205972Π119863119862

119898

1205971198872

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(29)



119863

119898is jointly






120597Π119863

119898

120597119908

=2 (2 minus 120574) (2119908 minus 119887119904 + 119888

119898) + 2Δ119904

1199042+ 120574 minus 2


+ 2119908 (120574 minus 2)

+ 2 (119888119898



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



2+ 120574 minus 2) + 119904

2minus 4])minus1

= 0

(31)




lowast119863


119894 satisfy

119901lowast119863

119894


2[

1198851

11988511

minus1198841

11988411

]

+119904 (1 minus 120574)

211988411

[2Δ1198721 + 1205741198722 + 2120573120574 (119888119898119904 minus 2Δ)

21198721+ 120574119872

2

]


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


2) 1198722


2)

1198723

= 8Δ + (Φ1+ Φ2minus 2119888119898)119904 1198701

= 4(120574 minus 1) + 1199042(2 minus 120574)

1198702= (2119888119898


1=


2)1198732= 8Δ+(Φ

1+Φ2minus2119888119898)1199041198841= 1199042+120574minus2

11988411


2minus 120574) 119885

1= 2 + 120574 minus 119904

2 11988511

=



2)




119894is invariable








= 119904119863119894and 119901






119895(119901119877119895

119901119877119894)) is increasing












119894(119901119894 119901119895) is increasing



119894 induce





119877119894)119876119894(119901119877119894 119901119877119895






2= 50 119888

119898= 20 Δ = 10 120574 = 04 or 120574 = 02

and 119904 = 01




0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

120574 = 02

120574 = 04

pi


0

5

10

15

20

25

30

35

40

45

50

pRi

0 02 04 06 08 1120573

120574 = 02

120574 = 04












0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 10

200

400

600

800

1000

1200

1400

1600

1800

2000

120573

120574 = 02

120574 = 04120587



0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

pi

120574 = 02

120574 = 04


0 02 04 06 08 1minus20

minus15

minus10

minus5

0

5

10

15

20

25

30

120573

pi

120574 = 02

120574 = 04







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120574 = 02

120574 = 04

120587









0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

Model CCModel DC

120573

pi


minus100

minus50

0

50

100

150

200

pRi

0 02 04 06 08 1

Model CCModel DC

120573



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

Model CCModel DC

times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

Model CCModel DC

120573

120587







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587




5 Conclusion




Notations








119876119894(119901119877119894 119901119877119895


119901119862

119894 119901119863


models 119862 and 119863

119901119862

119877119894 119901119863


119862 and 119863

Π119870






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













max119901119894119901119895119901119877119894119901119877119895

Π119862(119901119894 119901119895 119901119877119894 119901119877119895

)

=

2

sum

119894=1

((119901119894minus 119888119898)119863119894(119901119894 119901119895) + (Δ minus 119901

119877119894) 119876119894(119901119877119894 119901119877119895

))

(6)



119862

119895satisfy

119901lowast119862

119894=

(Φ119894+ Φ119895120573)

2 (1 minus 1205732)

+4 (Δ119904 minus 2119888

119898) (1 minus 120574) + 119904

2(Φ119894+ Φ3minus119894

)

41198831


41198832

(7)

119901lowast119862

119877119894=

Δ

2+ s[

(Φ119894+ Φ3minus119894

) minus (2119888119898


21198831


)

21198832

]

(8)


2(1+120573)


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)


|119867| =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus2 2120573 119904 minus119904120573

2120573 minus2 minus119904120573 119904

119904 minus119904120573 minus2 2120574

minus119904120573 119904 2120574 minus2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(10)



2| = 4(1 minus 120573) gt 0

|1198633| = minus2(1 minus 120573

2)(4 minus 119904

2) lt 0 and |1198634| = 16 + 16(120574


1205742))+8119904

2(1205732+120574120573(1minus120573

2)minus1)+119904



1198941199011198623minus119894

119901119862119895 and

119901119862



120597Π119862

120597119901119894

= Φ119894minus 2119901119894+ 2120573119901

119895+ 119904119901119877119894

minus 119904120573119901119877119895

+ (1 minus 120573) (119888119898

minus 119904Δ)

= 0

(11)

120597Π119862

120597119901119895


= 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)


lowast

119894and 119901

lowast


prices 119901lowast

119877119894and 119901

lowast

119877119895


119862

119894and 119901

119862





d119901119862119894

d120574= (1199042[1198832


minus1198832

2(Φ119894+ Φ3minus119894

+ (1 minus 120573) (Δ119904 minus 2119888119898))])

times (1198832

11198832

2)minus1

(15)



2(1+120573)



119862

119894and 119901

119862

119895are




119862

119877119894and 119901

119862




d119901119862119877119894

d120574= 2119904 (119883

2

2[Φ119894+ Φ3minus119894

minus (2119888119898

minus Δ119904) (1 minus 120573)]

minus1198832


)) times (1198832

11198832

2)minus1

(16)


2(1+120573)



119862

119877119894

and 119901119862


substitution effect





respect to 119901119894and 119901


prices 119901119894and 119901



2




119894is invariable




119894=1(Δ minus 119901

119877119894)119876119894(119901119877119894 119901119877119895


with respect to 119901119877119894



and 119901119877119895


2

119894=1(Δ minus 119901

119877119894)119876119894(119901119877119894 119901119877119895

)





max119901119894 119901119895

Π119863

119894= (119901119894 minus 119908)119863119894 (119901119894 119901119895) + (119887 minus 119901119877119895)119876119877119894 (119901119877119894 119901119877119895)

(17)


lowast119863



119877119894satisfy

119901lowast119863

119894=


11988411

+(Φ119894+ Φ3minus119894

) 1198841

211988411


) 1198851

211988511

(18)

119901lowast119863

119877119894=

119904 (Φ119894+ Φ3minus119894

)

211988411


)

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


2minus 120574)

1198851 = 2 + 120574 minus 119904


2 and119879 = 120574


2)



119894are

1205972Π119863

119894

1205971199012

119894

= minus21205972Π119863

119894

120597119901119894120597119901119895

= 119904

1205972Π119863

119894

120597119901119895120597119901119894

= 1199041205972Π119863

119894

1205971199012

119895

= minus119904120573

(20)


|119867| =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972Π119863

119894

1205971199012

119894

1205972Π119863

119894

120597119901119894120597119901119895

1205972Π119863

119894

120597119901119895120597119901119894

1205972Π119863

119894

1205971199012

119895

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(21)



2Π119863

1198941205971199012

119894= minus2 lt 0






120597Π119863

119894

120597119901119894


120597Π119863

119894

120597119901119877119894


(23)


119895are

120597Π119863

119895

120597119901119895


119877119895) = 0 (24)

120597Π119863

119895

120597119901119877119895


119877119895+ 120574119901119877119894

+ 119887 minus 119901119877119895

= 0

(25)




1= 1199042+120574minus2 lt 0119884


1199042minus120574) lt 0119885

1= 2+120574minus119904

2gt 0 and 119879 = 120574


2) lt 0




119877119894and




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)



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)


1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873

1= 120573(2 minus 119904


2) and

1198732= 8Δ + (Φ

1+ Φ2minus 2119888119898)119904


1205972Π119863

119898

1205971199082



1205972Π119863

119898

1205971198872

=41199042(2 minus 120574) (1 minus 120574)

2


1205972Π119863

119898

120597119908120597119887

=1205972Π119863

119898

120597119887120597119908

=41199042(2 minus 120574)



10038161003816100381610038161198671198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972Π119863

119898

1205971199082

1205972Π119863

119898

120597119908120597119887

1205972Π119863

119898

120597119887120597119908

1205972Π119863119862

119898

1205971198872

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(29)



119863

119898is jointly






120597Π119863

119898

120597119908

=2 (2 minus 120574) (2119908 minus 119887119904 + 119888

119898) + 2Δ119904

1199042+ 120574 minus 2


+ 2119908 (120574 minus 2)

+ 2 (119888119898



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



2+ 120574 minus 2) + 119904

2minus 4])minus1

= 0

(31)




lowast119863


119894 satisfy

119901lowast119863

119894


2[

1198851

11988511

minus1198841

11988411

]

+119904 (1 minus 120574)

211988411

[2Δ1198721 + 1205741198722 + 2120573120574 (119888119898119904 minus 2Δ)

21198721+ 120574119872

2

]


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


2) 1198722


2)

1198723

= 8Δ + (Φ1+ Φ2minus 2119888119898)119904 1198701

= 4(120574 minus 1) + 1199042(2 minus 120574)

1198702= (2119888119898


1=


2)1198732= 8Δ+(Φ

1+Φ2minus2119888119898)1199041198841= 1199042+120574minus2

11988411


2minus 120574) 119885

1= 2 + 120574 minus 119904

2 11988511

=



2)




119894is invariable








= 119904119863119894and 119901






119895(119901119877119895

119901119877119894)) is increasing












119894(119901119894 119901119895) is increasing



119894 induce





119877119894)119876119894(119901119877119894 119901119877119895






2= 50 119888

119898= 20 Δ = 10 120574 = 04 or 120574 = 02

and 119904 = 01




0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

120574 = 02

120574 = 04

pi


0

5

10

15

20

25

30

35

40

45

50

pRi

0 02 04 06 08 1120573

120574 = 02

120574 = 04












0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 10

200

400

600

800

1000

1200

1400

1600

1800

2000

120573

120574 = 02

120574 = 04120587



0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

pi

120574 = 02

120574 = 04


0 02 04 06 08 1minus20

minus15

minus10

minus5

0

5

10

15

20

25

30

120573

pi

120574 = 02

120574 = 04







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120574 = 02

120574 = 04

120587









0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

Model CCModel DC

120573

pi


minus100

minus50

0

50

100

150

200

pRi

0 02 04 06 08 1

Model CCModel DC

120573



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

Model CCModel DC

times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

Model CCModel DC

120573

120587







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587




5 Conclusion




Notations








119876119894(119901119877119894 119901119877119895


119901119862

119894 119901119863


models 119862 and 119863

119901119862

119877119894 119901119863


119862 and 119863

Π119870






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119862

119877119894and 119901

119862




d119901119862119877119894

d120574= 2119904 (119883

2

2[Φ119894+ Φ3minus119894

minus (2119888119898

minus Δ119904) (1 minus 120573)]

minus1198832


)) times (1198832

11198832

2)minus1

(16)


2(1+120573)



119862

119877119894

and 119901119862


substitution effect





respect to 119901119894and 119901


prices 119901119894and 119901



2




119894is invariable




119894=1(Δ minus 119901

119877119894)119876119894(119901119877119894 119901119877119895


with respect to 119901119877119894



and 119901119877119895


2

119894=1(Δ minus 119901

119877119894)119876119894(119901119877119894 119901119877119895

)





max119901119894 119901119895

Π119863

119894= (119901119894 minus 119908)119863119894 (119901119894 119901119895) + (119887 minus 119901119877119895)119876119877119894 (119901119877119894 119901119877119895)

(17)


lowast119863



119877119894satisfy

119901lowast119863

119894=


11988411

+(Φ119894+ Φ3minus119894

) 1198841

211988411


) 1198851

211988511

(18)

119901lowast119863

119877119894=

119904 (Φ119894+ Φ3minus119894

)

211988411


)

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


2minus 120574)

1198851 = 2 + 120574 minus 119904


2 and119879 = 120574


2)



119894are

1205972Π119863

119894

1205971199012

119894

= minus21205972Π119863

119894

120597119901119894120597119901119895

= 119904

1205972Π119863

119894

120597119901119895120597119901119894

= 1199041205972Π119863

119894

1205971199012

119895

= minus119904120573

(20)


|119867| =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972Π119863

119894

1205971199012

119894

1205972Π119863

119894

120597119901119894120597119901119895

1205972Π119863

119894

120597119901119895120597119901119894

1205972Π119863

119894

1205971199012

119895

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(21)



2Π119863

1198941205971199012

119894= minus2 lt 0






120597Π119863

119894

120597119901119894


120597Π119863

119894

120597119901119877119894


(23)


119895are

120597Π119863

119895

120597119901119895


119877119895) = 0 (24)

120597Π119863

119895

120597119901119877119895


119877119895+ 120574119901119877119894

+ 119887 minus 119901119877119895

= 0

(25)




1= 1199042+120574minus2 lt 0119884


1199042minus120574) lt 0119885

1= 2+120574minus119904

2gt 0 and 119879 = 120574


2) lt 0




119877119894and




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)



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)


1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873

1= 120573(2 minus 119904


2) and

1198732= 8Δ + (Φ

1+ Φ2minus 2119888119898)119904


1205972Π119863

119898

1205971199082



1205972Π119863

119898

1205971198872

=41199042(2 minus 120574) (1 minus 120574)

2


1205972Π119863

119898

120597119908120597119887

=1205972Π119863

119898

120597119887120597119908

=41199042(2 minus 120574)



10038161003816100381610038161198671198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972Π119863

119898

1205971199082

1205972Π119863

119898

120597119908120597119887

1205972Π119863

119898

120597119887120597119908

1205972Π119863119862

119898

1205971198872

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(29)



119863

119898is jointly






120597Π119863

119898

120597119908

=2 (2 minus 120574) (2119908 minus 119887119904 + 119888

119898) + 2Δ119904

1199042+ 120574 minus 2


+ 2119908 (120574 minus 2)

+ 2 (119888119898



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



2+ 120574 minus 2) + 119904

2minus 4])minus1

= 0

(31)




lowast119863


119894 satisfy

119901lowast119863

119894


2[

1198851

11988511

minus1198841

11988411

]

+119904 (1 minus 120574)

211988411

[2Δ1198721 + 1205741198722 + 2120573120574 (119888119898119904 minus 2Δ)

21198721+ 120574119872

2

]


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


2) 1198722


2)

1198723

= 8Δ + (Φ1+ Φ2minus 2119888119898)119904 1198701

= 4(120574 minus 1) + 1199042(2 minus 120574)

1198702= (2119888119898


1=


2)1198732= 8Δ+(Φ

1+Φ2minus2119888119898)1199041198841= 1199042+120574minus2

11988411


2minus 120574) 119885

1= 2 + 120574 minus 119904

2 11988511

=



2)




119894is invariable








= 119904119863119894and 119901






119895(119901119877119895

119901119877119894)) is increasing












119894(119901119894 119901119895) is increasing



119894 induce





119877119894)119876119894(119901119877119894 119901119877119895






2= 50 119888

119898= 20 Δ = 10 120574 = 04 or 120574 = 02

and 119904 = 01




0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

120574 = 02

120574 = 04

pi


0

5

10

15

20

25

30

35

40

45

50

pRi

0 02 04 06 08 1120573

120574 = 02

120574 = 04












0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 10

200

400

600

800

1000

1200

1400

1600

1800

2000

120573

120574 = 02

120574 = 04120587



0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

pi

120574 = 02

120574 = 04


0 02 04 06 08 1minus20

minus15

minus10

minus5

0

5

10

15

20

25

30

120573

pi

120574 = 02

120574 = 04







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120574 = 02

120574 = 04

120587









0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

Model CCModel DC

120573

pi


minus100

minus50

0

50

100

150

200

pRi

0 02 04 06 08 1

Model CCModel DC

120573



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

Model CCModel DC

times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

Model CCModel DC

120573

120587







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587




5 Conclusion




Notations








119876119894(119901119877119894 119901119877119895


119901119862

119894 119901119863


models 119862 and 119863

119901119862

119877119894 119901119863


119862 and 119863

Π119870






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894are

1205972Π119863

119894

1205971199012

119894

= minus21205972Π119863

119894

120597119901119894120597119901119895

= 119904

1205972Π119863

119894

120597119901119895120597119901119894

= 1199041205972Π119863

119894

1205971199012

119895

= minus119904120573

(20)


|119867| =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972Π119863

119894

1205971199012

119894

1205972Π119863

119894

120597119901119894120597119901119895

1205972Π119863

119894

120597119901119895120597119901119894

1205972Π119863

119894

1205971199012

119895

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(21)



2Π119863

1198941205971199012

119894= minus2 lt 0






120597Π119863

119894

120597119901119894


120597Π119863

119894

120597119901119877119894


(23)


119895are

120597Π119863

119895

120597119901119895


119877119895) = 0 (24)

120597Π119863

119895

120597119901119877119895


119877119895+ 120574119901119877119894

+ 119887 minus 119901119877119895

= 0

(25)




1= 1199042+120574minus2 lt 0119884


1199042minus120574) lt 0119885

1= 2+120574minus119904

2gt 0 and 119879 = 120574


2) lt 0




119877119894and




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)



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)


1198703= 119888119898(1199042minus 2)(120574 minus 2) + 2Δ120574119904 119873

1= 120573(2 minus 119904


2) and

1198732= 8Δ + (Φ

1+ Φ2minus 2119888119898)119904


1205972Π119863

119898

1205971199082



1205972Π119863

119898

1205971198872

=41199042(2 minus 120574) (1 minus 120574)

2


1205972Π119863

119898

120597119908120597119887

=1205972Π119863

119898

120597119887120597119908

=41199042(2 minus 120574)



10038161003816100381610038161198671198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972Π119863

119898

1205971199082

1205972Π119863

119898

120597119908120597119887

1205972Π119863

119898

120597119887120597119908

1205972Π119863119862

119898

1205971198872

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(29)



119863

119898is jointly






120597Π119863

119898

120597119908

=2 (2 minus 120574) (2119908 minus 119887119904 + 119888

119898) + 2Δ119904

1199042+ 120574 minus 2


+ 2119908 (120574 minus 2)

+ 2 (119888119898



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



2+ 120574 minus 2) + 119904

2minus 4])minus1

= 0

(31)




lowast119863


119894 satisfy

119901lowast119863

119894


2[

1198851

11988511

minus1198841

11988411

]

+119904 (1 minus 120574)

211988411

[2Δ1198721 + 1205741198722 + 2120573120574 (119888119898119904 minus 2Δ)

21198721+ 120574119872

2

]


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


2) 1198722


2)

1198723

= 8Δ + (Φ1+ Φ2minus 2119888119898)119904 1198701

= 4(120574 minus 1) + 1199042(2 minus 120574)

1198702= (2119888119898


1=


2)1198732= 8Δ+(Φ

1+Φ2minus2119888119898)1199041198841= 1199042+120574minus2

11988411


2minus 120574) 119885

1= 2 + 120574 minus 119904

2 11988511

=



2)




119894is invariable








= 119904119863119894and 119901






119895(119901119877119895

119901119877119894)) is increasing












119894(119901119894 119901119895) is increasing



119894 induce





119877119894)119876119894(119901119877119894 119901119877119895






2= 50 119888

119898= 20 Δ = 10 120574 = 04 or 120574 = 02

and 119904 = 01




0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

120574 = 02

120574 = 04

pi


0

5

10

15

20

25

30

35

40

45

50

pRi

0 02 04 06 08 1120573

120574 = 02

120574 = 04












0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 10

200

400

600

800

1000

1200

1400

1600

1800

2000

120573

120574 = 02

120574 = 04120587



0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

pi

120574 = 02

120574 = 04


0 02 04 06 08 1minus20

minus15

minus10

minus5

0

5

10

15

20

25

30

120573

pi

120574 = 02

120574 = 04







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120574 = 02

120574 = 04

120587









0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

Model CCModel DC

120573

pi


minus100

minus50

0

50

100

150

200

pRi

0 02 04 06 08 1

Model CCModel DC

120573



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

Model CCModel DC

times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

Model CCModel DC

120573

120587







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587




5 Conclusion




Notations








119876119894(119901119877119894 119901119877119895


119901119862

119894 119901119863


models 119862 and 119863

119901119862

119877119894 119901119863


119862 and 119863

Π119870






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120597Π119863

119898

120597119908

=2 (2 minus 120574) (2119908 minus 119887119904 + 119888

119898) + 2Δ119904

1199042+ 120574 minus 2


+ 2119908 (120574 minus 2)

+ 2 (119888119898



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



2+ 120574 minus 2) + 119904

2minus 4])minus1

= 0

(31)




lowast119863


119894 satisfy

119901lowast119863

119894


2[

1198851

11988511

minus1198841

11988411

]

+119904 (1 minus 120574)

211988411

[2Δ1198721 + 1205741198722 + 2120573120574 (119888119898119904 minus 2Δ)

21198721+ 120574119872

2

]


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


2) 1198722


2)

1198723

= 8Δ + (Φ1+ Φ2minus 2119888119898)119904 1198701

= 4(120574 minus 1) + 1199042(2 minus 120574)

1198702= (2119888119898


1=


2)1198732= 8Δ+(Φ

1+Φ2minus2119888119898)1199041198841= 1199042+120574minus2

11988411


2minus 120574) 119885

1= 2 + 120574 minus 119904

2 11988511

=



2)




119894is invariable








= 119904119863119894and 119901






119895(119901119877119895

119901119877119894)) is increasing












119894(119901119894 119901119895) is increasing



119894 induce





119877119894)119876119894(119901119877119894 119901119877119895






2= 50 119888

119898= 20 Δ = 10 120574 = 04 or 120574 = 02

and 119904 = 01




0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

120574 = 02

120574 = 04

pi


0

5

10

15

20

25

30

35

40

45

50

pRi

0 02 04 06 08 1120573

120574 = 02

120574 = 04












0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 10

200

400

600

800

1000

1200

1400

1600

1800

2000

120573

120574 = 02

120574 = 04120587



0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

pi

120574 = 02

120574 = 04


0 02 04 06 08 1minus20

minus15

minus10

minus5

0

5

10

15

20

25

30

120573

pi

120574 = 02

120574 = 04







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120574 = 02

120574 = 04

120587









0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

Model CCModel DC

120573

pi


minus100

minus50

0

50

100

150

200

pRi

0 02 04 06 08 1

Model CCModel DC

120573



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

Model CCModel DC

times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

Model CCModel DC

120573

120587







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587




5 Conclusion




Notations








119876119894(119901119877119894 119901119877119895


119901119862

119894 119901119863


models 119862 and 119863

119901119862

119877119894 119901119863


119862 and 119863

Π119870






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119895(119901119877119895

119901119877119894)) is increasing












119894(119901119894 119901119895) is increasing



119894 induce





119877119894)119876119894(119901119877119894 119901119877119895






2= 50 119888

119898= 20 Δ = 10 120574 = 04 or 120574 = 02

and 119904 = 01




0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

120574 = 02

120574 = 04

pi


0

5

10

15

20

25

30

35

40

45

50

pRi

0 02 04 06 08 1120573

120574 = 02

120574 = 04












0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 10

200

400

600

800

1000

1200

1400

1600

1800

2000

120573

120574 = 02

120574 = 04120587



0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

pi

120574 = 02

120574 = 04


0 02 04 06 08 1minus20

minus15

minus10

minus5

0

5

10

15

20

25

30

120573

pi

120574 = 02

120574 = 04







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120574 = 02

120574 = 04

120587









0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

Model CCModel DC

120573

pi


minus100

minus50

0

50

100

150

200

pRi

0 02 04 06 08 1

Model CCModel DC

120573



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

Model CCModel DC

times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

Model CCModel DC

120573

120587







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587




5 Conclusion




Notations








119876119894(119901119877119894 119901119877119895


119901119862

119894 119901119863


models 119862 and 119863

119901119862

119877119894 119901119863


119862 and 119863

Π119870






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

120574 = 02

120574 = 04

pi


0

5

10

15

20

25

30

35

40

45

50

pRi

0 02 04 06 08 1120573

120574 = 02

120574 = 04












0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 10

200

400

600

800

1000

1200

1400

1600

1800

2000

120573

120574 = 02

120574 = 04120587



0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

pi

120574 = 02

120574 = 04


0 02 04 06 08 1minus20

minus15

minus10

minus5

0

5

10

15

20

25

30

120573

pi

120574 = 02

120574 = 04







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120574 = 02

120574 = 04

120587









0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

Model CCModel DC

120573

pi


minus100

minus50

0

50

100

150

200

pRi

0 02 04 06 08 1

Model CCModel DC

120573



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

Model CCModel DC

times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

Model CCModel DC

120573

120587







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587




5 Conclusion




Notations








119876119894(119901119877119894 119901119877119895


119901119862

119894 119901119863


models 119862 and 119863

119901119862

119877119894 119901119863


119862 and 119863

Π119870






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 10

200

400

600

800

1000

1200

1400

1600

1800

2000

120573

120574 = 02

120574 = 04120587



0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

120573

pi

120574 = 02

120574 = 04


0 02 04 06 08 1minus20

minus15

minus10

minus5

0

5

10

15

20

25

30

120573

pi

120574 = 02

120574 = 04







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120574 = 02

120574 = 04

120587









0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

Model CCModel DC

120573

pi


minus100

minus50

0

50

100

150

200

pRi

0 02 04 06 08 1

Model CCModel DC

120573



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

Model CCModel DC

times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

Model CCModel DC

120573

120587







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587




5 Conclusion




Notations








119876119894(119901119877119894 119901119877119895


119901119862

119894 119901119863


models 119862 and 119863

119901119862

119877119894 119901119863


119862 and 119863

Π119870






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

120573

120574 = 02

120574 = 04

120587

times104


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

120573

120574 = 02

120574 = 04

120587









0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

Model CCModel DC

120573

pi


minus100

minus50

0

50

100

150

200

pRi

0 02 04 06 08 1

Model CCModel DC

120573



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

Model CCModel DC

times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

Model CCModel DC

120573

120587







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587




5 Conclusion




Notations








119876119894(119901119877119894 119901119877119895


119901119862

119894 119901119863


models 119862 and 119863

119901119862

119877119894 119901119863


119862 and 119863

Π119870






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 10

50

100

150

200

250

300

350

400

450

500

Model CCModel DC

120573

pi


minus100

minus50

0

50

100

150

200

pRi

0 02 04 06 08 1

Model CCModel DC

120573



0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2

Model CCModel DC

times104

120573

120587


0 02 04 06 08 1minus1000

minus500

0

500

1000

1500

2000

Model CCModel DC

120573

120587







0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587




5 Conclusion




Notations








119876119894(119901119877119894 119901119877119895


119901119862

119894 119901119863


models 119862 and 119863

119901119862

119877119894 119901119863


119862 and 119863

Π119870






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587


0 02 04 06 08 10

02

04

06

08

1

12

14

16

18

2times104

120573

120574 = 02

120574 = 04

120587




5 Conclusion




Notations








119876119894(119901119877119894 119901119877119895


119901119862

119894 119901119863


models 119862 and 119863

119901119862

119877119894 119901119863


119862 and 119863

Π119870






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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